From Static to Dynamic Metaheuristics: Design and ...

´ Paris Est Universite

`se d’Habilitation The par

Amir NAKIB

From static to dynamic metaheuristics

Soutenue le 09 décembre 2015 à Créteil devant le jury composé de: Pascal

Bouvry,

Rapporteur

Manuel

Gra˜ na,

Rapporteur

Abdelmalik

Taleb-Ahmed,

Rapporteur

Phlippe

Decq,

Examinateur

El-Ghazali

Talbi,

Examinateur

Hugues

Talbot,

Examinateur

Farouk

Yalaoui,

Examinateur

Patrick

Siarry,

Directeur

December 2015

”To my parents.”

Amir Nakib

Acknowledgements I would like to most sincerely thank the ”rapporteurs” of this thesis: Pascal Bouvry, Manuel Gra˜ na and Abdelmalik Taleb-Ahmed, who have kindly accepted to take part in this jury, to read this manuscript carefully and to produce a review of its content. Moreover, I would like to thank the rest of the members of the jury: Philippe Decq, El-Ghazali Talbi, Hugues Talbot and Farouk Yalaoui, for their interest in my work and for their participation in the jury. Thanks also to the scientific and administrative staff who made this thesis and associated events possible: Latifa Zeroual-Belou, Stéphane Bouton, Patricia Jamin, Katia Lambert and the Conseil Scientifique de l’Université Paris Est. Also, I would like to thank specially Patrick Siarry for all his precious advices. It is difficult to express in a few words my thanks to the members of LISSI, especially, its Director Yacine Amirat. That everyone finds here the expression of my friendship.

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Contents Acknowledgements

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Contents

iii

General Introduction

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1 Design of static metaheuristics for medical image analysis 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Image segmentation as an optimization Problem . . . . . . . . . . . . . . 1.3 Metaheuristics’ enhancement for medical image segmentation . . . . . . . 1.3.1 Hybrid Ant Colony System (ACS) . . . . . . . . . . . . . . . . . . 1.3.1.1 Segmentation criterion: Biased Survival exponential entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.2 Results and discussions . . . . . . . . . . . . . . . . . . . 1.3.2 Enhanced BBO for image segmentation . . . . . . . . . . . . . . . 1.3.2.1 Enhancement of biogeography based optimization (BBO) algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 The segmentation criterion: overall probability error . . . 1.3.2.3 Results and discussions . . . . . . . . . . . . . . . . . . . 1.3.3 Enhanced DE for image thresholding . . . . . . . . . . . . . . . . . 1.3.3.1 Segmentation criterion . . . . . . . . . . . . . . . . . . . 1.3.3.2 Results and discussions . . . . . . . . . . . . . . . . . . . 1.3.4 Metaheuristic for contours detection in 2D Ultrasound images . . . 1.3.4.1 Proposed Bone contour extraction method based on shortest path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.2 False alarm elimination using shortest path formulation . 1.3.4.3 Results and discussions . . . . . . . . . . . . . . . . . . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Publications related to this chapter . . . . . . . . . . . . . . . . . . . . . .

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Design of metaheuristics for Dynamic continuous optimization problems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bechmarks and performance evaluation . . . . . . . . . . . . . . . . . . . 2.2.1 The Moving Peaks Benchmark . . . . . . . . . . . . . . . . . . . . 2.2.2 The Generalized Dynamic Benchmark Generator . . . . . . . . . . 2.2.3 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . iii

11 13 13 15 17 17 18 20 21 21 23 24 25 26 26

29 29 31 31 33 35

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Contents 2.3

2.4 2.5

Proposed Multi-Agent based approach for dynamic optimization . 2.3.1 Multiagent Algorithm for Dynamic Optimization (MADO) 2.3.1.1 Overall description . . . . . . . . . . . . . . . . . . 2.3.1.2 Results on MPB and GDBG Benchmarks . . . . . 2.3.2 Covariance matrix adaptation MADO (CMADO) . . . . . . 2.3.2.1 Overall description . . . . . . . . . . . . . . . . . . 2.3.2.2 Results and discussions . . . . . . . . . . . . . . . 2.3.3 Prediction based MADO (PMADO) . . . . . . . . . . . . . 2.3.3.1 Overall description . . . . . . . . . . . . . . . . . . 2.3.3.2 Results and discussions . . . . . . . . . . . . . . . 2.3.4 Multiple Local Searches DO (MLSDO) . . . . . . . . . . . . 2.3.4.1 Overall description . . . . . . . . . . . . . . . . . . 2.3.4.2 Results and discussions . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Publications related to this chapter . . . . . . . . . . . . . . . . . .

3 Dynamic metaheuristics for brain cine-MRI 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proposed framework . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Segmentation problem . . . . . . . . . . . . . . . . . . . . 3.4.1.1 segmentation based on fractional differentiation 3.4.1.2 segmentation based on dynamic optimization . . 3.4.2 Geometric matching of the contours . . . . . . . . . . . . 3.4.2.1 Contours’ Alignment . . . . . . . . . . . . . . . 3.4.2.2 The deformation model . . . . . . . . . . . . . . 3.4.3 Cine-MRI registration as a DOP . . . . . . . . . . . . . . 3.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Publications related to this chapter . . . . . . . . . . . . . . . . .

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4 Particle tracking based on fractional integration 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fractional integration . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definition of fractional calculus . . . . . . . . . . . . . . . . 4.2.2 Discrete form: Gr¨ unwald approach . . . . . . . . . . . . . . . 4.2.3 Computation of the coefficients . . . . . . . . . . . . . . . . . 4.3 Statistical analysis of a fractionally integrated function . . . . . . . . 4.3.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Statistical analysis of the fractionally integrated path . . . . 4.3.2.1 Average value of the integrated function . . . . . . . 4.3.2.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . 4.4 Proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Enhancement of the path prediction using DFI . . . . . . . . . . . . 4.5.1 Decreasing of the archive size needed for accurated prediction 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 4.7

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Publications related to this chapter . . . . . . . . . . . . . . . . . . . . . . 79

General Conclusion and future work

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A Dynamic optimization using multiagent strategy

86

B Prediction based on fractional integration

128

C A framework for analysis of brain cine MR sequences

143

D Image thresholding based on Pareto multiobjective optimization

161

E A Learning-Based Resource Allocation Approach for P2P Streaming Systems 170 Bibliography

179

General Introduction First of all the work presented in this thesis extends from 2004 to 2014. This work were done at Laboratory images, signals and intelligent systems (LISSI, EA 3954), within SIMO (Signals, Images and Optimization) group, under the direction of Professor Patrick Siarry.

Since I get the position as the 61th section associate professor at the University Paris Est Créteil, I am a member of Laboratory Images, Signals and Intelligent Systems (LISSI), whose director is Prof. Yacine Amirat. Specifically, I do my work within the signals, images and optimization (SIMO) group.

The group’s research focuses on the development of morphological and parametric image analysis methods in cardiology and neurology, and imaging methods of multidiffusants environments. The group also develops methods based on metaheuristics to solve optimization problems especially in medical applications. My research is thus integrated into the actions of SIMO group. They consist of two complementary areas: metaheuristics and image processing.

The optimization problems arise in many areas of research and application. Whether the decision support, machine learning and classification, object recognition in images, either to regulate and control production systems, or to specify the dimensions and devices settings hardware, telecommunications, we may solve optimization problems as we seek most often to minimize cost or to maximize performance.

Overall, an optimization problem is defined by three components: a set of instances, a set of feasible solutions assigned to each instance and a function called objective function, which assigns each feasible solution a scalar value. An optimal solution for an instance of a problem is a feasible solution whose associated scalar value is minimum or maximum, 1


2

respectively, depending on whether a problem of minimizing or maximizing. An optimization algorithm for solving the problem is an algorithm for finding for each instance an optimal solution. If we are trying to minimize multiple objectives simultaneously, we are concerned with multi-objective optimization problems. The problems addressed here are mainly medical image processing optimization problems. Entities that defines the structure of problems, such as the nature of the images, the final aim of the treatment and the cost function that defines the problem. The medical image processing problems as the segmentation and the registration once modeled are NP-hard problems. It is known that most of the image processing issues are ill-posed, and generally, researchers tend to change the cost functions to make these well-posed problems and solve them with exact optimization methods. Therefore, unfortunately, the number of generic methods is very little and practically is a cost function by application, in particular in image segmentation. It follows that we do not know exact resolution methods that provide the optimal solution in a reasonable time, to different applications. It is therefore necessary to propose approximate methods produce satisfactory solutions, but in an acceptable time. Such ad hoc process applied to a particular problem is called a heuristic. When the method is expressed as an independent abstract level of the considered optimization problem, we call metaheuristic. These heuristics and metaheuristics are generally measured experimentally and validated by statistical evidence. The available formal tools do not allow for ensuring theoretically and a priori obtained result in practice. At the beginning of my research in the LISSI in 2004 (thesis work), metaheuristics had proven their efficiency in many areas but it has had to be adapted to the particularities of the image processing problems. To solve these image processing problems, we proposed to design and to hybridize different concepts and methods that can come from different fields of artificial intelligence, automation and operational research. Metaheuristics or heuristics are called hybrid, when the components of the algorithms include techniques from different topics. For example, we shall have to combine bio-inspired approaches with local research and then analyze their design in terms of collective problem solving. The way to relate approaches a priori different to found a method of solving problems nevertheless effective and broad enough to encompass most of the practical cases. This is an underlying concern in the work presented here. Of course, the variety of existing approaches to solving problems, their similarities and differences are that they can already be viewed as hybrid constructions. From one point of view, we only implement a common approach, but by making it explicit and applying it to specific problems in the field of medical image processing.


3

Generally, desired qualities of heuristics and algorithms for solving, tools and associated software are a good compromise between: • the performance and robustness in terms of quality of solutions, implementation and memory occupation,

• can be adapted into a dynamic stochastic context, • the possibility of implantation in a distributed environment, • simplicity and flexibility, • the modularity of the design process, • the generalization to other potential fields of application. In our work, we focus on the development of solution methods that are both effective (generating solutions close to the optimum in acceptable time), simple (easy to understand and implement) and also flexible (more easily extensible to more complex problems). To achieve this, three principles can guide the design: the use of metaphors, the pushing operation of the spatiality of the problem, and hybridization methods at different levels. This should result in a design of efficient algorithms that are intrinsically parallel to provide flexibility and robustness vis-a-vis unexpected events. The main research activity consists in four major contributions since I got the position in the LISSI. These extensions are the basis of a new methodology for metaheuristics and image analysis of: • adaptation of metaheuristics to image processing processing, • design and adaptation of dynamic metaheuristics, • solving image sequences analysis problems as a dynamic optimization problem, • fractional differentiation based prediction. Figure 1 summarizes the research topics tackled since my thesis. In the same figure, there are also the tools exploited for solving problems. Then, we present our work following four main chapters, we wanted that they reflect the different research directions taken since my PhD work within the LISSI. The first chapter is in the continuity of my PhD work, where we adapted most new efficient metaheuristics to solve medical image processing problems. Indeed, in my PhD

4


Figure 1: Research topics’ and tools tackled.

work we were interested by solving segmentation problem. In this context, we also proposed new criteria to model for this ill posed problem. Image segmentation methods can be divided looking to their final objective: regions extraction and edge detection. Image thresholding problem that we dealt with in our work allows to extract regions from images using gray-level information, in the case of many objects the problem becomes hard optimization problem. In this case, we adapted, enhanced and applied static metaheuristics to solve medical image problems, especially, brain MRI and CT-Scan images. Regarding the second group of methods, many mathematical approaches exist in the literature as deformable models, watersheds methods, etc. in most case, these methods are well solved using deterministic methods and iterative gradient descent methods. Our contribution in this group of method consists in re-formulating the problem of contour detection as shortest path problem and solving it using specialized heuristics. In parallel to this research, we were interested to a new topic where the objective function changes overs the time, called dynamic optimization. A summery of our work on in this area is presented in chapter 2 and 3. In chapter 2, in this context we started our research by adapting a classical efficient continuous optimization algorithms as CMAES [Hansen & Ostermeier, 2001], then, our conclusion was that these adaptations tend to make the algorithms more complex and hard to fit to deal efficiently with DOPs. Consequently, we proposed a new approach to design algorithms to deal with dynamic optimization problem. In first, specifications of an efficient algorithm were pointed out. In this context, our goal is to develop a dynamic optimization metaheuristics.


5

So, in this work our approach is based on the use of cooperative strategies composed by local search based agents. Two agent-based algorithms, called MADO MultiAgent algorithm for Dynamic Optimization and MLSDO Multiple Local Search algorithm for Dynamic Optimization, were proposed and validated using the two main benchmarks in dynamic environments: Moving Peaks Benchmark (MPB) Branke [1999b] and Generalized Dynamic Benchmark Generator (GDBG) Li et al. [2008]. Then, these algorithms are applied to real-world problems (chapter 3) in medical image sequence processing (segmentation and registration of brain cine-MRI sequences). The use of dynamic optimization algorithm to solve these medical image processing problems has rarely been explored. The performance gains obtained show the relevance of using the proposed dynamic optimization algorithms for this kind of applications. The analysis of real world application in DOPs shows that the predictability (possibility to predict) of the move of the global optimal solution can be discussed. In the literature some authors were interested with this issue using classical filters. However, many limitations were pointed out as the dependence of the quality of the prediction on the saved past positions. In our work, we interested by this issue: enhancing the performance of classical predictors and reducing the size of the archive of past positions. In this context, fractional differentiation Prospective that may arise from all of this research will be presented. The first set of prospective is a continuation of the main research that laid the foundation of an optimization methodology and analysis of medical images. The general pattern is as follows: optimization alone is not enough, it requires information on the nature of the images for the selection of the best cost function that describes well the problem. This model selection is an attractive approach to help the design; interesting in the sense that it forces the characterization of good solutions: promising starting points for the design of generic methods for image processing.

The second set of prospective consists in combining the efficiency of the fractional differentiation operator to enhance the predictability to design metaheuristics. Indeed, the anticipation in changing environments using metaheuristics have been proposed. The main goal of these approaches was to estimate the likelihood of particular future situations. Since information about the future typically was not available, it was attained through learning from past situations. Further prospective of my research activities than previously stated, will be presented.

Chapter 1

Design of static metaheuristics for medical image analysis 1.1

Introduction

Medical images, such as Computed Axial Tomography (CAT), Magnetic Resonance Imaging (MRI), Ultrasound, and X-Ray, in standard DICOM (Digital Imaging and Communications in Medicine) formats are often stored in Picture Archiving and Communication Systems (PACS) and linked with other clinical information in clinical management systems. Since 70’s research efforts have been devoted to processing and analyzing medical images to extract meaningful information such as volume, shape, motion of organs, to detect abnormalities, and to quantify changes in follow-up studies. Automated image segmentation, which aims at automated extraction of object boundary features, plays a fundamental role in understanding image content for searching and mining in medical image archives. A challenging problem is to segment regions with boundary insufficiencies, i.e., missing edges and/or lack of texture contrast between regions of interest (ROIs) and background. To address this problem, several segmentation approaches have been proposed in the literature, with many of them providing rather promising results. From the literature algorithms designed medical image segmentation are application dependent, imaging modality and type of body part to be studied. For example, requirements of brain segmentation are different from those of the thorax. The artifacts, which affect the brain image, are different partial volume effect is more prominent in brain while in the thorax region it is motion artifact which is more prominent. Thus while selecting a segmentation algorithm one is required to consider all these aspects. The problems common to both CT and MR medical images are:

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Chapter 1. Design of static metaheuristics for medical image analysis

7

• Partial volume effect • Different artifacts: example motion artifacts, ring artifacts, etc. • Noise due to sensors and related electronic system. It is well known that there is no a standard algorithm for the segmentation of all medical images. Each imaging system has its own specific limitations. For example, in MR imaging (MRI) one has to take care of bias field noise (intensity inhomogeneities in the RF field). It is obvious that some methods are more general as compared to specialized algorithms, and can be applied to a wider range of data. A brief survey of three generations of medical image segmentation techniques can be found in [DJ & ZJ., 2007]. The motivation of using metaheuristics is to design a new image segmentation techniques, that combine the flexibility of fitness functions with the power of metaheuristics for searching vast search spaces, in order to find the optimal solution. In our work, metaheuristics were improved to solve the continuous and combinatorial optimization problems.

1.2

Image segmentation as an optimization Problem

In this section, we show that the segmentation of an image can be reduced to an optimization problem, usually NP-hard [Monga, 1987]. Hence the need to use a metaheuristic. The segmentation of an image I using a homogeneity feature A is usually defined as a partition P = R1 , R2 , ..., Rn of I, where: 1. I =

S

Ri , i ∈ [1, n]

2. Ri is convex ∀i ∈ [1, n] 3. A(Ri ) = T rue, ∀ [1, n] 4. A (Ri ∪) = F alse, ∀i ∈ [1, n] for all connected regions(Ri , Rj ). One can notice that the uniqueness of the segmentation is not guaranteed by these four conditions. Indeed, the segmentation results depend not only on the information contained in the image, but also on the method used to process these information (method used to take a decision looking to the segmentation result). Generally, to reduce the problem of non-uniqueness of the solution, the segmentation problem is regularized by adding an optimization constraint function F characterizing the quality of a good segmentation. Then, a fifth condition is added to the first four:


8

5. F (P ∗ ) = M in F (P ) where F is a decreasing function and PA (I) is the set of all P ∈PA (I)

possible partitions of I. It is obvious that condition 5 does not entirely solve the problem of uniqueness of the segmentation. There are still cases where multiple segmentations can have the same optimal value. This explains the need to implement algorithms based on metaheuristics.

1.3

Metaheuristics’ enhancement for medical image segmentation

1.3.1

Hybrid Ant Colony System (ACS)

In this work, a new formulation of the image thresholding problem as a shortest path problem was proposed. Then, a hybrid ant colony system based algorithm to solve was used to solver it. Moreover, a new segmentation criterion was proposed, named the biased survival exponential entropy. This criterion considers the cumulative distribution of the gray level information and takes into account spatial quality of the segmentation result. It is well known that the ant colony optimization algorithms are slow; in order to solve this problem we enhanced the classical ant colony system by hybridizing it with a local search algorithm. The original Ant Colony System (ACS) [Dorigo & M., 1997] was applied in different real world applications. Since the formulation of the problem as a graph optimization problem, ACS can be applied by associating two measures to each arc: the closeness τ (i, j), and the pheromone trail η(i, j). ACS uses a mechanism based on three main operations: (1) the state transition rule provides a direct way to balance between exploration of new edges and exploitation of a priori and accumulated knowledge about the problem. (2) The global updating rule is applied only to edges that belong to the best ant tour. (3) While ants construct a solution, a local pheromone updating rule (local updating rule, for short) is applied. • ACS state transition rule In ACS the state transition rule is as follows: an ant positioned at the node r chooses the city v to move to by applying the rule given below.  n o b  ArgM ax q ≤ q0 u∈Jk (r) [τ (r, u)] · [η(r, u)] v=  s Otherwise

(1.1)

where q is a random number uniformly distributed in [0, 1], q0 is a parameter 0 ≤ q0 ≤ 1.

It determines the relative importance of exploitation versus exploration: when an ant


9

in node r has to choose a node s to move to, it samples a random number 0 ≤ q ≤ 1. b is a parameter that determines the relative importance of pheromones versus distance.

s is a random variable selected according to the probability distribution. It is given by:

pk (r, s) =

 

P

 0

[τ (r,s)][η(r,s)]b b u∈J (r) [τ (r,s)][η(r,s)] k

s ∈ Jk (r)

(1.2)

Otherwise

Jk (r) is the set of the neighborhood solutions to the current ant r. The state transition rule resulting from equations (1.1) and (1.2) is called pseudo-random proportional rule. This state transition rule, as with the previous random-proportional rule, favors transitions towards nodes connected by short edges and with a large amount of pheromone. If q ≤ q0 , then the best edge according to (1.1) is chosen (exploitation), otherwise an edge is chosen according to (1.2) (biased exploration). • ACS global updating rule In ACS, only the globally best ant is allowed to deposit pheromone. The pheromone level is updated by applying the following global updating rule: τ (r, s) = (1 − µ) · τ (r, s) + µ · ∆τ (r, s) where ∆τ (r, s) =

(

(Lgb )−1 if (r, s) ∈ gb 0

Otherwise

(1.3)

(1.4)

0 < µ < 1 is a pheromone decay parameter, and Lgb is the length of the globally best tour (gb) from the beginning of the trial. • ACS local updating rule After having crossed an edge (i, j) during the tour construction, the following local update rule is applied: τ (i, j) = (1 − ξ) + ξ · τ0

(1.5)

where 0 < ξ < 1, and τ0 are two parameters. The value for τ0 is suited to be the same as the initial value for the pheromone trails. In order to apply the EACS to solve the segmentation problem, it must be reformulated as a shortest path problem. Then, we define a stochastic rule of local choice of transition to carry out the good path research in this graph. It is also needed to fix the strategy of the deposit and the use of the different traces of pheromone. In our case, the graph is related and balanced. The nodes represent the various possible thresholds (255 thresholds). The weights will be


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Figure 1.1: Illustration of the problem formulation where M is the maximum graylevel in the image, Lo is the lowest gray-level and t∗ is an example of an optimal threshold. The dashed lines correspond to the shortest path.

placed on the arcs. The weight on an edge represents the value of the BSEE for the thresholds T bound by this edge (see figure 1.1). Then, we define: • A set of components C = T . • The whole L = T × T , either a total, simple and non controlled interconnection between the objects.

• The function of transition cost J(i, j) = pij . • The set of the solutions, that correspond to all possible thresholds that allow to segment the image, without violation of the sort constraint.

In order to solve the image segmentation problem, we specify the behavior of the set of the colony guided by the ACS to minimize the segmentation criterion. Initially, N ants are placed randomly on N nodes of the construction graph. Thus, each ant adds, in an incremental way, the threshold that minimizes the total segmentation criterion. Then, every ant moves to its neighborhood using a stochastic transition rule. This rule depends on the quantity of pheromone and heuristic information locally valid. In other terms, an ant having a constructed solution s chooses to move toward a node j according to the rule (1.1). As we consider that the timeliness of a solution depends on all the solutions (segmentation thresholds) already found, the heuristic information and the pheromone represent a total relation between the current solution and all the found solutions from the start S (all visited nodes).

τS (j) =

X

i∈S

τ (i, j)

(1.6)


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The EACS algorithm Data: image histogram, number of classes (N ) for each ant do Choose the first threshold randomly for i = 2 to N − 1 do Build a list of the candidates thresholds Choose a threshold that minimizes the segmentation criterion r = rand if r < 0.3 then Apply a Tabu search in the neighbor of the current threshold end Local update of the pheromone end Global update of the pheromone end Retrun The best ant Figure 1.2: The EACS algorithm.

The heuristic information corresponds to the value of the segmentation criterion. Then, the candidate is more desirable with the decrease of its fitness function. The proposed algorithm (figure 1.2) consists in applying the proposed EACS (hybridization of ACS with Tabu Search (TS)). The different steps of the proposed segmentation algorithm are presented in the algorithm in figure 1.2. Indeed, the EACS principle consists in applying a local search based on TS for 30% of the ant colony. The aim of this local search is to accelerate the convergence of the algorithm and to avoid the local optima. Indeed, the tabu search is used for the intensification procedure as a local search and ACS is used with a specific fitting that allows a diversification.

1.3.1.1

Segmentation criterion: Biased Survival exponential entropy

In this section, the information measure of random variables, called Survival Exponential Entropy (SEE), is recalled. Here, we show how using a biased version of this measure can allow to segment images. Let X = (X1 . . . Xm ) be a random vector in x to mean that |Xi | >

xi f orxi ≥ 0, i = 1, . . . , m.

The multivariate survival function F |X| (x) of the random vector |X| with an absolutely

continuous distribution with probability density function f (x) is defined by:


F |X| (x) = P (|X1 | > x1 , ..., |Xm | > xm )

12

(1.7)

m . For a discrete distribution, the survival cumulative distribution function where x ∈ R+

can be expressed as:

F¯ (x) = 1 −

x X

p(i)

(1.8)

i=0

For the random vector X in 0 and α 6= 0 , where m denotes the number of dimensions for X. The SEE

uses the density function with the cumulative distribution that is more regular than the density function. The SEE has several advantages over the Shannon entropy and differential entropy (extension of Shannon entropy to the continuous case): it is consistently

defined in both the continuous and discrete domains, it is always nonnegative and it is easy to compute from sample data. Whereas the Shannon entropy is based on the density of the random variable, that may not exist in some cases and, when it exists, must be estimated. For the application in image segmentation, we use the well known property of measures of entropy, which states that the entropy of the joint distribution is equal to the sum of entropies of the marginal distributions under the assumption of independence.Thus the proposed biased survival exponential entropies (BSEE) associated with different image classes’ distributions are defined below: • the BSEE of the class m − 1 can be computed through: (m−1)

BMα



= ω(m−1) · 

tX n −1

j=tn−1

1/(1−α)

α F (i, j) 

(1.10)

• the BSEE of the class m can be computed through: (m)

BMα



= ω(m) · 

tn+1 −1

X

j=tn

1/(1−α) α F (i, j) 

(1.11)

(m) where ω(m) = log(Ni ), so that the first term ω(m) · Mα is high for non-homogeneous

regions (typically, the large ones), while Ni denotes the number of pixels in the class i, and ω(1) = 1. For the convenience of illustration, two threshold values t0 = 1 and


13

tN = 255 were added, where t0 < ... < tN . Then the total BSEE is: (T )

BMα =

N −1 X

(i+1)

BMα

(1.12)

i=0

According to the minimum survival exponential entropy principle that corresponds to the maximum Shannon entropy principle, the optimal vector (t∗0 < ... < t∗N ) should meet:

n o (T ) (T ) BMα = ArgMin BMα

(1.13)

where 1 < t1 < ... < 255. In the case of one threshold (N = 2), the computational complexity for determining the optimal vector t∗ is O(L2 ) where L is the total number of gray level. However, it is too time-consuming in the case of multilevel thresholding. For the n-thresholding problem, it requires O(Ln+2 ). In this work, we used a the hybrid ACS, previously presented, for solving the problem formulated in (1.13) efficiently.

1.3.1.2

Results and discussions

In our experiments, the value of α was equal to 100, in order to have stable performances. The different values of the parameters of EACS algorithm were fixed empirically as follows: Population size= 100, Number of Cycles=2 × N , µ = 0.5, b = 2, TL = 3 × N ,

Sn = 10. The different images were acquired using Siemens Avento 1.5T, the resolution was 256 × 256 and the field of view equal to 250. The figure 1.3 presents an illustration

of a segmentation results obtained on two different pathologic MRI cases. Indeed, in this paper the pathology studied the hydrocephalus, that consists in atrophy in the ventricular system. Then, our goal was to calculate the volume of the cerebrospinal fluid (CSF) inside only the ventricle. The proposed algorithm was used as a pre-processing step to calculate the volume. The obtained result through the application of the segmentation algorithm when N = 2, shows that the use of the BSEE allows to segment satisfactorily the brain MR images.

1.3.2

Enhanced BBO for image segmentation

Biogeography is known as a popular method of studying geographical distribution of biological organisms, whose earliest works can be traced back to the days of Alfred Wallace and Charles Darwin [Simon, 2008]. The mathematical models of biogeography are available which describe the governing laws about migration of specifies from one island to another island, the arrival of new species and the extinction of some existing species.


(a)

14

(b)

Figure 1.3: Illustration of the segmentation results using BSEE-EACS on two different MRI (a) and (b).

However, only very recently a population based optimization technique has been proposed employing the basic nature of biogeography and it has been named biogeography based optimization (BBO) [Simon, 2008]. The mathematical models of biogeography are available which describe the governing laws about migration of specifies from one island to another island, the arrival of new species and the extinction of some existing species. However, only very recently a population based optimization technique has been proposed employing the basic nature of biogeography and it has been named biogeography based optimization (BBO) [Simon, 2008]. In biogeography models, the fitness of a geographical area (called ”island” or ”habitat”) is judged on the basis of habitat suitability index (called HSI). A habitat with a high HSI indicates that it is more suited for species to reside here. Similarly a habitat with a low HSI indicates that it is less suited for species to reside there. It is natural that higher the HSI of a habitat, it is likely that more number of species will be present there. The variables that characterize habitability, e.g. rainfall, vegetation, temperature etc, are called suitability index variables (SIVs). The dynamics of the movement of the species among different habitats is mainly governed by two parameters, called immigration rate (λ) and emigration rate (µ) and these two parameters are functions of the species count in a habitat. Fig. 1.4 shows the species model of a single habitat [Simon, 2008; MacArthur & Wilson, 1967]. The curve shows a special case when maximum immigration rate (I) and maximum emigration rate (E) are equal. However, strictly speaking, there is no such constraint and the condition of equality can be easily relaxed. In this curve the maximum number of species that a habitat can host is considered to be Smax . When the number of species (S) is small, there is more possibility of immigration of species from neighboring habitats and less possibility of emigration of species from this habitat to


15

neighboring habitats. With the increase in the population of species, the possibility of immigration to the habitat decreases and the possibility of emigration from the habitat increases. This is characterized by the two curves for immigration rate and emigration rate in figure 1.4. When S = S0 , the equilibrium condition is reached i.e. in a given time span, the same number of species immigrate to and emigrate from the habitat.

Figure 1.4: Curves for immigration and emigration rates in basic BBO algorithm.

Let the probability that the habitat contains S species at time t be given by Ps , then, the dynamic equations of the probabilities of species count in the habitat can be defined using a matrix relation.   S=0   − (λs + µs ) Ps + µs+1 Ps+1 ˙ Ps = − (λs + µs ) Ps + λs−1 Ps−1 + µs+1 Ps+1 1 ≤ S ≤ Smax − 1    − (λ + µ ) P + λ P S = Smax s s s s−1 s−1

(1.14)

As there is a possible maximum of Smax number of species in the habitat, one can obtain a matrix relation governing the dynamic equations of the probabilities of species count in the habitat:           1.3.2.1

P˙0 P˙1 .. . .. . P˙Smax





        =        

− (λ0 + µ0 ) .. . .. . .. . 0

··· .. . .. . .. .

0 .. . .. . µn

· · · − (λn + µn )

         

P0 P1 .. . .. . PSmax

         

(1.15)

Enhancement of biogeography based optimization (BBO) algorithm

These basic ideas of biogeography have been utilized to design a population based optimization procedure that can be potentially used to solve many engineering and other


16

BBO Inputs: ItM ax: maximum number of iterations while i < ItM ax do Evaluate the HSI (fitness) of each solution Compute S, λ, andµ for each solution Modify habitats (Migration) based on λ and µ Mutation based on probability Perform elitism to keep only the best solutions end Figure 1.5: Classical biogeographic optimization algorithm

optimization problems. As there are a lot of similarities between the mathematical model of biogeography and the population based optimization algorithms. The BBO algorithm proposed in [Simon, 2008] designate each habitat H as a potential m×1 decision variable vector, where H ∈ SIV m i.e. each habitat or solution comprises m SIV s. For each habitat H, its HSI corresponds to the fitness function in population-based algorithms.

A habitat with a higher HSI indicates that it is a better candidate for the optimum solution. It is considered that the ecosystem has n habitats i.e. the ecosystem is H n . In the context of population based metaheuristics, it means there are a total of n possible candidate solutions (i.e. the population size). The overall scheme of BBO is presented in figure1.5. In the basic BBO algorithm, the immigration and emigration rates follow as linear variations of species count. They are described using the following equations: λS = I ∗ 1 − and µS =

S Smax

E∗S Smax

(1.16)

(1.17)

In our work, we proposed an enhancement of BBO by implementing nonlinear variations of immigration rate and emigration rate with the number of species in a habitat. Indeed, the basic spirit of these variations will not violate the original considerations i.e. the immigration rate should decrease and emigration rate should increase with number of species. Our proposed variations for these two rates can be described as: λS = I ∗ 1 − and µSnlin = E ∗

S Smax S Smax

p1

(1.18)

p2

(1.19)

Chapter 1. Design of static metaheuristics for medical image analysis Theoretically both p1 and p2 can be chosen in the range [0,

17

∞[. The basic BBO

algorithm is a special case of these improved variants proposed in (6) and (7), where

p1 = p2 = 1.0. Fig. 2 shows typical variations of these proposed immigration and emigration rates for the case, when their maximum permissible values are same i.e. E = I. However, this condition is not really a constraint and it can be relaxed, if needed. The proposed modifications allows to have more flexibility and, depending on the image(s) under consideration, one can choose an appropriate set accordingly.

1.3.2.2

The segmentation criterion: overall probability error

The maximum fuzzy entropy measure was proposed in the works of Tao et al. [Tao et al., 2007, 2003], which were inspired by an earlier work of Zhao et al. [Zhao et al., 2001], where an entropy function was used to measure the compatibility between the fuzzy c-partition (FP) and the probability partition (PP). In Tao’s work [Tao et al., 2003] the entire image is classified into three partitions of dark pixels, medium pixels and bright pixels and each partition is characterized by a fuzzy membership function (MF). The dark pixels are characterized by a Z-shaped MF, the medium pixels are characterized Q by a -shaped MF and the bright pixels are characterized by an S-shaped MF. An example of three membership functions are shown in figure 1.6. In this method, the

three classes are associated to three membership functions, respectively, µd function of the class dark, the function µm , and tbe µb function. Then, the segmentation problem consists in finding the optimal thresholds vector that allows to maximize the total fuzzy entropy.

Figure 1.6: Membership function graph.

1.3.2.3


The proposed segmentation method was used to segmentation CT-Scan images. These images were acquired from the face CT scan of a volunteer in ”Centre Hospitalier Universitaire (CHU) Henri Mondor, Créteil (France)”. The image acquisition system employed was the PHILIPS famous multi-slice CT scanner. The resolution was 0.4883 mm


18

per pixel, each slice was 0.9mm thick, and the spacing between the consecutive slices is 0.45mm. The obtained results are illustrated by the one in figure 1.7. More results can be found in [Chatterjee et al., 2012]. In this work, we demonstrated that some form of the enhanced BBO could largely outperform both basic BBO and other metaheuristics for the posed problem. It is recommended that the choice of a specific enhanced BBO based variant should be carried out on a trial and error basis. However, our results have provided a thumb rule which can be adopted in providing a possible search direction for choosing the specific improved BBO based variant, where it is expected to obtain satisfactory segmentation performance.

(a)

(b)

Figure 1.7: Illustration of the segmentation results using Enhanced BBO. (a) Original CT-Scan slice, (b) Segmented image using Enhanced BBO.

1.3.3

Enhanced DE for image thresholding

Differential Evolution algorithm (DE) [Price et al., 2005] is one of the most popular metaheuristic for solving continuous optimization problems. Recently, it has gained much popularity in different kinds of applications, because of its simplicity and robustness in comparison with other evolutionary algorithms [J.Vesterstrom & R.Thomsen, 2004]. DE has very few parameters to adjust, making it particularly easy to implement for a diverse set of optimization problems [A.Bast¨ urk & E.G¨ unay, 2009; W-D.Chang, 2006; Liu et al., 2007]. This work proposes the development of a new optimal multilevel thresholding algorithm based on image histograms by employing an improved version of DE called EDE. After fitting the Gaussian curves using EDE, the optimal threshold is calculated by minimizing the overall probability error between these Gaussian distributions.


19

In this section we briefly describe the EDE, an enhanced version of basic DE. EDE uses the concepts of opposition based learning, random localization and has a one population set structure. The working of EDE is as follows. Population initialization EDE starts with a population S = {X1 , X2 , ..., XN P } of N P solutions: Xi = x(1,i) , x(2,i) , . . . , x(n,i) with i = 1, ..., N P , where the index i denotes the ith solution of the population. For this we randomly construct a population P 1 of N P

solutions, using the following rule: xi,j = xmin,j + rand(0, 1) × (xmax,j − xmin,j )

(1.20)

where xmin,j and xmax,j are lower and upper bound for j th component respectively and rand(0, 1) is a uniform random number between 0 and 1. We construct another population P2 of N P opposite solutions to those in the population P1 using the following rule: yi,j = xmin,j + xmax,j − xi,j

(1.21)

where xi, j is the component of solution Xi of population P1 . Now the initial population S is constructed by taking NP best solutions from union of P1 and P2 . Mutation The mutation operator of EDE applies the vector difference between the existing population members for determining both the degree and direction of perturbation applied to the individual subject of the mutation operation. The mutation process at each generation begins by randomly selecting three solutions {Xr1 , Xr2 , Xr3 } from the population

corresponding to target solution Xi . A tournament is then held among the three solutions and the region around the best point is explored. That is to say if Xr1 is the point having the best fitness function value, then the region around it is searched with the hope of getting a better solution. Assuming that Xtb = Xr1 , the mutation equation is given as: Vi = Xtb + F × (Xr2 − Xr3 )

(1.22)

where r1 , r2 , r3 ∈ 1, ..., N P are randomly selected such that r1 6= r2 6= r3 6= i, and F

is the control parameter such that F ∈ [0, 1]. This variation gradually transforms itself

into search intensification feature for rapid convergence, when the points in S form a cluster around the global minima. Crossover The Crossover operator of EDE is same as that of DE. According to it, once the perturbed individual Vi = (v1,i , . . . , vn,i ) is generated, it is subjected to crossover operation with target individual Xi = x( 1, i), x(2,i) , . . . , x(n,i) , that finally generates the trial


20

solution, Ui = (u1,i , ..., un,i ), as follows:

uj,i =

(

if randj ≤ Cr ∨ j = k

vj,i xj,i

Otherwise

(1.23)

where j = 1, . . . , n, k ∈ 1, . . . , n is a random parameter index, chosen once for each i. The crossover rate, Cr ∈ [0, 1], is set by the user.

Selection The selection operator of EDE for new solutions is different from that of DE. After the generation of a new solution, selection operation is performed between it and its corresponding target solution by the following equation:

Xi0

=

(

Ui Xi

if f (Ui ) ≤ f (Xi ) Otherwise

(1.24)

If the new solution is better than the target solution then it replaces the old one in the current population. While in DE, the better one of these two solutions is added to an auxiliary population, two populations (current and auxiliary) are considered simultaneously in all the iterations, which results in the consumption of extra memory and CPU time. On the other hand, in EDE, only one population is maintained and the individuals are updated when a better solution is found. Also, the newly found better solution that enters the population instantly takes part in the creation of new solution.

1.3.3.1

Segmentation criterion

The segmentation criterion used in this work is based on the approximation of the image histogram by a Gaussian mixture model. Indeed, over the years, many authors have proposed several algorithms to solve Gaussian mixture model for multi-level thresholding. Besides, Snyder et al. [Synder et al., 1990] presented an alternative method for fitting curves based on a heuristic method called tree annealing; we also proposed a fast scheme for optimal thresholding using simulated annealing algorithm [Nakib et al., 2007; A.Nakib et al., 2008]; Zahara et al. [Zahara et al., 2005] proposed a hybrid Nelder-Mead Particle Swarm Optimization (NM-PSO) method and more recently a hybrid method based on Expectation Maximization (EM) and Particle Swarm Optimization (PSO+EM) was proposed in [Fan & Lin, 2007]. All these metaheuristics based methods are efficient in solving the multi-level thresholding problem and could provide better effectiveness than the other traditional methods (local search and deterministic methods). However, curve fitting is usually time-consuming, which indicates that improved methods are yet needed to enhance the efficiency, while maintaining effectiveness, and these methods have many parameters that must be well fitted.


21

Table 1.1: Parameters of EDE. N = 3 × D where D is the number of classes (to be fixed by the user).

Parameter Population size Scaling factor F Crossover rate Cr Maximum number of iterations

Value 10 × N 0.25 0.25 200

This segmentation approach consists in two optimization problems: the first is that of the Gaussian curve fitting problem. While the second is the minimization the overall probability error of miss classification of the pixels. The first problem is a continuous optimization problem, while the second is a discrete optimization problem.

1.3.3.2


The proposed algorithm EDE has only 4 parameters that must be well fitted. We have done preliminary testing for the purpose of getting suitable values of these parameters and results are listed in Table 1.1, and the initial population is generated randomly under some considerations. Moreover, the stopping criterion we used for the algorithm is the maximum number of iterations. The obtained results through the application of our segmentation algorithm are illustrated through the original brain MRI in Fig. 1.8. Fig. 1.8 shows the original images and its multilevel classification (segmented) version when the number of thresholds is 4 (5 classes segmentation). The number of classes is an input parameter of the segmentation algorithm. A non-supervised technique for determining the number of classes was proposed in [A.Nakib et al., 2008] that can be used at initialization. Our goal is to detect the different spaces and th white matter surrounding the ventricular space quickly.

1.3.4

Metaheuristic for contours detection in 2D Ultrasound images

In Computer Assisted Orthopedic Surgery (CAOS) systems, the intra-operative image modality of choice is often Computed Tomography (CT) or fluoroscopy (X-rays projection). These image modalities are not completely safe for the patient, and for the surgeon who uses them everyday. Within the last decade, ultrasounds (US) became an interesting alternative for orthopedic surgeons. It is well known that US devices are not too expensive, and portable ; it also can be used in real-time intra-operatively and it is non-invasive. However, the US images are difficult to analyze for the surgeon, because of the high level of attenuation, shadow, speckle and signal dropouts [Jain & Taylor, 2004].


(a)

22

(b)

Figure 1.8: Illustration of the segmentation results using Enhanced DE. (a) Original pathologic MRI 2D image, (b) 5 classes segmented image using Enhanced DE.

In the literature, the extraction of the bone surface in US images was studied in [Heger et al., 2005]. The authors used a A-mode ultrasound pointer. The probe was tracked mechanically, and it was used to register the distal femur in total hip replacement. The A-mode of ultrasound probes consists in using only one ultrasound beam. Then, the output image is a one dimensional vector. Usually, the used mode is the B-mode, where the resultant image is a matrix, and the number of beams is greater than one. In CAOS systems, ultrasounds can be used to collect some sample points on bone surface [Beek et al., 2006], or to perform intra-operative registration, extracting the full 3D model [Zhang et al., 2002]. Manual segmentation of the bone surface in US images is highly operator dependent and time consuming [Barratt et al., 2006]. Moreover, the thickness of the response can reach 4mm in some cases [Jain & Taylor, 2004], and it can lead to a high error. In [Foroughi et al., 2007] developed an automatic segmentation method of bone surface in US images using dynamic programming. This method depends on a threshold value, the obtained average error was between 2.10 pixels to 2.67 pixels at the comparison between automatic and manual segmentation; the average time of computation per image were 0.55 seconds. In this work, our main interests lies in the use of US images in computer assisted intramedullary nailing of tibia shaft fractures. If a surgeon choose to heal a tibia shaft fracture using an intramedullary nail, then he has to lock the nail in the bone. Normand et al. proposed to use some measures on the healthy symmetric tibia to assist the surgeon during the locking of the nail [Normand et al., 2010]. To do so, the 3D position of some anatomical landmarks is needed (malleolus, trochlea, femoral condyles, ...), and the healthy tibia should not be cut. Then, the authors proposed to use the US probe as a subcutaneous pointer. The main goal of this new method is to extract automatically,


23

and in real-time, the bone surface from US images, and particularly, the anterior femoral condyles. The proposed method consists in two main steps. In the first step, a vertical gradient is applied to extract potential segments of bone from 2D US images. In the second step, a new method based on shortest path is used to eliminate all pixels that do not belong to the final contour. Finally, the contour is closed using polynomial interpolation.

1.3.4.1

Proposed Bone contour extraction method based on shortest path

Let I : Ω ⊂ N2 → I ⊂ N be an image (two dimensional (2D) real function). Segmenting

bone surface in I consists in extracting {Pi |i = 1, .., n} a subset of contiguous points in

I, where Pi = (xi , yi ) ∈ Ω, ∀i = 1, .., n. Considering ultrasound properties of bones [Jain

& Taylor, 2004], we admit that ∀(i, j) ∈ [1, n]2 such that i 6= j, yi 6= yj .

Then, the proposed segmentation method consists in three steps : in the first step, original images are filtered, a vertical gradient is computed, and an extraction of some potential segments of bone contour is performed. Then, the second step consists in characterizing these segments of contour, in order to eliminate those that a priori do not belong to the bone contour. Final step consists in closing the contour using least square polynomial approximation. It is well known that ultrasound images are highly textured, mainly with speckle. Then, the first step consists in a pre-processing step where a low-pass filter is applied to the original image (an example of images at hand is presented in figure 1.9) in order to eliminate noise and to strengthen interesting features. Once the filtered image Is is computed, the vertical gradient is applied to the filtered image, we denote the result image Ig . The choice of the vertical gradient was motivated by ultrasound propagation properties, where the bone contours are mainly horizontal. It was shown in [Jain & Taylor, 2004] that it is suitable that the bone contour lies on the top of the fiducial surface. Then, we only keep high values of the gradient, we called this image IBW . Then, using properties of ultrasound imaging on bones [Jain & Taylor, 2004], we can extract from IBW a first subset of potential contour points {Qi |i = 1, .., c}, where c is

the number of columns in the original image I. We denote Qi = (xi , i), ∀i = 1, .., c in the rest of this section. The subset of Qi was built by taking the first non-zero point in

each column of IBW starting from the bottom of the image. The next step consists in characterizing these points to determine whether or not they belong to the bone contour.


24

Figure 1.9: Original US image.

1.3.4.2

False alarm elimination using shortest path formulation

The subset of points {Qi |i = 1, .., c} are potentially part of the bone contour. To

select those that belong to the bone contour, we consider them as segments by grouping

contiguous points. Two points are considered to be contiguous if they belong to the same neighborhood. In this step, all small segments that are very likely noise, and segments that are too close to the skin are eliminated automatically, and not considered. For each segment k, where k = 1, .., M , the first point is designated by Qak and the last point by Qbk , where ak and bk are the column of Qak and Qbk , respectively. To define segments that belong to the bone contour, we define G(N , E) as an oriented graph from the M segments (figure 1.10) : N

= {ni | i = 1, .., 2M } = {ak | k = 1, .., M } ∪ {bk | k = 1, .., M }

(1.25)

is the set of all nodes in the graph, where the node index ni is defined by :

∀i ∈ [1, 2M ], ni =

 a i+1 2

b i

2

if i is odd if i is even

(1.26)


! (2,7)

25

! (6,9)

! (2,5) n1 = a1

! (1,2)

n2 = b1

! (2,3)

n3 = a2

! (3,4)

n4 = b2

! (4,5)

n5 = a3

! (5,6)

n6 = b3

! (4,7)

! (6,7)

n7 = a4

! (7,8)

n8 = b4

n2 M = bM

! (4,9)

Figure 1.10: The construction of the graph G. We distinguish nodes called ”start of segments” which are the ak nodes and the nodes called ”end of segments” which are the bk nodes.

We define also the set of edges in the graph as : ∀(i, j) ∈ [1, 2M ]2 ,

  1  if i is odd and j = i + 1 (b j − a j )   2 2 2 E(i, j) = kQb i+1 − Qa j+1 k if i and j are even, and if i < j ≤ min(2M, i + 6)  2 2    0 otherwise (1.27)

Then, G is a graph with two types of node: the first nodes of segments, which are

{ak | k = 1, .., M } with an only one child, bk and the weight of the edge between them is 1 2 (b 2j

−a j ), and the last nodes of segments which are {bk |k = 1, .., M } with at most three 2

children which are {bl | l = k + 1, .., min(k + 3, M )}, and the weight of edges between

them are kQb i+1 − Qa j+1 k. We penalize the intra-segment distance because the path to 2

2

be minimized is the inter-segment one. For each segment k, ak is used to include the

segment length in the shortest path computation, and bk is used to eliminate non-bone segments. Then, to solve the shortest path problem any metaheuristic can be applied to solve it. In our work, genetic algorithms, ant colony optimization algorithm and Dijkstra’s algorithm were tested. Finally, the closure of the contour is performed by a polynomial approximation using least square method. The contour is computed using points that belong to the remaining set of segments.

1.3.4.3


To show the efficiency of the proposed method, different tests were performed on several series of ultrasound images. The probe and the used beamformer were provided by Telemed (Vilnius, Lituania), and a software developed by Aesculap SAS (Echirolles, France) was used for the acquisition. The acquisition protocol consists in putting the


26

probe under the patella, and to perform a scan of the femoral condylar region rotating the probe up and down. Figure.1.11 illustrates the different steps of the proposed framework and especially the contours detection step. One can see that the proposed approach to detect the contours is efficient and low complex because it allows to track the contours in real time and quality of the results was validated on large database. More details about this contribution can be found in [Masson-Sibut et al., 2012].

1.4

Conclusion

In this chapter, we first formulated the image segmentation problem as an ill posed problem and its formulation as an optimization problem. Then, we outlined our main contributions. Over the last few years, our work has focused on two areas: improving the performance of metaheuristics and developing new segmentation criteria. Our contributions presented in this chapter can be summarized on the following points: • New procedure for initializing metaheuristics based on lows discrepancy sequences. • Adding tabu memory to ACS. • New formulation of the image thresholding problem as a graph cut problem. • New formulation for contours detection based on shortest path optimization problem.

Moreover, a new segmentation criteria and formulation based on survival exponential entropy and a new method to extract contours in the case of US images were proposed.

1.5

Publications related to this chapter

• A. Masson-Sibut, A. Nakib, ”Real-time assessment of bone structure positions via

ultrasound imaging”, Journal of Real-Time Image Processing, DOI 10.1007/s11554015-0520-8, 2015.

• A. El Dor, J. Lepagnot, A. Nakib , P. Siarry, ”PSO-2S Optimization Algorithm

for Brain MRI Segmentation”, in Proc. of The Seventh International Conference on Genetic and Evolutionary Computing, ICGEC 2013, Prague, Czech Republic, Vol. 238, pp 13-22, August 25-27, 2013.


27

(a)

(b)

(c)

(d)

(e) Figure 1.11: Extraction of femoral condyles and trochlea in an ultrasound image, after the calculation of the bone contour using proposed method. (a) Ultrasound image of femoral condyles, (b) First set of the potential bone contour pixels, (c) Results of the shortest path based step, (d) Result after interpolation step, (e) Extraction of the femoral condyles and trochlea in an US image.

• A. Masson-Sibut, A. Nakib, E. Petit, F. Leitner, ”New automatic landmarks extraction framework on ultrasound images of the femoral condyles” in Proc. of the SPIE Medical Imaging conference, San Diego, CA, USA, Feb. 4-6, 2012. • A.Chatterjee, P. Siarry, A. Nakib, R. Blanc, ”An improved biogeography based


28

optimization approach for segmentation of human head CT-scan images employing fuzzy entropy”, Engineering Applications of Artificial Intelligence, Elsevier, vol. 25, no. 8, pp. 1698-1709, 2012. • A. Nakib, B. Daachi, P. Siarry, ”Hybrid differential evolution using low-discrepancy

sequences for image segmentation”, in Proc. Of the 26th IEEE Int. Parallel and Distributed Processing Symposium, IPDPS’2012, Shanghai, China, May 21-25, 2012, pp. 627-633.

• A. Masson-Sibut, A. Nakib, E. Petit, F. Leitner, ”Computer Assisted Intramedullary Nailing using Real-Time Bone Detection in 2D Ultrasound Images”, in Machine

Learning in Medical Imaging (MLMI), Workshop at the 14th International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI 2011, Toronto Canada September 18-24, 2011. • S. Hajjem, A. Nakib, H. Oulhadj, P. Siarry, ”Brain MRI segmentation based on shortest path and biased survival exponential entropy”, in Proc. of the 10th Biennal Int. Conf. on Artificial Evolution, EA 2011, Angers, France, Oct. 24-26, 2011. • A. Nakib, R. Blanc, H. Oulhadj and P. Siarry, ”Artificial ants for magnetic reso-

nance images segmentation”. Artificial Ants, Ed. by N. Monmarché, F. Guinand and P. Siarry. Chapter 10, pp. 205-218. Edition ISTE - John Wiley & Sons. 2010.

• A. Nakib, H. Oulhadj, P. Siarry, ”Image thresholding based on Pareto multiob-

jective optimization”, Engineering Applications of Artificial Intelligence, Elsevier, vol. 3, no. 2, pp. 313-320, 2010.

• A. Nakib, R. Blanc, H. Oulhadj and P. Siarry, ”Les fourmis pour la segmentation d’images médicales par résonance magnétique”, book title: ”Fourmis artificielle 1, des bases de l’optimisation aux applications industrielle ”, Hermes-Lavoisier Collection Traité IC2, pp. 241-254, 2009. • A. Nakib, H. Oulhadj, P. Siarry, ”Non-supervised image segmentation based on

multiobjective optimization”, Pattern Recognition Letters, Elsevier, vol. 29, no. 2, pp. 161-172, 2008.

Chapter 2

Design of metaheuristics for Dynamic continuous optimization problems 2.1

Introduction

Dynamic metaheuristics of optimization (DMO), or the study of applying metaheuristics algorithms to dynamic optimization problems (DOPs), is an active research topic and has increasingly attracted interest from the metaheuristic community. The field is relatively young as most of the studies have been made in the last 20 years with the exception of a few early works. Due to its relatively young age, the field still has a lot of open areas with open research questions, of which perhaps one of the most important questions is about how well academic DMO research reflects the common characteristics of DOPs and if there are any types of DOPs that have not been covered by current academic research. The main purpose of our work is to investigate this important question and to propose solutions. So, a dynamic optimization problem can be expressed as in (3.6), where f (~x, t) is the objective function of a minimization problem, hj (~x, t) denotes the j th equality constraint and gk (~x, t) denotes the k th inequality constraint. All of these functions may change over time (iterations), as indicated by the dependence on the time variable t.

min f (~x, t) s.t.

hj (~x, t) = 0 for j = 1, 2, ..., u gk (~x, t) ≤ 0 for k = 1, 2, ..., v 29

(2.1)

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 30 In this work, our interest is focused on dynamic optimization problems (DOPs), which are solved online by an optimization algorithm. In other terms, a time-dependent problems that is solved in a dynamic way, i.e. new solutions are produced to react to changes over the time are considered. The other cases where the future changes can be completely integrated into a static objective function, or a single robust-to-changes solution can be provided, or only the current static instance of the time-dependent problem is taken into account, then the problem can be solved using static optimisation techniques and hence is no longer of interest to MDO. The performance of dynamic optimization algorithms in the literature is improving, and many research directions are left to be further investigated in order to obtain even more efficient algorithms. In our work, a new algorithms for dynamic continuous optimization were proposed. The approach used belongs to the class of cooperative search strategies for DOPs. It makes use of a population of coordinated local searches to explore the search space. In the literature many metaheuristics were adapted to solves DOPs, we will not provide a literature review on the topic, the interested reader is referred to these two recent surveys [Nguyen et al., 2012; Cruz et al., 2011] and the website http:// dynamic-optimization.org for more information. However, some authors investigated also the use of local search as a main feature of a dynamic optimization algorithm. We can cite [Zeng et al., 2007], where authors propose an algorithm based on the use of local searches to explore the search space. The local optima found are archived, in order to be tracked when a change occurs, using additional local searches. In [Moser & Hendtlass, 2007; Moser & Chiong, 2010], Moser and Hendtlass use extremal optimization (EO) to determine the initial solution of a local search procedure. EO does not use a population of solutions, but improves a single solution using mutation. In [Pelta et al., 2009a,b], Pelta et al. propose a multi-agent decentralized cooperative strategy, where multiple agents cooperate to improve a set of solutions stored on a grid. Then, Gonzalez et al. presented a centralized cooperative strategy for DOPs that uses several tabu search based local searches [Gonzalez et al., 2010]. These local searches are controlled by a coordinator that keeps a memory of the found local optima. Besides, Wang et al. propose in [Wang et al., 2009] a memetic algorithm based on the cooperation and competition of two local search procedures, combined with diversity maintaining techniques.

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 31

2.2

Bechmarks and performance evaluation

In dynamic environments, the goal is to quickly find and follow the global optimum over the time. Then, the measures used to evaluate the performance of dynamic optimization algorithms take into account this fact. In the literature, benchmark problems can be divided into two categories: combinatorial problems based and real functions based. In the first category, some stationary problems are initially defined, then the environment switches between them. In this category, we can cite the Dynamic Bit-matching, the dynamic Knapsack problem, and the dynamic vehicle routing problem. This list is not exhaustive and other benchmarks can be found in the literature. The second category of benchmark problems uses a basic function which changes to build the dynamic environments. In the literature, few benchmarks were proposed, we can cite: Moving Peaks (MPB)[Branke, 1999b], Moving Parabola [PeterJ., 1997], Dynamic Optimization Problems Generators [Yang & Yao, 2005], Problem generator based on deceptive functions [Yang, 2004], Generalized Dynamic Benchmark Generator (GDBG)[Li & Yang, 2008; Li et al., 2008]. A recent survey the different benchmarks in the literature can be found in [Nguyen et al., 2012]. In our work, we were interested by the most used benchmark: MPB, and GDBG this choice is motivated by their common utilisation by the authors in the literature. Then, they are presented in the follwing.

2.2.1

The Moving Peaks Benchmark

Table 2.1 presents a summary of competing methods available in the literature and gives the test problems used by each one. The test problems gathered in the first column of the table 2.1 are those used by the competing methods. These competing methods are described in the references listed in the third column. From table 2.1, one can remark that the most commonly used testbed is the Moving Peaks Benchmark (MPB) [Branke, 1999b]. This benchmark is becoming the standard for testing dynamic optimization algorithms, and is claimed to be representative of real world problems [Branke, 1999a]. To compare our algorithm to the competing ones, this testbed was adopted. MPB consists of a number of peaks that vary their shape, position and height randomly upon time. At any time, one of the local optima becomes the new global optimum. MPB generates DOPs consisting of a set of peaks that periodically move in a random direction, by a fixed amount s (the change ”severity”). The movements are autocorrelated by a coefficient 0 ≤ λ ≤ 1, where 0 means uncorrelated and 1 means highly autocorrelated. The peaks change position every α iterations (function evaluations), and α is called time span. The fitness function for the landscape of MPB is formulated as follows:

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 32 Table 2.1: Summary of DOP algorithms and corresponding bachmarks used to test their performances. Test problems MPB1 and the dynamic Rastrigin function various continuous dynamic problems [Dr´ eo & Siarry, a computer vision problem MPB MPB MPB a set of discrete dynamic problems [Yang, 2003; Yang a set of discrete dynamic problems [Yang, 2003; Yang various continuous dynamic problems [Dr´ eo & Siarry, MPB MPB MPB a testbed proposed by the authors MPB 1

2006]

& Yao, 2005] & Yao, 2005] 2006]

Base algorithm PSO2 [Du & Li, 2008] ACO3 [Tfaili & Siarry, 2008] EAs4 2008 [Rossi et al., 2008] Hybrid PSO/EAs [Lung & Dumitrescu, 2008] Hybrid PSO/EAs [Lung & Dumitrescu, 2007] EO5 [Moser & Hendtlass, 2007] GA6 [Tinos & Yang, 2007] GA [Yang, 2006] ACO [Dr´ eo & Siarry, 2006] PSO [Li et al., 2006] PSO [Blackwell & Branke, 2006] DE7 [Mendes & Mohais, 2005] EAs [Huang & Rocha, 2005] PSO [Blackwell & Branke, 2004]

The Moving Peaks Benchmark [Branke, 1999b]. 2 Particle Swarm Optimization. 4 Evolutionary Algorithms. 5 Extremal Optimization. 6 Genetic Algorithm.

3 7

Ant Colony Optimization. Differential Evolution.

v  u d uX F (~x, t) = maxi=1,...,m Hi (t) − Wi (t)t (xj − Xij (t))2  



(2.2)

j=1

where m is the number of peaks, d is the number of dimensions, Hi (t) is the height of ~ i (t) is the ith peak at the time t, Wi (t) is the width of the ith peak at the time t and X the position of the ith peak at the time t. The figure 2.1 illustrates an MPB landscape before and after a change (after one time span).

Figure 2.1: An MPB landscape before and after a change.

In order to evaluate the performance, the ”off-line error” is used. Off-line error (oe) is defined as the average of the errors of the best points evaluated during each time span. It is defined by:   N e(j) Nc X X 1 ∗   1 oe = fj∗ − fji Nc N e(j) j=1

i=1

(2.3)

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 33 Parameter Number of peaks Dimension d Peak heights Peak widths Change cycle α Change severity s Height severity Width severity Correlation coefficient Number of changes Nc

Scenario 2 10 5 [30, 70] [1, 12] 5000 1 7 1 0 100

Table 2.2: MPB parameters in scenario 2.

where N c is the total number of fitness landscape changes within a single experiment, N e(j) is the number of iterations performed for the j th state of the landscape, fj∗ is the ∗ is the current best fitness value of the optimal solution for the j th landscape and fji

value found for the j th landscape. We can remark that this measure has some weaknesses: it is sensitive to the overall height of the landscape, and to the number of peaks. It is important for an algorithm to find the global optimum quickly, to minimize the off-line error. Hence, the most successful strategy is a multi-solution approach that keeps track of every local peak [Moser & Hendtlass, 2007]. In [Branke, 1999b], three sets of parameters, called scenarios, were proposed. It appears that the most commonly used set of parameters for MPB is scenario 2 (see Table 2.2), hence, it will be used in this paper.

2.2.2

The Generalized Dynamic Benchmark Generator

The Generalized Dynamic Benchmark Generator (GDBG) is the second benchmark used in this paper, it is described in [Li & Yang, 2008; Li et al., 2008]. It was provided for the CEC’2009 Special Session on Evolutionary Computation in Dynamic and Uncertain Environments. It is based on the Sphere, Rastrigin, Weierstrass, Griewank and Ackley test functions that are commonly used in the literature. They are presented in detail in [Li et al., 2008]. These functions were rotated, composed and combined to form six problems with different degrees of difficulty: F1 : rotation peak function (with 10 and 50 peaks) F2 : composition of Sphere’s function F3 : composition of Rastrigin’s function F4 : composition of Griewank’s function

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 34 Parameter Dimension d (fixed) Dimension d (changed) Change cycle α Number of changes Nc

Value 10 [5, 15] 10000 × d 60

Table 2.3: GDBG parameters used during the CEC’2009 competition.

F5 : composition of Ackley’s function F6 : hybrid composition function

A total of seven dynamic scenarios with different degrees of difficulty was proposed:

T1 : small step change (a small displacement) T2 : large step change (a large displacement) T3 : random change (Gaussian displacement) T4 : chaotic change (logistic function) T5 : recurrent change (a periodic displacement) T6 : recurrent with noise (the same as above, but the optimum never returns exactly to the same point) T7 : changing the dimension of the problem

The basic parameters of the benchmark are given in Table 2.3, where the change cycle corresponds, as for MPB, to the number of evaluations that makes a time span.

There are 49 test cases that correspond to the combinations of the six problems with the seven change scenarios (indeed, function F1 is used twice, with 10 and 50 peaks respectively). For every test case, the tested algorithm is run several times. The number of runs of the tested algorithm is equal to 20 in our experiments.

As defined in [Li et al., 2008], the convergence graphs, showing the relative error ri (t) of the run with median performance for each problem, are also computed. For the maximization problem F1 , the formula used for ri (t) is defined in equation 2.4, and for the minimization problems F2 to F6 , it is defined in equation 2.5:

ri (t) =

fi (t) fi∗ (t)

(2.4)


ri (t) =

fi∗ (t) fi (t)

(2.5)

where fi (t) is the value of the best found solution at time t since the last occurrence of a change, during the ith run of the tested algorithm, and fi∗ (t) is the value of the global optimum at time t. The marking scheme proposed by the authors of GDBG works as follows: a mark is calculated for each run of a test case, and the average value of this mark is denoted by markpct . The sum of all marks markpct gives a score that corresponds to the overall performance of the tested algorithm, denoted by op. The percentage of the mark of each test case in this final score is defined by a coefficient markmax . It is also the maximum value of markpct that can be obtained by the tested algorithm on each test case. The authors of GDBG have set the values of the coefficients markmax such that the maximum value of op is equal to 100. This score is a measure of the performance of an algorithm in terms of both convergence speed and solution quality. It is based on the value of an approximated off-line error, and on the value of the best relative error, calculated for each time span. More details about this marking scheme are given in [Li et al., 2008].

2.2.3

Performance evaluation

Finding the best performance measure of a dynamic optimization algorithm (DOA) is not such easy. Indeed, a perfect metric must quantify and describe what the user the developer attend from a DOA. Moreover, it should be low complex in order to be easy to implement, and to use. Some classical measures from non-stationary (static) problems, like off-line error used in MPB [Branke, 1999b] (see eq. 2.3) and accuracy [Weicker, 2003], have been modified and adapted to DOPs. Theres is also measures that have been specially designed for DOPs, like collective mean fitness [Morrison, 2003]. Despite the existance of may other measures: window accuracy [Weicker, 2003], best known peak error [Bird & Li, 2007b], and peak cover [Branke, 1999b], the most current studies tend to use the off-line error and a visual analysis of the algorithm running performance. Recently, the authors in [Sarasola & Alba, 2013] proposed also a new metric, degradation metric linear regression, that allows to quantify the the ability of an algorithm to be able to track the moving optima for a big number of periods(where a period is an interval of time without changes in the problem definition). Consequently, there is no general consensus about what measure to use. While the great majority of studies use an average of the best current fitness, it is obvious that a single value provided by is often not sufficient, since two different distributions of the scores

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 36 can have the same average value. However, in order to have a quantitative comparison of the proposed algorithms to those of the literature, we used the performance measure proposed in the two benchmarks presented previously: GDBG and MPB.

2.3

Proposed Multi-Agent based approach for dynamic optimization

The idea of this new approach for metaheuristics is based on the way that the information flows in multiagent systems. The parallel between a metaheuristic and multiagent system consists in considering an agent (exploratory function) as a local search, then, a set of agents is a set local searches algorithms. So, we called this approach MultiAgent (MA) based Dynamic Optimization. Four dynamic algorithms based on this approach were proposed and evaluated. In the next, we present theses algorithms briefly. To develop these algorithms the following considerations were taken into account : in the case of a multimodal environment, the tracking each local optimum is recommended, in order to overcome the case where the global optimum ”jumps” from one of them to another. The found optima are archived in a memory. Then, this memory can be used when a change is detected. It is common that real world problems have time costly evaluation of fitness functions. Hence, the computational cost of the proposed algorithm should be made as short as possible and expressed in terms of number of evaluations. The proposed algorithm makes use of the following inspirations: • keeping information about the previous positions allows to prevent wasting evaluations in unpromising zones of the search space;

• using a sampling of candidate solutions that optimally cover the local landscape may reduce the number of evaluations per trajectory step, without decreasing performance; • the sampling of candidate solutions is adaptable to the local landscape; • avoiding the case where some search trajectories explore the same area of the search space (reducing the recovery areas).

The overall scheme of an algorithm based on the proposed is illustrated in figure 2.2. As it is shown, all the interactions between the memory manager and the agent manager are through the coordinator. Moreover, all global decisions are taken by the coordinator. The memory manager maintains the archive of local optima, that are provided by the

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 37 coordinator. The agent manager informs the coordinator about the found optima, and receives its instructions for creating, deleting, and repositioning agents.

Figure 2.2: Overall scheme of MA based optimization algorithm.

The three modules that compose an MA algorithm are : 1. Memory module: in case of a multimodal environment, a dynamic optimization method needs to keep track of each local optimum found, since one of them can become the new global optimum after a change. We propose to use a memory to archive the found optima. 2. Agent manager: it contains all the agents, and manages their execution, creation and deletion. Agents are nearsighted (they have only a local vision of the search space). More precisely, agents are only performing local search, they jump from their current position to a better one, in their neighborhood, until they cannot improve their current solution, reaching thus a local optimum. 3. Coordinator: it counterbalances the nearsightedness of the agents. The coordinator has indeed a global vision of the search performed by the agents, and it is able to prevent them from searching in unpromising zones of the search space. The coordinator supervises the whole search, and manages the interactions between memory and agents modules. In our approach agent consists in a single solution based local search that has a memory. Then, agents can do their local search independently of each other. The flowchart of the search procedure of an agent is illustrated in Figure 2.3. One can see that two special states, named ”check point A” and ”check point B”, appear in this flowchart. These states mark the end of one step of the procedure of an agent. Hence, if one of


Figure 2.3: flowchart of the main procedure of an agent in MA based metaheuristic.

these states has been reached, then the agent halts its execution until all other agents have reached one of these states. Afterwards, the execution of the agents is resumed; i.e., if an agent halts on check point A (check point B, respectively), then it resumes its execution on check point A (check point B, respectively). This special state allows the parallel execution of the agents. Then, we proposed three different algorithms that differ in the way of exploring the search space. In the following, we present brievlfy these three methods.

2.3.1 2.3.1.1

Multiagent Algorithm for Dynamic Optimization (MADO) Overall description

The exploration strategy of the agents

Agents explore the search space step-by-step, moving from their current position to a better one in their neighborhood, until they reach a local optimum. Hence, to precisely describe the behavior of the agents, we first explain the used kind of neighborhood, and

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 39 how the agents make use of it. As agents are nearsighted, they can only test candidate solutions for their next move in a delimited zone of the search space, centered on their current position. This zone must be bounded. Without any information about the local landscape of an agent, a good choice is to make it isotropic. Then, the most adapted topology is a ball. To keep small the number of fitness function evaluations, we need a sampling that optimally covers the local landscape. Afterwards, we maximize the distances between all candidate solutions inside this ball. It leads to sample them on the boundary of the ball (on a hypersphere centered on the current solution of an agent). This delimited zone of the search space must be adaptive to the local landscape, to increase the efficiency of the agents. It is done by using an adaptive radius for the ball that defines this zone. From all these considerations, we define the neighborhood of the agents as a set of N candidate solutions placed on the hypersphere of radius R centered on the current position of an agent (N is a parameter of the algorithm). Figure 2.4 illustrates this neighborhood in dimension 2. Another set of N points are generated and stored at the beginning of the MADO algorithm. These points are calculated in order to meet the following constraints: • the smallest distance between them is maximized; • the distances between them and the origin of the space are normalized; • the number of dimensions of the space in which they are defined match the number of dimensions of the search space.

This computation is done using the well known electrostatic repulsion heuristic [Conway & Sloane, 1998], that considers points as charged particles repelling each other. Starting from an arbitrary distribution, it uses point repulsion, where all points are considered to repel each other according to a 1/r2 force law, with r the distance between two points, and dynamics are simulated. The algorithm runs until the satisfaction of a stopping criterion, and the resulting set of points is returned. We use a stopping criterion that is satisfied when no point can be moved by a distance greater than , typically equal to 10−4 . Then, this set of points is used by agents to get the positions of the candidate solutions of their current neighborhood. The procedure used to get the exact positions of candidate solutions is presented in algorithm. The adaptation of the radius R makes use of trajectory information gathered along the steps of an agent. A low complexity version of the ”cumulative path length control” described in [Hansen & Ostermeier, 2001] was proposed to adapt the step size. The figure 2.5 illustrates the two possible cases of ”bad” trajectories that an agent may follow.: (1) when one of these cases is detected, an increase or a decrease of the step


The agent is depicted by a grey-filled circle. Candidate solutions are represented by black squares.

Figure 2.4: The sampling of candidate solutions of an agent, when d = 2 and N = 8.

size R is performed. The first case (figure 2.5 (a)) may mean that the agent is turning around an optimum, without being able to reach it directly. This is due to a too large step size, leading to back and forth displacements, which cancel each other out. Thus, in this first case, a decrease in R is needed. On the contrary, the displacements on figure 2.5 (b) are oriented in the same direction, which may mean that the agent is hill climbing a large peak. Then, to avoid too slow moves to the top of this peak, an increase in R is performed.

Figure 2.5: The two kinds of trajectories that lead to a step size adaptation. (a) Agent turning around an optimum. (b) Agent hill climbing a large peak.

The diversity maintaining strategy To prevent several agents from exploring the same zone of the search space, and to prevent them from converging to the same local optimum, an exclusion radius is attributed to each agent. Hence, when several agents are close, only the agent with the best fitness, among the detected agents and the agent having detected them, is allowed to continue its search. All the other agents have to start a new search elsewhere.

Local optima archiving strategy

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 41 The memory manager maintains the archive of local optima found by the agents. This archive is bounded and its size is fixed by the user. Thus, all the found optima cannot be included into this archive, only best ones will be stored. When the archive is full, the following condition to update the archive are applied: • If the new optimum is better than the worst optimum of the archive, or its fitness

value is at least equal to the one of this worst optimum, then this worst optimum is replaced by the new one;

• If there is one or several other optima in the archive that are ”too close” to the new optimum, then it is possible that all these optima close to each other are in fact

scattered around the top of one peak. Thus, this subset of solutions is replaced by the best optimum among them. Hence, if there are other optima in the archive that are ”too close” to the newly found one, this new optimum will replace them, only if it is better, or if its fitness is at least equal to the fitness of the best one. The process of detecting possible ”too close” optima is presented in algorithm. The change detection and the tracking of the optima Changes in the environment are detected by re-evaluating the fitness of the best optimum of the archive. When any optimum has been found yet, we re-evaluate the current fitness of an agent randomly chosen, and compare it to its previous value. If these values are different, a change is supposed to have occurred, and a tracking of the stored optima is performed. In order to track the optima, i.e. to hill climb the peaks of the previously stored optima to find their new position. A large initial step size allows to these agents to leave the peaks of the tracked optima. Thus, they will explore other zones of the search space rather than staying on the peaks they have to hill climb.

2.3.1.2

Results on MPB and GDBG Benchmarks

The comparison, on MPB, of MADO with the other leading optimization algorithms in dynamic environments is summarized in Table 2.4. The off-line errors and the standard deviations are given, and the algorithms are sorted from the best to the worst. Results are averaged on 50 runs of the tested algorithms, and the maximum number of fitness evaluations per run is fixed to 5 × 105 , i.e. 100 changes per run. Results gathered in Table 2.4 for competing algorithms are those given in the references listed in the first column. As it can be seen, MADO is the third best algorithm in this classification. Although MADO is rated behind MMEO [Moser & Chiong, 2010] on scenario 2 of MPB, MMEO produces worse results than MADO on some other instances of MPB, i.e. those


80

70,76

69,73 65,21

70

58,09

60

57,57

Op

50 38,29 40 30 20 10 0 MADO

Brest et al. 2009

Korosec et al. 2009

Yu et al. 2009

Li et al. 2009

França et al. 2009

Figure 2.6: Comparison with competing algorithms on GDBG.

with a higher number of peaks or dimensions. The analysis of obtained results showed that MADO is more robust and can be applied to highly multimodal DOPs and DOPs with high dimensionality [Lepagnot et al., 2009, 2010]. Algorithm Moser and Chiong, 2010 [Moser & Chiong, 2010] Novoa et al., 2009 [Novoa et al., 2009] MADO [Lepagnot et al., 2009, 2010] Moser and Hendtlass, 2007 [Moser & Chiong, 2010; Moser & Hendtlass, 2007] Yang and Li, 2010 [Yang & Li, 2010] Liu et al., 2010 [Liu et al., 2010] Lung and Dumitrescu, 2007 [Lung & Dumitrescu, 2007] Bird and Li, 2007 [Bird & Li, 2007a] Lung and Dumitrescu, 2008 [Lung & Dumitrescu, 2008] Blackwell and Branke, 2006 [Blackwell & Branke, 2006] Mendes and Mohais, 2005 [Mendes & Mohais, 2005] Li et al., 2006 [Li et al., 2006] Blackwell and Branke, 2004 [Blackwell & Branke, 2004] Parrott and Li, 2006 [Parrott & Li, 2006] Du and Li, 2008 [Du & Li, 2008]

Offline error 0.25 ± 0.08 0.40 ± 0.04 0.59 ± 0.10 0.66 ± 0.20 1.06 ± 0.24 1.31 ± 0.06 1.38 ± 0.02 1.50 ± 0.08 1.53 ± 0.01 1.72 ± 0.06 1.75 ± 0.03 1.93 ± 0.06 2.16 ± 0.06 2.51 ± 0.09 4.02 ± 0.56

Table 2.4: Comparison with competing algorithms on MPB using standard settings (see Table 2.2).

The comparison on GDBG is presented on Figure 2.6, where the obtained score using the performance evaluation metric used in this set of benchmarks, celled op, is showed. op measures the performance of an algorithm and provides a score in the range 0 100. One can remark that MADO is ranked first on this benchmark. However, the use of an isotropic neighborhood (a hypersphere) in local search agents does not allow MADO be suitable for ill-conditioned problems. To improve the performance of MADO, we propose to incorporate a technique as it allows to adapt to this type of problems.


2.3.2

Covariance matrix adaptation MADO (CMADO)

2.3.2.1

Overall description

It was shown that the proposed MADO did not allow to solve the ill-conditioned problems. Then, to deal with this kind of problems, a new version of MADO was proposed. The idea in this algorithm is enhance what did not work well in MADO when are face to ill-conditioned problems, is to modify the local search by changing the step-size update. We recall that an ill-conditioned problem is a problem where small perturbation of the current solution causes a great variation pf the objective function. The proposed improved version of MADO is called CMADO, Covariance matrix adaptation for multi-agent Dynamic Optimization. The used method consists in adapting a covariance matrix, denoted C, at each displacement of an agent. Each agent has its own matrix C that is initialized at the identity matrix when creating or repositioning the agent.

2.3.2.2


The comparison between the obtained results via MADO and CMADO on MPB and the Rosenbrock function (well known as ill-conditioned) showed that the performances of CMADO are better than those of MADO. Indeed, the emprove was significant in the case of Rosenbrock, but in the case of MPB there was only a small enhancement. Nevertheless, the addition of this method to adapt MADO to ill-conditioned problems increases significantly its complexity: the calculation of covariance matrices, and the need to compute the eigenvalue and eigenvectors matrices. Algortihm CMADO MADO

Offline error for MPB 0.58 ± 0.09 0.59 ± 0.11

Table 2.5: Comparison between MADO and CMADO on MPB.

2.3.3 2.3.3.1

Prediction based MADO (PMADO) Overall description

In order to improve the performance of the proposed CMADO algorithm, a technique to predict the regions of interest of the objective function, to locate a local search agent, has been developed. This new version of the M ADO algorithm is called P M ADO. It is based on the procedure:

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 44 Iterations between 2 successive changes

0

1

0

1

0

1 Iterations

Figure 2.7: Time-line representing the periods of activation of predictive process performed in parallel to local research agents CMADO. (1) prediction procedure is ON, (0) prediction procedure is OFF.

1. During the same timespan (number of consecutive evaluations of the objective function where the landscape of the objective function does not change), the past of candidate solutions evaluated by the agents, as well as value the objective function of these solutions, are archived. 2. Once this archive, denoted AP , reached a sufficient size, a process is activated in parallel to the research carried out by agents. This process aims to Predict, based on the archive AP , a region of interest of the objective function. In other terms, the prediction procedure starts when the predictor receives the content of AP . Then, AP emptied to archive new candidate solutions. (a) This process starts by learning the objective function means of a neural network using AP as the training base. The estimated objective function is then optimized using a fast local search algorithm (local search used by agents CMADO). (b) Following this optimization of the estimated objective function, an optimum, is found (Op ). It is transmitted to the coordinator (the coordination module of CMADO agents), and then the learning process is disabled until AP again reached a sufficient size. 3. Then, once an agent CMADO should start a new local search, the coordinator provides it Op as the initial solution. Once Op sent to the agent, the coordinator deletes Op and is therefore never used as an initial solution for other agents. 4. Then return to step 1, for a new learning could be done using a new training base. The activations cycle of the predictor is shown in Figure 2.7. One can notice that the estimated function is simpler than the real function. Its role is to help and guide the search to promising areas of the search space.

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 45 2.3.3.2


To allow a good learning shape of the objective function, the number of neurons in the hidden layer must be sufficiently large. In the following tests, we used 30. To test the performance of PMADO by CMADO report, we use the Rastrigin function. The obtained results on this function showed the advantage of using predictor to track the optimas in dynamic optimization. However, we did not explore more this issue because of the high increase of the complexity of the algorithm. So, we will see later that we will expect developing efficient low complex predictors in order tu use them in dynamic optimization.

2.3.4

Multiple Local Searches DO (MLSDO)

The analysis of MADO showed that the approach using multiple local searches is promissing. However, MADO does not allow to obtain a good performance in the benchmarks. The complexity and the sensitivity analysis pointed out that MADO is too complex and consumes lot of evaluations of the objective function. Then, MLSDO is proposed to solve these problems. So, in MLSDO global flowchart is that of MADO. In the following we present only the enhancement on MADO.

2.3.4.1

Overall description

An MLSDO agent explores the search space step-by-step, moving from their current ~c to a better one S ~c0 in its neighborhood, until it reaches a local optimum. solution S These step-by-step displacements are performed according to a step size R, adapted during the local search of the agent. An agent can be created for two reasons: to explore the search space or to track an archived optimum, when a change in the objective function is detected. In the first case, the agent is called an “exploring agent”, and in the second case, it is called a ”tracking agent”. These two kinds of agents only differ in their initialization:

• Exploring agents: At the initialization of an exploring agent, its current solution ~c is provided by the coordinator. The initial value of its step size R is equal to S re , a parameter of MLSDO to be fixed. • Tracking agents: Their current solutions are initialized using the best solutions of the archive containing the found local optima. The initial value of R is a parameter of MLSDO that has to be lower bounded.

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 46 The adaptation of the step size R is based on the cumulative dot product, that makes use of trajectory information gathered along the steps of the agent (using the successive direction vectors of the displacements of the agent). The diversity maintaining strategy The procedure ot maintain the diversity is the same as that used in MADO: If an agent has found a local optimum, then this optimum is transmitted to memory through the coordinator. Afterwards, the coordinator gives the agent its new position, in order to perform a new local search. The relocation of the agents If an agent has terminated its local search (it has found an optimum), or if it has been found too close to other agents, then the coordinator can either destroy the agent, or let the agent start a new local search at a given position. This heuristic generates several random locations uniformly distributed in the search space, selects one of them and returns it. The selection mechanism of this heuristic is as follows. For each generated location, the distance to the closest one in the set of current solutions of the agents, and of those stored in Am and Ai , is calculated. Then, the generated location that has ~new for the agent is the greatest calculated distance is selected. If the new position S in an unexplored zone of the search space (if it is not too close to another agent, to an archived optimum or to an archived initial position), then the agent is relocated at ~new . Otherwise, the search space is considered saturated and the coordinator destroys S the agent. The decision of destroying the agent is also taken if more than na agents exist. It happens if agents are created to track archived optima after the detection of a change in the objective function. The change detection and the tracking of the optima Changes in the objective function are detected by reevaluating the fitness of a randomly chosen agent or archived optimum, and comparing it to its previous value. If these values are different, a change is supposed to have occurred, and the following actions are taken: the fitnesses of all agents and archived optima are reevaluated; then, the procedure of the creation of additional agents is executed. Archive management This procedure is very important in MLDSO and was modfied from that of MADO. Then, the capacity of the archive was reduced, and depends on the ratio between the problem dimension and the exclusion radius.

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 47 The archive is updated following rules that was defined based on experimentations: ~ c , is better than the worst optimum in the 1. If the new optimum, denoted by O archive, or its value is at least equal to the one of this worst optimum, then: (a) If there is one or several optima in the archive that can be replaced, then the ~ c; worst of them is replaced by O ~ c. (b) otherwise, the worst optimum of the archive is replaced by O ~ c , then 2. If there is one or several optima in the archive that are ”too close” to O all these optima close to each other are considered to be dominated by the best of them. Thus, this subset of solutions is replaced by only their best one.

2.3.4.2


The comparison, on MPB, of MLSDO with the other leading optimization algorithms in dynamic environments is summarized in Table 2.4. These competing algorithms were tested by their authors using the most commonly used set of MPB parameters (see Table 2.2). The off-line errors and the standard deviations are given, and the algorithms are sorted in increasing order of offline error. Results of competing algorithms are given in the references listed in the first column. As we can see, MLSDO is the second ranked algorithm in terms of off-line error. The first ranked algorithm on MPB is the one proposed by Moser and Chiong in 2010, called Hybridized EO [Moser & Chiong, 2010]. Indeed, MLSDO and Hybridized-EO use restarted local searches approach in the search space, and store the found local optima in order to track them after a change. However, Hybridized-EO does not perform several local searches in parallel and to generate an initial solution for a local search in a promising area of the search space, it evaluates several candidate solutions and uses the best one as the initial solution of the local search (it samples every dimension of the search space in equal distances). Thus, for each start a local search, it has first to evaluate 10 × d candidate solutions, this can be

a problem when the dimension of the problem arises. In comparison, MLSDO uses only the archived solutions to produce an initial solution for a local search, i.e. it does not evaluate any additional one.

Analysis of the balance between exploring and tracking agents

The evolution of the number of exploring and tracking agents over the time is shown in Figure 2.8, for the first 40000 evaluations of a run of MLSDO on MPB (scenario 2). The evolution of the number of archived local optima is also shown in Figure 2.9, as it is

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 48 Algorithm Moser and Chiong, 2010 [Moser & Chiong, 2010] MLSDO Novoa et al., 2009 [Novoa et al., 2009] Lepagnot et al., 2009 [Lepagnot et al., 2009, 2010] Moser and Hendtlass, 2007 [Moser & Chiong, 2010; Moser & Hendtlass, 2007] Yang and Li, 2010 [Yang & Li, 2010] Liu et al., 2010 [Liu et al., 2010] Lung and Dumitrescu, 2007 [Lung & Dumitrescu, 2007] Bird and Li, 2007 [Bird & Li, 2007a] Lung and Dumitrescu, 2008 [Lung & Dumitrescu, 2008] Blackwell and Branke, 2006 [Blackwell & Branke, 2006] Mendes and Mohais, 2005 [Mendes & Mohais, 2005] Li et al., 2006 [Li et al., 2006] Blackwell and Branke, 2004 [Blackwell & Branke, 2004] Parrott and Li, 2006 [Parrott & Li, 2006] Du and Li, 2008 [Du & Li, 2008]

Off-line error 0.25 ± 0.08 0.36 ± 0.08 0.40 ± 0.04 0.59 ± 0.10 0.66 ± 0.20 1.06 ± 0.24 1.31 ± 0.06 1.38 ± 0.02 1.50 ± 0.08 1.53 ± 0.01 1.72 ± 0.06 1.75 ± 0.03 1.93 ± 0.06 2.16 ± 0.06 2.51 ± 0.09 4.02 ± 0.56

Table 2.6: Comparison with competing algorithms on MPB using standard settings (see Table 2.2).

Agents 6 4 2 0

0

1E4

Exploring agents

2E4

t

3E4

4E4

Tracking agents

Figure 2.8: Evolution of the number of exploring and tracking agents during the first 40000 evaluations of a run of MPB.

Solutions 6 4 2 0

0

1E4

2E4

3E4

4E4

t Figure 2.9: Evolution of the number of solutions stored in the archive of the found local optima, during the first 40000 evaluations of a run of MPB.

linked to the number of tracking agents created after a change, i.e. the highest number of tracking agents created after a change is equal to the number of archived optima. As we can see, during the first time span, five local optima are found among ten. Then, the remaining ones are progressively detected. The tracking agents require few evaluations to locate the new position of the detected local optima. Hence, most evaluations

Chapter 2. Design of metaheuristics for Dynamic continuous optimization problems 49 of a time span of MPB are used to explore the search space, in order to find the local optima that remain to be detected.

Performances on GDBDG The comparison, on GDBG, of MLSDO with the other leading optimization algorithms in dynamic environments is summarized in Figure 2.10. The algorithms are ranked according to their overall performance. As we can see, MLSDO is the first ranked algorithm on this benchmark. 90

81,22

80

70,76

69,73 65,21

70

58,09

60

57,57

Op

50 38,29 40 30 20 10 0 MLSDO

MADO

Brest et al. 2009

Korosec et al. 2009

Yu et al. 2009

Li et al. 2009

França et al. 2009

Figure 2.10: Comparison with competing algorithms on GDBG.

2.4

Conclusion

We have described in this chapter the basis our work in DOPs. New dynamic optimization algorithms were proposed, these algorithms are based on coordinated multi-agent approach. Then, to explore the search space, a population of coordinated local search agents is used. The best local optima found by these agents are stored in memory in order to accelerate the convergence of the algorithm. The diversity is maintained thanks to the Coordination. It allows agents to explore the best the search space and to ensure the diversity. Detecting a change in the objective function is done via a revaluation of a solution. When a change is detected, the current solutions of the agents and the content of the archive are revalued. Then, agents are created and positioned on the optima of the archive. Once these new agents created, the archive is emptied. This allows to track, over the variations of the objective function, the local optima found before. Comparisons with competing algorithms on the well known Moving Peaks Benchmark, as well as on the Generalized Dynamic Benchmark Generator (GDBG), show the efficiency of the proposed algorithms, especially MLSDO.


2.5


• E. Alba, A. Nakib, P. Siarry, ”Metaheuristics for Dynamic optimization”, Springer Series: Studies in Computational Intelligence, Vol. 433, 2013.

• J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”A multiple local search algorithm for continuous dynamic optimization”, Journal of Heuristics, vol. 19, no. 1, pp. 35-76, 2013. • J. Lepagnot, A. Nakib, H. Oulhadj and Patrick Siarry, ”A new multiagent algorithm for dynamic continuous optimization”, International Journal of Application Metaheuristic Computing (IJAMC), Vol. 1, no. 1, pp. 16-38, 2010. • J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”A Dynamic multi-agent algorithm

applied to challenging benchmark problems”, in Proc. Of the IEEE Congress on Evolutionary Computation, CEC 2012, as a part of the 2012 IEEE World Congress on Computational Intelligence, WCCI 2012, Brisbane, Australia, June 10-15, 2012, pp. 2508-2515.

• J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”Performance analysis of MLSDO dynamic optimization algorithm”, in Proc. Of the Int. Conf. on Metaheuristics

and Nature Inspired Computing, META 2012, Port El-Kantaoui- Sousse, Tunisia, October26-31, 2012 • J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”A Multi-Agent based Algorithm

for Continuous Dynamic Optimization”, In EU/MEeting 2010 workshop, Lorient, France, June 2010.

• J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”Performance analysis of MADO

dynamic optimization algorithm”, in Proc. Of the 9th IEEE Int. Conf. on Intelligent Systems Design and Applications, ISDA’09, Pisa, Italy, Nov. 30 - Dec. 2, 2009, pp. 37-42.

• J. Lepagnot, A. Nakib, H. Oulhadj and P. Siarry, ”Réduction du nombre de

paramètres de la métaheuristique d’optimisation dynamique MADO”, Congrès de la Société Fran¸caise de Recherche Opérationnelle et d’Aide à la Décision (ROADEF’10), 24-26 Feb.’10, Toulouse, France.

Chapter 3

Dynamic metaheuristics for brain cine-MRI 3.1

Introduction

Recently, a new technique for obtaining brain images of cine-MR (Magnetic Resonance) type has been developed by Hodel et al. [Hodel et al., 2009]. The principle of this technique is to synchronize the MRI signal with the ECG (Electrocardiographic) signal. The MRI signal provides three dimensional images and cuts of high anatomical precision, and the ECG signal is obtained from the heart activity. An image of brain cine-MRI is therefore built by making the average of the MRI signals acquired during the R-R period of the ECG signal. This technique allows to have a good visualization of the movements of the walls of the third ventricle during the cardiac cycle. For more details about this method see. In this work, we were interested in the automation of the assessment of the movements of the walls, which allows a better understanding of physiological brain functioning and the provision of aid to diagnosis and therapeutic decision. Here, we are not be interested in the cerebrospinal fluid (CSF), the reader can have more details about analysis of the motion using CSF in [Kurtcuoglu et al., 2007b,a]. Several methods for the movement quantification have been proposed in the literature for myocardium images. This image processing application requires the partitioning of the image into homogeneous regions. Then, the fundamental process used is called image segmentation, that plays an important role in entire image analysis system [Mayer & Greenspan, 2009]. It is generally performed before the analysis and the decisionmaking process in many image analyses, such as the quantification of tissue volumes, diagnosis, and localization of disease, the study of anatomical structures, the matching and motion tracking. Since the segmentation is considered by many authors to be an 51

Chapter 3. Dynamic metaheuristics for brain cine-MRI

52

essential component of any image analysis system, this problem has received a great deal of attention; thus any attempt to completely survey the literature would be too space consuming. However, surveys of most segmentation methods may be found in [Paragios & Osher, 2003; He, 2008]; another review of thresholding methods is in [Sezgin & Sankur, 2004]. In [He, 2008; Budoff et al., 2009; Sundar et al., 2009] the authors proposed level set based methods to assess cardiac movements. These methods cannot be applied directly to our brain sequences. Indeed, firstly, the amplitude of movements in the heart is much greater than that of the walls of the third ventricle. Secondly, due to the presence of cerebrospinal fluid, one cannot properly segment the entire ventricle and quantify its movements in the sequence, because the stopping criteria are adapted to our images and the algorithmic complexity is high. The goal in this work was to develop a low computation complexity method to assess and quantify these movements to be used as a clinical routine. Then, we proposed a framework that consists of two phases: 1. the extraction of contour of the walls of the 3rd ventricle is performed. 2. the contours’ registration is done to achieve the deformation model. In the segmentation step (phase 1), images are segmented by a fractional differentiation based method in [Nakib et al., 2009a,b]. It is followed by a technique aimed at optimizing the differentiation order. This phase allows to enhance the contrast and to extract the contours of a selected region of interest (ROI) at different times of the cardiac cycle. In the second step, the information provided by these contours is combined through their mapping. This registration procedure allows us to track the movements of each point of the contour of the ROI over time, and to obtain a better mathematical modeling of the movement. The parameters of this model are calculated over all the sequence. As we perform optimization several times for every couple of images, then the landscape of the function to be optimized (objective function) changes. Thus, we talk about dynamic optimization.

3.2

Problem formulation

The goal behind this work is to build an atlas of the movements of the healthy ventricles in the context of the hydrocephalus pathology. Then, in this paper, our objective is to segment the walls of the third ventricle, and quantify their movements. We have tested several image segmentation methods based on edge detection approach: the method of Canny, derivative methods and more robust methods such as the Level-set approach [He,


53

2008]. All these methods give similar unconvincing results because they do not reproduce precisely the contours of the third ventricle on the images of the used sequences. This is due to the fact that they are very noisy, due to the presence of CSF, and variation of their contrasts on the same subject (third ventricle). For our application, the segmentation of the entire walls is not possible (differences between the pathologic and sane cases) and is not necessary. Consequently, we decided to work only on a region of interest (ROI). In figure 3.1(a), we present an example of an ROI: Lamina Terminalis (LT). It is the first image of a brain cine-MRI sequence in a pathologic case (hydrocephalus). The sane case is presented in figure 3.1(b). Hydrocephalus is usually due to the blockage of CSF outflow in the ventricles. Patients with hydrocephalus have abnormal accumulation of CSF in the ventricles. This may be source of the increase of the intra-cranial pressure inside the skull and progressive enlargement of the head, convulsion, and mental disability. The goal of developing this framework is to contribute to the study of the diagnosis of endoscopic third ventriculostomy (ETV) patency. To validate the proposed method, sixteen age-matched healthy volunteers were explored with the same MR protocol (12 women, 4 men; average age 38.5 years, interquartile range: 25.5–54). This study was approved by the local ethics committee; written informed consent was obtained for all the patients.

(a)

(b)

Figure 3.1: Illustration of the first image of the original sequence. Pathologic case (Hydrocephalus), (b) sane case.

3.3

Data acquisition

Data were obtained using 1.5-T MR (Siemens, Avanto, Erlangen, Germany). The protocol included a retrospectively gated cine true/ Fast Imaging with Steady-state Precession Magnetic Resonance (cine True FISP MR) sequence: mid-sagittal plane was defined on a transverse slice from the center of the Lamina Terminalis to the cerebral aqueduct [Hodel


54

et al., 2009]. This acquisition technique provides only a sequence of 2D image, then a 3D image segmentation technique cannot be applied.

3.4

Proposed framework

In this section we present the proposed framework that consists in two phases: the first phase is the extraction of contours of the image ROI with a segmentation technique, based on two dimensional digital-fractional integration (2D-DFI). In the second phase, we combine the information provided by the contours of the image sequence through a registration. The steps of our framework are illustrated in figure. 3.2.

Figure 3.2: Overall scheme of the cine-MRI framework analysis.

3.4.1 3.4.1.1

Segmentation problem segmentation based on fractional differentiation

In order to segment every image of the sequence, a segmentation method based on two dimensional digital fractional integration (2D-DFI), where the fractional order is negative (in the interval (−1, 0]). The fractional integration (also called fractional differentiation (FD) with negative order) is based on the works of Leibniz and Hospital in 1965. The applications of this method are numerous, it is used in automatics [Benchellal et al., 2006; Mansouri & Bettayeb, 2008], in signal processing [Nakib & Ferdi, 2002] and in image processing [Nakib et al., 2009a,b; Mathieu et al., 2003b]. The FD of Riemann-Liouville is defined as follows:


D

−α

1 f (x) = Γ(α)

Z

c

55

x

(x − ξ)α−1f (ξ) dx.

(3.1)

where f (x) is a real and causal function, x > 0, α the order of the FD, c the interval of the integral and the function of Euler-Lagrange. In the case of integration, the order α is negative and in the discrete domain, the approximation of the DFI is given by:

g α (x) = D−α f (x) ≈

M 1 X ((βk )(α)f (x − kh)) hα

(3.2)

k=0

where h is the sampling step, M the number of samples, x = M × h and βk (α) are defined by:

β0 (n) = 1, βk+1 (α) =

(k + 1) − α − 1 βk (α), k = 0, 1, 2, M − 1 (k + 1)

(3.3)

The equation 3.2 is equivalent to the Riemann-Liouville equation when h tends to zero. The 2D-DFI is given, for a real and bounded function f (x, y) by:

D

−α

∂ 1 ∂ f (x, y) = ( )α ( )α f (x, y) ≈ 2α ∂x ∂y h

b −M/2c

X

b −N/2c

X

k=b−M/2c l=b−N/2c

(p(k, l)f (x − hk, y − lh)) (3.4)

M and N are the number of elements of f taken into account for calculating the differential image, M ×N represents the size of the mask,p(k, l) = βk (α)×βl (α) are the elements

of the matrix ,pαM,N (p(k, l)) calculated from the equation (3), which corresponds to the horizontal and vertical components, respectively. bxc denotes the integer part of x. The optimal segmentation of an image corresponds to finding the optimal 2D-DFI order. In order to find the best value, a criterion that characterizes the best segmentation was used, then the best value is that providing a segmentation that optimizes the defined criterion. Indeed, a good result means that all regions (connected components) are homogeneous and their number will not be under 2 and greater than 3. Less than 2 (equal to 1), it means that there is only one region; in this case, the segmentation result is bad. Greater than 3, it means that there are too many regions and the image is over segmented, thus the segmentation result is bad. In order to obtain the optimal order for the segmentation process, the well known uniformity criterion is used [Cocosco & Evans, 2003].


56

Optimal DFD order Inputs: α = −0.9

for i = −0.9 to −0.1 do Apply 2D-DFI at the order αto the original Image (I) Compute t that corresponds to the order α Calculate the segmentation quality measure criterion corresponding to t

end Find topt optimizes the segmentation quality measure criterion. Print results: optimal order αopt , optimal threshold topt and segmented image. Figure 3.3: Search for the optimal αopt .

3.4.1.2

segmentation based on dynamic optimization

Before using this criterion we must fit the histogram of the image to be segmented to a sum of Gaussian probability density functions (pdf’s). This procedure is named Gaussian curve fitting, more details about it are given below. The pdf model must be fitted to the image histogram, typically by using the maximum likelihood or meansquared error approach, in order to find the optimal threshold(s). For the multimodal histogram h(i) of an image, where i is the gray level, we fit h(i) to a sum of d probability density functions. The case where the Gaussian pdf’s are used is defined by: "

d X

(x − µi )2 p (x) = Pi exp − σi2 i=1

#

(3.5)

where Pi is the amplitude of Gaussian pdf on µi , µi is the mean and σi2 is the variance of mode i, and d is the number of Gaussian used to approximate the original histogram and corresponds to the number of segmentation classes. Given an image histogram h(j) (observed probability of gray level j), it can be defined as follows: g (j) h(j) = PL−1 i=0 g (i)

(3.6)

where g(j) denotes the occurrence of gray-level j over a given image ranges [0, L − 1]. Our goal is to find a vector of parameters, Θ, that minimizes the fitting error J, given by the following expression : Minimize J =

X i

|h(i) − p(Θ, xi )|2

(3.7)

where i ranges over the bins in the measured histogram. Here, J is the objective function to be minimized with respect to Θ, a set of parameters defining the Gaussian pdf’s

Chapter 3. Dynamic metaheuristics for brain cine-MRI and the probabilities, given by Θ = {Pi , µi , σi ;

57

i = 1, 2, · · · , d}. After fitting the

multimodal histogram, the optimal threshold could be determined by minimizing the overall probability of error, for two adjacent Gaussian pdf’s, given by : e(Ti ) = Pi

Z

Ti

pi (x) dx + Pi+1

−∞

Z

∞

pi+1 (x) dx

(3.8)

Ti +1

with respect to the threshold Ti , where pi (x) is the ith pdf and i = 1, . . . , d − 1. Then the overall probability to minimize is:

E(T ) =

d−1 X

e(Ti )

(3.9)

i=1

where T is the vector of thresholds: 0 < T1 < T2 < ... < T(d−1) < L − 1. In our case L

is equal to 256.

The criterion in (3.9) has to be minimized for each image, as we are in the case of a sequence, then the fitting criterion becomes:

Minimize J(t) =

X i

|h(i, t) − p(Θ(t), xi )|2

(3.10)

where t is the number of the current image in the cine MRI sequence, Θ(t) is same as Θ defined before but here is dependent on the image. h(i, t) is the histogram of the image number t.

3.4.2

Geometric matching of the contours

The obtained contours in the segmentation phase will be used to assess the movement of the contours of the ROI over time. This step requires matching these contours. However, several false matches appear. To eliminate this problem, we have separated the obtained contours after segmentation and indexing and we have only kept the contour corresponding to the third ventricle (in the case of the Lamina Terminalis, this is in the right side). To evaluate the ventricle deformation, a contour matching operation is required after the segmentation phase in order to track the position of points belonging to the contours of the region of interest (ROI) over time. This operation is carried out in two steps: first, a rigid registration, called alignment, takes into account the displacement of the global membrane. Then, a morphing process performs accurate elastic matching of the successively aligned contours.

Chapter 3. Dynamic metaheuristics for brain cine-MRI 3.4.2.1

58

Contours’ Alignment

This step consists in looking, for each point in the curve of a reference image, at the nearest point in the curve in the destination image, based on a predefined minimum distance. The different steps of that phase matching are summarized as follows: For each point of the source curve: 1. Calculate the distance between this point and all points of the destination curve. 2. Match this point with the nearest point of the destination curve. In this step, the goal is to associate each point of the initial contour to one point of the second contour. We assume that each point of the first contour can be associated with at least one point of the second contour. The alignment procedure realizes a registration between these two curves through a geometrical transformation controlled by a dissimilarity criterion based on 3 parameters. A non linear model can also be considered as a deformation model. In the matching phase, a point is mapped to the nearest point, but the direct application of the matching method to the ROI contour without indexing produces false matches. Other false matches are due to the presence of several equidistant points from the point to match. Zhang [Zhang, 1994] proposed a method using an adaptive statistical threshold, which is based on the statistical characteristics of the distances between matched points, such as the average and the standard deviation. We used this method to eliminate false matches and to assess the quality of the registration obtained. False matches of this type appear only if an alignment takes the first curve as a reference, i.e. by matching the contour of each sequence with the first points of the contour of the first image.

3.4.2.2

The deformation model

In order to approximate the deformation model, we use a registration technique. The idea of the registration is to combine information from two (or more) images of the same scene, but obtained at different moments, with different views or different acquisition methods. Then, the aim of a registration system is to determine the mapping information (positions, gray-scale, structures, etc.) representing a physical reality on these different images. The goal of image registration is to find the best transformation T 0 among all the transformations T applied to an image I that looks at best like the image J. It is quite difficult to review all the image registration methods that have been proposed in the


59

literature, a survey of many methods is presented in [Zitová & Flusser, 2003]. According to the used primitives (attributes), in our work we considered the geometric approach of image registration. In this approach an extract of the geometric primitives (points, curves or surfaces) is performed, and then a geometric similarity criterion is used to find the best correspondence. In the case where the primitives are points, the Euclidean distance is usually the most used [Noblet et al., 2006]. In the case of surfaces and curves, the most widely used algorithm is the Iterative Closest Point (ICP) algorithm [Zhang, 1994; Besle & McKay, 1992; Malandin et al., 2004; Delhay et al., 2004]. Other criteria are based on geometric calculations cards, Chamfer distances or Hausdorff distance [Noblet et al., 2006; Baty et al., 2006]. Geometric approaches have the advantage of holding high-level information, but remain vague regarding the extraction of primitives. The optimization is a very important step in image registration. It aims at determining an optimum processing according to a similarity criterion. The optimization process is iterative. It must find at each stage the parameters T of the processing that ensure the best match between the two images until the convergence to the optimal solution. Broadly speaking, the optimization problem is formulated by:

Topt = argτ eΩ maxζ(J, T (I))

(3.11)

where: I is the original image, J the image to register, Ω:search space for possible transformations, ζ : similarity criterion chosen, Topt : optimum processing. Among the optimization methods, we find numerical methods (without use of the gradient), such as Simplex and Powell [Baty et al., 2006] methods, and gradient based methods, such as gradient descent, conjugate gradient, the method of Newton and Levenberg-Marquardt [Noblet et al., 2006; Baty et al., 2006]. In our works, we proposed to use dynamic optimization metaheuristics to solve this optimization problem. Recently, static evolutionary algorithms were also used to find the optimal registration model parameters’ [Santamariaa et al., 2011]. In order to assess the deformation model, the contours of the indexed images of the sequence were matched by taking at each time two contours of two successive images. The similarity criterion that measures the distance between two successive contours must be minimized.

Considering the distortion models that exist, we assume, for instance, that the movements of the third ventricle are governed by an affine transformation. This model is


60

characterized by a rotation θ , two translations (tx ,ty ) and two scaling factors (s1 ,s2 ) according to x and y:       x2 S1 . cos θ −S2 .sinθ tx x2        y2  =  S1 . sin θ S2 . cos θ ty  .  y2        1 0 0 1 1

(3.12)

P1 (x1 ,y1 ) is a point of the reference primitive and P2 (x2 ,y2 )is the point obtained with the geometric model.

The parameters of this transformation are defined by minimizing the squared error among all points of the curve and those obtained with the model given by this function: N1 X 0 SE(X) = ( ([C2 (i) − T x (C1 (i))]2 )

(3.13)

i=1

N1 is the cardinal of all the points of the contour C1 and x ≡ (s1 , s2 , θ , tx , ty )is

the vector of parameters. Several authors such as [Zhang, 1994] use an iterative local search algorithm to solve this problem. To avoid local minima, the optimization criteria are modified by adding weight terms inversely proportional to the distances between matched points. This makes the optimization algorithm more complex.

3.4.3

Cine-MRI registration as a DOP

The registration of a cine-MRI sequence can be seen as a dynamic optimization problem. Then, the dynamic objective function optimized by a dynamic optimization algorithm changes according to the following rules:

• The criterion in ( 3.12) has to be minimized for each couple of contours, as we are in the case of a sequence, then the optimization criterion becomes:

T 0 Lt 0 X Ct (j) − TΦ(t) (Ct (j)) Ct (j) − TΦ(t) (Ct (j)) M SE(Φ(t), t) = Lt

(3.14)

j=1

where t is the index of the contours on which the transformation TΦ(t) is applied, also equal to the index of the current couple of contours in the sequence. Φ(t), Ct0 (j), Ct (j) and Lt are the same as Φ, C10 , C1 and L1 defined before, respectively, but here are dependent on the couple of contours.


61

• Then, the dynamic optimization problem is defined by: Min M SE(Φ(t), t)

(3.15)

• If the current best solution (transformation) found for the couple t cannot be

improved anymore (according to a stagnation criterion), the next couple (t + 1) is treated.

• The stagnation criterion of the registration of a couple of successive contours is satisfied if no significant improvement in the current best solution is observed.

• Thus, the end of the registration of a couple of contours and the beginning of the registration of the next one constitute a change in the objective function.

This formulation (introduction of the time variable t to get an objective function that changes over the time) allows the use of dynamic optimization algorithms to solve this problem, rather than having to restart a static algorithm to register a sequence of images. Then, information acquired on the objective function during the registration of several couples of contours, in a sequence, can be used by the dynamic optimization algorithm to accelerate the registration of the next couples (the correlations between the images of the sequence can be taken into account).

3.5

Results and Discussions

In this section, we illustrate the results of the extraction of the ROI’s contours (for the Lamina Terminalis), followed by the results for the registration and the quantification of the movement. Finally, we will present the results obtained for the quantification of other ROIs of the sequences. The results of the proposed quantification method have been clinically validated by an expert and compared to those in [Hodel et al., 2009] obtained using a manual segmentation. The first phase of the framework consists in the extraction of the ROI. It is carried out in two steps: the segmentation and the indexation. We proposed two algorithms to segment the different images of the sequence. Then, only perimeters of the segmentation results are considered. In order to illustrate this phase, we consider the original sequence in figure 3.4. The obtained segmentation results are presented in figure 3.5.


62

Figure 3.4: Original sequence.

From the segmentation results and after the matching process, the representation of movement of the ROI can be shown. In figure 10, we present the case of the example in figure 3.6. The registration process allows to have all parameters of the deformation for all the different couples of images (i.e. deformation of the contour in image1 towards that of image 2, etc.).

To quantify the movement of the ROI, we have aligned all the contours of the sequence with the contour of the first image. Figure 3.7 shows the amplitudes of displacements of each point over time in the case of the Lamina Terminalis (figure 3.4). The movements that we are interested in this work are those that have maximum amplitude. In the sequence used, the maximum movement is of 2.57 mm. This has been clinically validated by an expert in this study and is in the same range as the results published in [Hodel et al., 2009]. The horizontal and vertical displacements are given in Table 3.1. Further tests were made on other image sequences of patients and sequences of witnesses with no abnormalities in the third ventricle. We found that our method depends on the


63

Figure 3.5: Illustration of the segmentation result for the sequence in 3.4.



spatial resolution of images used. When the resolution is low, the quantification result is better. After several tests, we noticed that, for a good quantification result of the movement of the walls of the third ventricle, it is necessary that the spatial resolution


64

Table 3.1: Obtained set of parameters of the deformation model.

Images imag01-02 imag02-03 imag03-04 imag04-05 imag05-06 imag06-07 imag07-08 imag08-09 imag09-10 imag10-11 imag11-12 imag12-13 imag13-14 imag14-15 imag15-16 imag16-17 imag17-18 imag18-19 imag19-20

s 0.9917 0.9999 1.0044 0.9985 0.9977 1.0018 0.9984 0.9985 0.9967 0.9959 1.0018 1.0004 1.0087 0.9843 1.0033 1.0086 1.0047 1.0028 1.0073

θ 0.0023 0.0015 0.0042 0.0025 0.0028 0.0117 0.0076 -0.0016 -0.0011 -0.0015 -0.0004 -0.0010 0.0085 0.0070 -0.0033 -0.0086 0.0047 0.0028 -0.0073

tx 0.615 0.2151 0.2884 -0.2611 0.0736 0.5290 0.3330 0.1376 0.1136 0.2835 -0.2613 0.1992 0.1123 0.4257 -0.5332 0.371 0.1351 -0.1303 0.2596

ty 1.4813 -0.639 0.4406 0.4406 0.1055 0.1088 -0.1900 -0.2963 -0.1026 -0.3654 -0.5670 -0.4103 -0.6622 0.7800 -0.3712 0.1734 -0.1494 0.9025 0.9034

of a pixel is smaller than or equal to 0.6 mm. More details about these contributions can found in the publications listed in the section 3.7.

3.6

Conclusion

In this work, we were interested in quantifying the movements of the third cerebral ventricle in a cine-MRI sequence. To solve the problem of quantifying the movements of the third cerebral ventricle, we proposed a method for quantification of movement based on fractional integration and dynamic metaheuristics. A new segmentation methods based on the fractional integration and a new formulation of the segmentation problem as a dynamic optimization problem were proposed to segment ”quickly” all sequences. In this step, the thresholding method provided good results for detecting the contours of the images. For the registration step, a covariance matrix adaptation evolution strategy, called D-CMAES and MLSDO were proposed to allow, first, to build a distortion model representing the distortion of the ROI and, secondly, to quantify its movements, without restarting the optimization process from the beginning. The obtained results were considered good and represent the movement of the ROI rather well. In order to


65

take into account the third dimension, the design of a new acquisition technique is under progress to enhace the quality the cine-MRI.

3.7


• A. Nakib, P. Siarry, P. Decq, ”A framework for analysis of brain cine MR sequences”, Computerized Medical Imaging and Graphics, Elsevier, vol. 36, no. 2, pp. 152-168, 2012. • A. Nakib, Y. Schluze, E. Petit, ”Image thresholding framework Based on Two-

Dimensional Digital Fractional Integration and Legendre moments’”, IET image processing, vol. 6, no. 6, pp. 717-727, 2012.

• J. Lepagnot, A. Nakib, H. Oulhadj, and P. Siarry, ”Elastic registration of brain cine-MRI sequences using MLSDO dynamic optimization algorithm”, Book title:

”Metaheuristics for Dynamic optimization”, Guests editors: E. Alba, A. Nakib and P. Siarry. Springer 2013. • J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”Brain cine-MRI Sequences Registra-

tion using B-spline Free-Form Deformations and MLSDO Dynamic Optimization Algorithm”, in 6th International Conference on Learning and Intelligent Optimization Conference, LION 6, Paris, France, Jan. 16-20, 2012, pp.443-448.

• A. Nakib, Y. Scheluze, E. Petit, ”Optimal fractional filter for image segmentation”, in IS&T- SPIE 24th Annual Symposium on Electronic Imaging, San Francisco, CA, USA, Jan. 22-26, 2012. • J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”Brain cine MRI segmentation based on a multiagent algorithm for dynamic continuous optimization”, in Proc. of the

IEEE Congress on Evolutionary Computation, CEC’2011, as a part of the 2011 IEEE World Congress on Computational Intelligence, WCCI 2011, New Orleans, LA, USA, June 2011, vol. 1, pp. 1695-1702. • J. Lepagnot, A. Nakib, H. Oulhadj, P. Siarry, ”Brain cine-MRI registration using MLSDO dynamic optimization algorithm”, in Proc. of the 9th Metaheuristics International Conference, MIC’2011, Udine, Italy, July 2011, vol. 1, pp. 241-249. • A.Nakib,F.Aiboud,J.Hodel,P.Siarry,P.Decq,”Thirdbrainventricledeformationanalysis using fractional differentiation and evolution strategy in brain cine-MRI”, in Proc.

of the SPIE Medical Imaging conference, San Diego, USA, Feb. 13-18, 2010, vol. 7623, pp. 76232I-1 to 76232I-10.

Chapter 4

Particle tracking based on fractional integration 4.1

Introduction

As seen in previously, in the literature dynamic metaheuristics, most of the authors propose techniques that maintain a suitable degree of population diversity (set of candidate solutions well distributed in the search space) to guarantee an efficient exploration, and to avoid the algorithm to be blocked on the current optimum. A premature convergence would make the algorithm miss important changes in the environment, while a diversified population can adapt to changes more easily. Recently, several works regarding the anticipation in changing environments using metaheuristics have been proposed. The main goal of these approaches was to estimate the likelihood of particular future situations and to decide what the algorithm should do in the present situation. Since information about the future typically was not available, it was attained through learning from past situations. The classical scheme of a search for a moving optimum can be divided into two phases: 1. The algorithm has no previous information about the optimum, and should explore the search space in order to find it. This phase can be considered a standard search problem in a time-changing environment. The result of this phase is the optimum (or near-optimum). 2. Tracking phase: the algorithm must be able to follow the moving optimum when it moves. In this phase, under the assumption that the underlying movement law is nonrandom, the information about the current optimum can be used in order 66

Chapter 4. Particle tracking based on fractional integration Predicted position

Start

Predictor

67

Real past positions

Dynamic Metaheuris3c

End

Figure 4.1: Illustration of the application of the principle of coupling a dynamic metaheuristic (metaheuristic for DOPs) and a predictor.

to find the new one, which will be in a neighborhood whose size depends on the velocity of change. The transition between search and tracking is not well defined and it is the issue that we are dealing with in this work. The information about past optima can be used to refine the search. During the tracking phase the movement law can be estimated by looking at the sequence of the optimum solutions found. Such an estimation can be used to predict where the optimum will be, and help the tracking process, directing the search toward the region where most likely the new optimum will be. Note that in principle, if the current optimum and the motion estimation were perfectly determined, one could simply compute the new optimum. In order to help the algorithm to track the moving optima, the common idea underlying all techniques is to generate or privilege positions that are close to the estimated position at time t. Indeed, new possible positions (candidate solutions) are generated starting from those were evaluated according to the information available at time t − 1, but they will be evaluated with the new fitness at time t, when the optimum has moved.

One can notice, that the add of predictor to a metaheuristic allows to enhance its performances, as showed in [Branke & Mattfeld, 2000; Stroud, 2001; Schmidt et al., 2005; Michalewicz et al., 2007; Bosman & La Poutre, 2007; Simões & Costa, 2009a,b; Lepagnot, 2011]. However, the complexity of the algorithm will also increase drastically and the user must fit more parameters moreover that of the metaheuristic: new archive added, mechanism to manage it, frequency of the prediction update, the predictor order. In this work, we propose a new technique to enhance the performance of the classical target tracking techniques that are low complex. We particularly take advantage of properties of the digital fractional integration (DFI) to enhance the performance of the classical techniques such as linear predictors and Kalman filters. Indeed, these methods are not efficient when environment is noisy and, when multiple targets need to be tracked. Those cases are usual in dynamic optimization problems.

Chapter 4. Particle tracking based on fractional integration

68

Different techniques to bias the search for a moving optimum, making use of information provided by some external mechanism. This work was motivated by the need to improve tracking capabilities in a metaheuristics-based [Dakkak et al., 2011] indoor location algorithm used to locate and track a moving terminal (target, local optima). However, the discussion is not limited to target tracking, as general principles can be formulated that can be applicable to any moving-optimum metaheuristic (dynamic metaheuristics). Besides, predicting or filtering in this context is equivalent to optimal state estimation in dynamic environment. The aim is to use indirect noisy measurements to estimate hidden signal consisting of sequence of states. The prediction is optimal in the sense that it minimizes the expected estimation error. This is done on-line, which means that updated state estimate is available as soon as new measurements (real positions) are obtained. In Bayesian sense filtering, this is equivalent to a recursive estimation of posterior distribution of states. Clearly, the tracking problem is a form of filtering where the hidden signal consists of states of moving or stationary targets (position, velocity, etc.). Therefore, the purpose of tracking is to estimate the position of these targets as accurately as possible using only indirect, and noisy measurements. Moreover to these kinematic measurements, there are also attributes that are most useful in the process of target measurement association by providing information on types of targets in area. In order to achieve a high-level of accuracy, the prediction algorithm should be able to track all existing targets, and clutter insensitive such that there are no extra targets in addition to existing ones. Kalman Filters and Particle Filters are commonly used in the domain of target tracking [Guvence et al., 2003; Kumar et al., 2006; Gilks et al., 1996; Doucet et al., 2001, 2000] as real-time linear predictors. The particle filters can be seen as the best alternative rather than the classical approach using the model-based Kalman filter techniques [Kailath et al., 2000; Gustafsson, 2000]. Linear predictor and Kalman filtering methods underperform when multiple hypotheses need to be tracked. In this case, the prediction process becomes more approximate, and the achieved accuracy may be insufficient for critical applications. In [Kushki et al., 2010], authors introduce the Nonparametric Information (NI) filter, a novel state-space Bayesian filter for target tracking. The filter provides a recursive position estimate over time by fusing information from two sources: measurements and a motion model. Authors propose an intelligent tracking system that mimics the characteristics of a cognitive dynamic system. Cognitive dynamic systems learn rules of behavior through interactions with the environment to deal with uncertainties in a robust and reliable manner [Haykin, 2007].


69

In this work, we propose to enhance the performance of the linear prediction filter and Kalman filters to allow their use. In order to have an idea about the degree of predictability of a path, it is common to use the autocorrelation function as metric to measure the redundancy and the relationship between samples of a path. The autocorrelation refers to the correlation of a path with its own past and future values. Moreover, autocorrelated path are probabilistically predictable because future values depend on current and past values. Then, a long memory path is characterized by a correlation function which decreases slowly when the lag increases, and vice-versa. From a geometric point of view, the regularity can be also used to characterize the predictability of a path. So, it is easy to see that a regular path that present a smooth curvature can be predicted with a good accuracy. Consequently, to enhance the performance of a predictor, one has just to increase its autocorrelation and the regularity of the path at its input. To do so, the use the Digital Fractional Integration based filter was proposed. The basic idea is to use the Digital Fractional Integration based filter to enhance the performance of a predictor, and to increase its autocorrelation. A remarkable merit of fractional differentiation operators is that they may still be applied to functions that are not differentiable in the classical sense. Unlike the integer order derivative, the fractional order derivative at point x is not determined by an arbitrary small neighborhood of x. In other words, the fractional derivative is not a local property of the function. There are several well-known approaches to unification of differentiation and integration notions, and their extension to non-integer orders [Prodlubny, 2002]. A general survey on the different approaches is given in [Miller & Ross, 1993; Samko et al., 1993; Oustaloup et al., 2000].

The theory of fractional integrals was primarily developed as a theoretical field of mathematics. More recently, fractional integration has found applications in various areas: in control theory it is used for path planning [Oustaloup & Linares, 1996; Oustaloup et al., 2006, 1999, 1995; Banos et al., 2011; Matli et al., 2011]; it is also used to solve the inverse heat conduction problem [Battaglia et al., 2001]; other applications are reported for instance in modeling using fractal networks [Ramus-Serment et al., 2002], in image processing for edge detection [Mathieu et al., 2003a], in biomedical signal processing [Ferdi, 2011; Sommacal et al., 2007], in thermal systems [Victor et al., 2010] and in biological tissues [Ionescu et al., 2011; Sommacal et al., 2007].


4.2

Fractional integration

4.2.1

Definition of fractional calculus

70

We consider a fractional integral operator that has the function y(t) as output and x(t) as the input function. Then, the equation defining this operator is : y(t) = J α x(t)

(4.1)

where, J is the fractional integration operator and α is the fractional integration order. The word fractional is used for historical reasons and means any real, imaginary, or complex number. In the case of fractional integration the real part of α is negative. We denote the αth integral operator by J α . In this paper, we recall the definitions introduced in [Oldham & Spanier, 1974] to make this paper self-consistent. Then, the operator J α proposed by Riemann-Liouville for fractional differentiation is defined by the formula: 1 J x(t) = Γ(α) α

Zt c

(t − ξ)α−1 x(ξ) dξ

(4.2)

where: x(t) is a real function, t > 0, Re(α) > 0, c the integral reference and Γ the Euler-gamma function. Whereas the fractional differentiation Dα (the left sided fractional integration or the inverse operation) is given as :

Dα = Dm J m−α , m − 1 < α ≤ m, m ∈ N

(4.3)

where Dm denotes the ordinary derivative of order m. We also define: J 0 = D0 = I

(4.4)

where I is the identity operator. In particular, we have:

Jx(t) =

Zt

x(ξ) dξ

(4.5)

c

We note the semi-group property: J α J β = J α+β for α ≥ 0, β ≥ 0

(4.6)


71

and the expression for α ≥ 0: J α Dα = I

(4.7)

J α Dα x(t) = x(t)

(4.8)

that means

4.2.2

Discrete form: Gr¨ unwald approach

The first derivative of a function x(t) is given by: x(t) − x(t − h) h→0 h

D1 x(t) = lim

(4.9)

Using a finite sampling step h of the time t, i.e. t = Kh, leads to: D1 x(t) =

x(Kh) − x((K − 1)h) h

(4.10)

Introducing the delay operator q −1 , defined by: q −1 x(Kh), we can write: D1 x(t) =

1 − q −1 x(Kh) h

(4.11)

The same calculus is achieved at the order 2: 1 − q −1 D x(t) = h2 2

2

x(Kh)

(4.12)

The generalization to any order (real or complex) is immediate: 1 − q −1 D x(t) = hα α

α

x(Kh)

(4.13)

where α is real or complex. Developping (1 − q −1 )α from the Newton binomial formula gives: 1 D x(t) = α h α

∞ X k=0

k α(α

(−1)

− 1)....(α − k + 1) k!

!

x(Kh)

(4.14)


72

Since q −k x(Kh) = x((K − k)h) = x(t − kh), another representation of the fractional

derivative is:

Dα x(t) =

∞ 1 X α(α − 1)....(α − k + 1) x(t − kh) (−1)k α h k!

(4.15)

k=0

As x(t) = 0 fot t < 0, one has x(t − kh) = 0 for t − kh < 0, the finite sum in the previous equation can be reduced to:

K 1 X α(α − 1)....(α − k + 1) D x(t) = α (−1)k x(t − kh) h k! α

(4.16)

k=0

where K is as defined before. It is obvious to see that when a negative real part for the fractional order of differentiation α is chosen, the fractional integral (4.2) can be computed. Definition (4.16) shows that the fractional integral of a function takes into account the past of the function x(n). For more details about the definition of the fractional integration, we suggest to the reader to see [Oldham & Spanier, 1974; Prodlubny, 2002].

4.2.3

Computation of the coefficients

To implement the fractional differentiation, the computation of its coefficients is necessary: (α)

ωk

= (−1)k

α(α − 1)....(α − k + 1) k!

(4.17)

One of the possible approaches is to use the recurrence relationships: (

(α)

ω0

=1

(α) ωk

= 1−

α+1 k

(α)

ωk−1 ,

k = 1, 2, 3, . . . , K

(4.18)

In the case of a fixed value of α, this approach allows to create an array of coefficients that can be used to differentiate (integrate) various functions. However, in some problems the most suited value of α must be found: various values of α are considered, and for (α)

each particular value of α the coefficients ωk must be computed separately. In such case the fast Fourier transform method can be used [Oldham & Spanier, 1974; Prodlubny, 2002]. (α)

The coefficients ωk function (1 −

can be considered as the coefficients of the power series for the

z)α : ∞ X (−1)k (1 − z)α = k=0

α k

!

zk =

∞ X k=0

(α)

ωk z k

(4.19)


73

where z = q −1 is the delay operator. Substituting z = e−iϕ , we have: 1 − e−iϕ (α)

and the coefficients ωk

=

∞ X

(α)

ωk e−ikϕ

(4.20)

k=0

are expressed in terms of the Fourier transform: (α)

ωk where fα (ϕ) = 1 − e−iϕ

α

α

=

1 2πi

Z

2π

fα (ϕ)eikϕ dϕ

(4.21)

0

. Then, any technical implementation of fast Fourier trans(α)

form can be used to compute the coefficients ωk .

4.3

Statistical analysis of a fractionally integrated function

In this section the properties of the fractionally integrated path are studied. We start by the geometric properties in terms of regularity, then the statistical properties are presented.

4.3.1

Regularity

Indeed, the regularity of the path provides information on how the path is varying over the time: a path that has low regularity means that the target is changing its direction frequently and it would be hard to predict its position. On the contrary, a target path that is highly regular means that the target does not often change its direction, then, its path would be easier to track (predict). In this paragraph, we show that the application of the DFI with a negative order allows enhancing the regularity of a target path and, consequently, enhancing its predictability. To illustrate our proposition, the modifications of the regularity at the application of the fractional integration on a positive function (a path) with negative DFI order, we consider an example of a given path X(n) (Fig.4.2a), where the different coordinates (positions or measures) are positive. Now let us see what happens when we differentiate fractionally the path. So, three different orders: -0.1, -0.3, and -0.5 are considered and, the corresponding results are presented in Fig.??b, c, and d, respectively. One can see that the path becomes smooth when the absolute value of the DFI order increases. In Fig.??b the path is very smooth (regular) and tends to be linear. However, in Fig.??c

18

20

16

15

14

10

12

5

D−0.1X(n)

X(n)


10

0

8

−5

6

−10

4

−15

2 0

10

20

30

40

50

−20 0

60

10

20

n

15

15

10

10

5

5

0

−5

−10

−10

−15

−15

30 n

(c)

50

60

40

50

60

0

−5

20

40

(b) 20

D−0.5X(n)

D−0.5X(n)

(a)

10

30 n

20

−20 0

74

40

50

60

−20 0

10

20

30 n

(d)

Figure 4.2: Illustration of the application of the DFI to a given path X(n). (a) original trajectory coordinates X(n), (b) DFI of X(n) path with α = −0.1, (c) DFI of X(n) path with α = −0.3, (d) DFI of X(n) path with α = −0.5, where n is the number of the samples and the distance between the points is expressed in meter.

and d we can remark that it is smoother (regular). Then, we can say that for a given path an optimal DFI order exists where the path becomes regular and can be predicted more accurately. The application of the fractional integration with negative order (differentiation) provides a compressed or regular path, where the amplitude range is decreasing with the increase of the absolute value of α. On the contrary, as it can be derived from the property of the differentiated path given in expression (4.8) that, when α is positive, the amplitude range of the integrated function increases. This property can also be explained from the frequency properties of the fractional differentiation filter point of view.


4.3.2

75

Statistical analysis of the fractionally integrated path

In this subsection, we present the statistical effect of the application of the fractional integration on a given path. Indeed, we know that theoretically the fractional integration is reversible but in the discrete case, some errors can appear due to the approximation. To do so, we studied the average value of the fractionally differentiated path. Then, as pointed out before, we analyze the autocorrelation coefficients of the fractionally differentiated path to show their increase with the application of the DFI.

4.3.2.1

Average value of the integrated function

The Z− transform function of the discrete filter given in (4.16) can be written as: H(z) =

G(z) = (1 − z)α F (z)

(4.22)

From (4.22), we can easily show that the average value of the output path, i.e. the integrated function, is given by: µg = H(0) × µf

(4.23)

where µf is the average value of the function f . The value of µg is infinite when the DFI order is positive. Consquently, to keep constant the average value of the path, the original trajectory is centered before applying the DFI.

4.3.2.2

Autocorrelation

Proposition: The application to a function of the DFI with a negative order increases the autocorrelation of a function and consequently reduces the prediction error. The opposite case happens when a DFI with a positive order is applied to this function. The proof of this proposition can be found in [Nakib et al., 2013].


4.4

76

Proposed method

The proposed method, to enhance the path prediction procedure, consists in adding on downstream the digital fractional integration operator to any classical prediction filter. Fig. 4.3 illustrates the principle of this procedure.

Figure 4.3: The principle of applying DFI to classical prediction filters.

The main idea takes profit from the properties of the DFI, and the first step consists in applying the DFI with a negative order α to the current path, which forms the entry of any classical prediction filter. Reciprocally, we apply the DFI with inversed order (−α) to the output (estimated trajectory), which consists of the predicted locations of the target. This idea is based on two assumptions: 1. the data association problem is solved, 2. the path of the target can be seen as a positive function P (i). Then, locations P (i) = (x1 , ..., xD ), where D is the dimension of the space, of a target are separated into D positive coordinates trajectories X (j) (i) where j = 1, ..., D. For example, in the case of D = 2: X (1) (i) expresses the abscissa trajectory, and X (2) (i) is the ordinate one. Then, the DFI is applied to X (1) (i) and to X (2) (i), separately. Let X (1) (i) = X (1) (1), X (1) (2), ..., X (1) (N ) be abscissa of the target discrete trajectory

locations. Then, the DFI of this trajectory, for a given DFI order α (α < 0), is defined by the formula (4.16). The obtained vector is: Dα X(i) =

Dα X (1) (1), Dα X (1) (2), ..., Dα X (1) (N ) . After-

wards, Dα X (1) (i) is used as the input of any prediction filter to estimate the future α X (1) (i). Then, to get the estimated postions of the target, a differentiated positions Db differientation with the opposite order is applied. Finally, the same operation is applied on each dimension.


4.5

77

Enhancement of the path prediction using DFI

Based on the properties of the DFI seen previously (section 4.3.2.2), the application to a function of the DFI with α negative increases the autocorrelation of the considered function. Fig. 4.4 illustrates the use of the linear prediction (LP) filter. One can see the small difference between a DFI path and its prediction (low prediction error, see Fig. 4.4(b)). However, the prediction error is significant between the path and its prediction (see Fig. 4.4(a)). The prediction errors between the original trajectory and its estimated one using only LP of order 3, and the prediction error values with the estimated trajectory using DFI-LP are presented in Fig. 4.4(c). One can notice that the linear predictor estimates better the transformed trajectory using the DFI than the original trajectory. This can also be seen when looking to the Fig. 4.4(c) where the amplitude of the prediction error on each point using the DFI is less than that of LP alone. In Fig. 4.4(d) we present the variation of root mean square error (RMS) for different values of α. As it can be seen, the decrease of the RMS is linear with the increase of the absolute value of the DFI order |α|. It can be seen that

best DFI order is −0.9 for this path. From this, one can conclude that the best order is always for any path, unfortunately, the best DFI order must be optimized for each path.

As stated before, instead of the original path, the input of prediction filters is the fractionally differentiated trajectory, therefore the prediction error value decreases. The classical prediction filters, as Kalman filter (KF), and LP have two main computational steps, the first step consists of the update time, while the second one is the measurement update.

4.5.1

Decreasing of the archive size needed for accurated prediction

Another consequence of using DFI when predicting an optimum path: the reduction of the number of the past positions that will be used to predict the future positions of a moving optimum. It is obvious that the benefit of using a short-archive when predicting is the decrease of the number of locations used for the estimation of the path. Moreover, it allows to eliminate the influence of the accumulated approximation errors during long-time tracking. The use of small archive leads to use a lower number of positions that will be taken into account to predict the position of the optimum: therefore, in our experiments, we considered the last 5, 10, 15 and 25 positions. Our experiments showed the decrease of the archive size to predict the position of an optimum path, a performance analysis has been done using different kinds of paths.


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(a)

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Root mean squared error

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Figure 4.4: Illustration of using LP and DFI-LP on given path, (a) Original trajectory (solid line) and its estimation using LP (dashed line), (b) DFI of the path and its estimation with α = −0.5, (c) Prediction error values using LP (solid line) and prediction error using DFI-LP (dashed line), (d) Variation of the root mean squared error with the increase of the DFI order (α).

4.6

Conclusion

In this work, it was showed that under some assumptions, the digital fractional integration allows increasing the autocorrelation of a real positve function (i. e. moving model of an optimum in the case of a dynamic problem). Then, this property is exploited to track an optimum and to predict its future positions accurately. Particularly, the paradigm of digital fractional integration (DFI) is used to significantly reduce the number of past positions that will be used at predictions procedure. Indeed, the proposed scheme allows to enhance the performance of existing prediction techniques as linear predictors and the Kalman filter. It was found that the DFI based approach allows reducing the archive size that leads in many cases to a decrease in the spiraling effect


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of accumulated rounding error that builds up over a long tracking time, due to smaller number of addends.

4.7


• A. Nakib, B. Daachi, M. Dakkak, P. Siarry, ”Mobile Tracking Base On Fractional

Integration”, IEEE Transactions on Mobile Computing, vol.13 (10), pp. 23062319, 2013.

• M. Dakkak, A. Nakib, B. Daachi, P. Siarry, J. Lemoine, ”Mobile indoor location based on fractional differentiation”, in Proc. Of the IEEE Wireless Communi-

cations and Networking Conference, WCNC 2012, Paris, France, April 2012, pp. 2003-2008.

Conclusion and future work In this dissertation I1 summarized the most significant, in my opinion, research work I have done in metaheuristics and image processing over the last 6 years. It is exciting to work in optimization because the designed algorithms can be applied to many different fields (smart grid, computer networks, parallel systems, etc.) in addition to image processing. Then, this document is focus on the design of metaheuristics for dynamic optimization in order to solve only medical image processing problems. My wish is keep always this need to contribute to optimization areas that has an untold number of applications. As my research has been project driven, it would seem natural to look at the prospective projects to see what is to come. It is obvious that I can only present my current ideas at this time, without been sure what will truly occur. As I would like to continue working on the area of metaheuristics and, especially, for the image processing problems. In the following, I present what I am planning to work on in the future. Future work It is well-known that performances of metaheuristics decrease with the increase of problems’ dimension. So, the design of efficient metaheuristics for large-scale problem is still an open problem. So, on the following I propose some idea in order to enhance the performance of metaheuristics and to design news algorithms. Prediction based Dynamic metaheuristics Under the assumption that a given dynamic environment have a repeated behavior, it was shown that the use of a prediction procedure to enhance to the performance of memory-based metaheuristics. Indeed, the use of past data, can provide accurate predictions and consequently, the algorithm can anticipate the change and adapts its self to this event. In the literature different predictors were used to estimate when the next change would occur: linear regression, and nonlinear regression. The first one was tested 1

In this section I will switch away from the academic “we”

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General conclusion and futur work

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using unlimited and limited archive size of the observations, enclosed by a time-window. It was shown that the linear predictor using all the collected data, performed well for cyclic and patterned change periods, but failed in the presence of nonlinear change periods. On the other hand, the nonlinear predictor provided accurate predictions in different kind of environments, except for the nonlinear change periods obtained by functions unknown to the module. The use of a time-window in the linear predictor was the most robust and efficient method: it allowed the metaheuristics to obtain the best performances in all types of change periods, with the minimum computational cost. One can point out that the prediction based algorithms have disadvantages due to training errors. The origin of these errors might be from a wrong training set: if the algorithm has not well performed in the previous time periods, the collected observations for the training might not be helpful for the prediction and then drive the algorithm to a wrong direction. Another problem related to the use of this approach is the case of the algorithms that need a large enough set of training data to produce good results. It means that the prediction can only be started after collecting enough training data. In the case of dynamic optimization where there is a need of finding/tracking the optima as quick as possible, this might be a problem that must be solved. These previous works showed the importance of the size of the past observations to store is an important issue and the profit obtained from predictor depends on this parameter. Then, the first obvious prospective is to include the designed predictors based on fractional integration within our MLSDO algorithm to enhance its performance. Another issue in this case that we did not explore intensively is the use of neural networks to learn the move of the optima and predicting their position, when a change in the objective functions occurs. Indeed, the use of learning methods to predict movement of the optima will impose to store many information from the past. indeed, the use of learning methods to learn the fitness landscape and anticipate the change is an important issue. We obtained encouraging results on the use of a predictor based neural network. However, more investigation are needed to dress all the questions: what is the most adapted neural network for all kind of environments ? we mean by the best low complex, free of parameters, and high prediction performance. One solution would consist in using possibility theory to deal with this problem. Design of metaheuristics based on Learning algorithms The idea behind is to provide a general framework of metaheuristics. In first a decomposition of the different algorithms should be performed. This means that a general framework in which we identify the components that have a clear and distinct functionality will be created.


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In order to create the decomposition, one can distinguish at least two levels of precision. For example at the first level, we define the most general framework that has become common in all approaches within metaheuristics and as such is found in every kind of a metaheuristic algorithm. This decomposition of metaheuristics is illustrated in figure 5.1(a). One can see, as is the case in almost optimization algorithms, we have an initialization step. The first set of solutions is generated (most often at random) and their fitness is obtained. Once this is done, we can build the body of the algorithm. As it can be seen in the figure 5.1(a), this amounts to checking first whether the algorithm should be terminated. If this is true, then it stops. Otherwise, an evolution step is performed, bringing the population to the next iteration. At this decomposition level we do not need to specify further how this is established. All we require is that any new chromosome in the population be evaluated, so, as to establish the invariant that every member of the population has been evaluated (so its fitness value is known) when the termination condition is to be checked. After this evolving step, the algorithm has reached the next iteration and the termination condition is checked again. From this first level model for metaheuristics algorithms one can derive a few components that are part of the decomposition. In the figure 5.1(a), this has been indicated by suggestively by ”Stopping conditions”. It is clear that one can vary the termination (stopping) conditions and they can be based upon any information from the metaheuristic. Furthermore, the initialization process could be modeled within a component. Its role is to generate new offspring at the beginning of the algorithm. This step is the most important and in most cases is done randomly. However, in some cases the random initialization is not suitable, in the literature many techniques were proposed and would be used. The part from figure 5.1(a) that has not been discussed yet, is the Evolving procedure. We cannot derive a new component from this, because it is a composite step. In our work, we propose the second level of precision presented in figure 5.1(b) that consists in decomposing the evolving system. We are not yet able to transform the state of the metaheuristic from one iteration to the next. This needs a more detailed decomposition. In order to obtain such decomposition, we could propose a unified approach to metaheuristics where the different components that compose the evolving procedure are identified i.e the EA decomposition proposed by Back [1996]. In figure 5.1(b) a new scheme of decomposition is presented. then, the main question is: each of the method from a set of those at hand would provide the best performance for a given problem? In order to answer this question, one of the different issues is the use of machine learning techniques may be the best issue. This answer opens also the door for many other questions such as: each learning techniques?

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(a)

(b)

Figure 5.1: Metaheuristic decomposition. (a) high level view of a metaheuristic, (b) Proposed 3 levels decomposition.

The complexity of the algorithm? Time needed to learn from the past iterations? Each criterion to evaluate the on-line performances? One can see that we can add another decomposition level where the fitting of the different components of the metaheuristic for a given combination of the components. This issue was already treated widely in the literature using machine learning techniques. So, in our future work, we will assume that the different components are well fitted, even, if this assumption can be discussed. A first attempt to work on this issue was already done and the preliminary result are promising. In this attempt, we focused only on evolutionary algorithms and the obtained results showed the efficiency of the approach compared to more than fifteen algorithms [Nakib et al., 2015]. Fractal analysis based metaheuristics The question that I would like to answer in this approach is: how to locate the particle inside the search space? and how to be sure that the already visited regions will not be re-visited ? The idea to design an efficient metaheuristic based on geometric characterization of the search space using geometric fractals to solve continuous optimization problems. The goal is to use of different search strategies (local searches) inside a subspaces defined by the fractal decomposition, then, go from one region to another until finding the global optimum. As I my search space is modeled, the diversification can be done easily if the algorithm stagnates.Then, the problem is what is the best fractal dimension ? Based on previous ideas, I started to design this algorithm and the primary results are excellent, I don’t yet submit these results for publication, and I hope continue to explore this issue in the near future.


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Multilevel optimization I am also interested to design metaheuristics to multilevel optimization problems. Indeed, some works on the use of metaheuristics to solve bilevel problems were proposed in the literature Talbi [2013]. The multilevel optimization problems can be defined as an extension of bilevel problems to any number of levels problems. Then, multilevel optimization problems are mathematical programs that have a subset of their variables constrained to be an optimal solution of order programs parameterized by their remaining variables. It is obvious that the complexity of these problems increases drastically when the number of levels is greater than two. It finds many applications in daily life. The first application that I would like to deal with is the resource allocation Aiyoshi & Shimizu [1981]problem. I started collaboration with my colleague H-N Nguyen from BULL SA, to this NP-Hard problem, in the case of cloud computing systems. The second application that I am interested with is to formulate some image processing problems as segmentation problem as a multilevel optimization problem and then design stochastic optimization algorithms to solve them. I end this section with some remarks about the DOPs and designed algorithms until know. So, it is obvious that a lack of a clear link between dynamic metaheuristics in academic research and real world scenarios has lead to some criticisms on how realistic current academic problems are. Moreover, the predictability of changes. The number of studies in this topic is still relatively small. The general purpose benchmark problems are non-time-linkage that means the algorithm does not influence the future dynamics. In most work optimality is the primary goal or the only goal in majority of academic dynamic metaheuristic papers. So, how about the end user? In terms of methodology research, new efficient approaches are still greatly needed to address different types of DOPs, and finally, the theoretical studies on metaheuristics are quite limited so far.

Content of the appendix This appendix contains the following documents 5 significant publications.

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Appendix A

Dynamic optimization using multiagent strategy This article was published in Journal of Heuristics. @articleJOH13, author = Julien Lepagnot and Amir Nakib and Hamouche Oulhadj and Patrick Siarry, title = A multiple local search algorithm for continuous dynamic optimization, journal = J. Heuristics, volume = 19, number = 1, pages = 35–76, year = 2013,

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Journal of Heuristics manuscript No. (will be inserted by the editor)

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Julien Lepagnot · Amir Nakib · Hamouche Oulhadj · Patrick Siarry

Received: date / Accepted: date

Abstract Many real-world optimization problems are dynamic (time dependent) and require an algorithm that is able to track continuously a changing optimum over time. In this paper, we propose a new algorithm for dynamic continuous optimization. The proposed algorithm is based on several coordinated local searches and on the archiving of the optima found by these local searches. This archive is used when the environment changes. The performance of the algorithm is analyzed on the Moving Peaks Benchmark and the Generalized Dynamic Benchmark Generator. Then, a comparison of its performance to the performance of competing dynamic optimization algorithms available in the literature is done. The obtained results show the efficiency of the proposed algorithm. Keywords dynamic · non-stationary · time-varying · continuous optimization · multi-agent · metaheuristic · Moving Peaks 1 Introduction Recently, optimization in dynamic environments has attracted a growing interest, due to its practical relevance. Many real-world problems are dynamic optimization problems (DOPs), i.e., their objective function changes over time. For instance, changes can be due to machine breakdowns, to the changing quality of raw materials, or to the appearance of new jobs to be added to a schedule. In dynamic environments, the goal is not only to locate the global optimum, but also to follow it as closely as possible. A dynamic optimization problem can be expressed as in (1), where f (x,t) is the objective function of a minimization problem, h j (x,t) denotes the jth equality Patrick Siarry Laboratoire Images, Signaux et Systèmes Intelligents, LISSI, E.A. 3956 Université Paris-Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil, France E-mail: [email protected]

2

Julien Lepagnot et al.

constraint and gk (x,t) denotes the kth inequality constraint. All of these functions may change over time (iterations), as indicated by the dependence on the time variable t. min f (x,t) s.t. h j (x,t) = 0 for j = 1, 2, ..., u gk (x,t) ≤ 0 for k = 1, 2, ..., v

(1)

The performance of dynamic optimization algorithms in the literature is improving, and many research directions are left to be further investigated in order to obtain even more efficient algorithms. In this paper, a new algorithm for dynamic continuous optimization is proposed, called MLSDO (Multiple Local Search algorithm for Dynamic Optimization). It belongs to the class of cooperative search strategies for DOPs. It makes use of a population of coordinated local searches to explore the search space. The use of local searches provides a fast convergence to the local optima, and the strategies used to coordinate these local searches enable the algorithm to widely explore the search space. The local optima found during the optimization process are archived, in order to be used when a change is detected. The set of heuristics used to coordinate and control all the local searches, and to manage the archived optima, constitutes a new efficient way to deal with DOPs. The rest of this paper is organized as follows. Section 2 discusses some related work. Section 3 describes the fundamentals of the proposed MLSDO algorithm. Section 4 explains in detail each strategy used in MLSDO. Section 5 presents the benchmark sets used to test the algorithm. Experimental results are discussed in Section 6. Conclusions and work under progress are presented in section 7. A nomenclature of all the variables used in this paper is given in Appendix. 2 Related work Almost all algorithms for DOPs are population-based metaheuristics, which are generally bioinspired algorithms. Most bioinspired algorithms belong to the classes of evolutionary algorithms (EAs) and particle swarm optimization (PSO), though ant colony optimization and artificial immune systems have also been investigated [8, 10, 15, 34]. In order to perform better on DOPs, some authors have tried hybrid algorithms, like Lung and Dumitrescu in [24] and [25], who propose hybrid PSO/EAs algorithms. However, the results of those algorithms are not significantly different from those of pure PSO or EAs algorithms. The use of local search as a main feature of a dynamic optimization algorithm has also been studied. We will now describe competing dynamic metaheuristics that have been proposed in the literature for continuous environments, and focus on the cooperative techniques used in each of these algorithms. Then, we highlight the main strategies and cooperative techniques used to deal with DOPs. A significant number of existing dynamic optimization algorithms are based on EA principles. EAs can be indeed well suited to optimization in changing environments, since they are inspired by the principles of natural evolution. It has been shown

A Multiple Local Search Algorithm for Continuous Dynamic Optimization

3

that multipopulation EAs allow to provide good results on multimodal DOPs. Promising results have been obtained by a self-organizing scouts algorithm proposed by Branke et al. in [6]. In this algorithm, a parent population searches over the entire search space while child populations track the local optima. The creation and merging of these child populations, as well as the adjustment of the number of individuals in each population, are typical examples of cooperative techniques that can be used in a multipopulation algorithm for DOPs. In [26], Mendes and Mohais propose a multipopulation differential evolution (DE1 ) algorithm based on some techniques to maintain diversity. Especially, if the best individuals of several subpopulations have a separating distance less than a predefined threshold, called the exclusion radius, then only the population with the highest performance is kept, and the others are reinitialized. The use of an exclusion radius is another common technique to coordinate several subpopulations. Among EAs, this algorithm achieves the best results on the Moving Peaks Benchmark (MPB) [4, 5]. Another multipopulation DE based algorithm is proposed by Brest et al. in [7] that also makes use of an exclusion radius to reinitialize overlapping subpopulations. It obtains the best results among EAs on the Generalized Dynamic Benchmark Generator (GDBG) [18, 20]. The different techniques used in EAs to deal with DOPs are classified in [13] into the following groups: 1. Generating diversity after a change: if a change in the environment is detected, then explicit actions are taken to increase diversity and to facilitate the shift to the new optimum. 2. Maintaining diversity throughout the run: convergence is avoided over the time and it is hoped that a spread-out population can adapt to changes more efficiently. 3. Memory-based approaches: the optimization algorithm is supplied with a memory to be able to further recall useful information from the past. In practice, good solutions are archived in order to be reused when a change is detected. More sophisticated memory-based techniques also exist [37]. 4. Multipopulation approaches: dividing up the population into several subpopulations, distributed on different optima, allows tracking multiple optima simultaneously and increases the probability of finding new ones. Another widely used class of algorithms for DOPs is PSO. PSO is a populationbased approach in which simple software agents, called particles, move in the search space. The particle dynamics are inspired by models of swarming and flocking [14]. Dynamic PSO based algorithms are mainly multi-swarm, i.e. they make use of several sub-swarms, as several sub-populations can be used in EAs. A multi-swarm algorithm using a speciation process is proposed by Parrott and Li in [21, 30], where spatially close particles form a particular species. Each species corresponds to a sub-swarm and, if the number of particles in a sub-swarm is higher than a predefined threshold, then the worst particles in this sub-swarm are reinitialized. This technique can be 1 The strategy of DE consists in generating a new position for an individual, according to the differences calculated between other randomly selected individuals.

4


also considered as a cooperative one that enables the regulation of the number of particles in a sub-swarm. Several improved variants of this algorithm are proposed in [1, 22, 31]. Du and Li propose another PSO based algorithm in [9]. Here, two swarms of particles are used: the first one is for diversification, and the second is for intensification. The exchanges of information about the velocities and positions of the particles between the swarms enable their cooperation. Li and Yang propose in [19] a multi-swarm approach that makes use of clustering techniques to create sub-swarms, and overlapping sub-swarms are merged. Best solutions found until a change occurs are archived, in order to be added as new particles if a change is detected. It is yet the only PSO algorithm analyzed using GDBG. A simplified version of this algorithm is benchmarked using MPB [36]. The use of concepts from the physics domain in PSO has been also widely investigated. In [2], Blackwell and Branke propose two multi-swarm algorithms based on an atomic model. The first algorithm uses multiple swarms, composed of a subswarm of mutually repelling particles, orbiting around another sub-swarm of neutral, or conventional PSO particles. The second algorithm is based on a quantum model of the atom, where the charged particles (electrons) do not follow a classical trajectory, but are instead randomized within a ball centered on the swarm attractor. If several swarms converge to the same local optima, then an exclusion radius is used in order to randomize the worst ones. Both of these algorithms place their swarms on each of the localized optima, then each swarm tracks a different optimum. The same approach is used in [3], [22] and [29], hybridized with other techniques to increase the diversity and to track the optima. Among PSO algorithms, the best results obtained on MPB are achieved by Novoa et al. [29], thanks to the use of a simple controlling mechanism that determines how the position and velocity of a particle have to be updated: if a particle performs a fixed number of successive non improving moves in the search space, then it is relocated at the best position found by its swarm, and its velocity is multiplied by a random number in [0, 0.5]. Otherwise, its position and velocity are updated using the classical equations of PSO. This way, the particles that tend to waste evaluations in unpromising regions of the search space are relocated in more promising ones. Some authors investigated also the use of local search as a main feature of a dynamic optimization algorithm. We can cite [39], where Zeng et al. propose an algorithm based on the use of local searches to explore the search space. The local optima found are archived, in order to be tracked when a change occurs, using additional local searches. A specialized local search procedure is proposed in order to provide a fast convergence to the local optima. In [28], Moser and Hendtlass use extremal optimization (EO) to determine the initial solution of a local search procedure. EO does not use a population of solutions, but improves a single solution using mutation. This algorithm uses a “stepwise” sampling scheme that samples every dimension of the search space in equal distances. Then, the algorithm takes the best candidate as the initial solution of a hill-climbing phase, in order to fine-tune the solution. Finally, the solution is stored in memory, and the algorithm is applied again on another randomly generated solution. An improved and generalized variant of this algorithm is proposed in [27], termed as Hybridised EO. Compared to the algorithm of Zeng et


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al. [39], where the exploring local searches are randomly initialized, Hybridised EO makes use of a special seeding strategy to initialize the local searches in promising areas of the search space. However, this algorithm has been specifically designed for MPB and no study has been done yet to determine whether there is a potential of wider applicability. In [16, 17], Lepagnot et al. propose a multi-agent centralized cooperative algorithm for DOPs, called MADO. Local searches are used also to explore the search space, and to track the local optima stored in memory. As Hybridised EO, it makes use of a special seeding strategy to initialize the local searches in promising areas of the search space. However, in MADO, the local searches are performed by a population of agents that are coordinated. In [32, 33], Pelta et al. propose a multiagent decentralized cooperative strategy, where multiple agents cooperate to improve a set of solutions stored on a grid. Then, Gonzalez et al. presented a new centralized cooperative strategy for DOPs that uses several tabu search based local searches [12]. As in MADO, these local searches are controlled by a coordinator that keeps a memory of the found local optima. Besides, Wang et al. propose in [35] a memetic algorithm based on the cooperation and competition of two local search procedures, combined with diversity maintaining techniques.

In this paper, the proposed MLSDO algorithm makes also use of the main existing cooperative techniques in a centralized framework, implemented in a different way, i.e. a dynamic number of agents perform local searches in parallel; the found local optima are archived in order to be tracked when a change is detected; a seeding strategy is used to initialize the exploring local search agents in promising, or unexplored, areas of the search space; an exclusion radius is used to prevent several local searches to converge to the same local optimum. These techniques provide a maintained and enhanced diversity throughout the execution of the algorithm, in order to widely explore the search space and detect the local optima. They also enable a fast convergence to the local optima, using several parallel local searches, and their tracking. This way, the algorithm can adapt and react to changes more efficiently. In [17], we proposed to combine these main ideas to solve DOPs in a preliminary work. It is obvious that these techniques can be implemented in different ways that can lead to different performances. In this paper, new heuristics are proposed to implement them, and it constitutes a new efficient way to deal with DOPs. In MLSDO, our motivation is to propose more suitable heuristics for dynamic optimization. Each heuristic in MLSDO is designed by a trial and error approach, using the main two benchmarks available in dynamic optimization to evaluate, and retain an algorithm. It leads to novel algorithms as the seeding mechanism used to initialize local search agents, the stopping criterion of their local search procedure, as well as an adaptation mechanism used to adapt the step size of the performed local searches. All these algorithms will be detailed in Section 4, and analyzed in Section 6, along with a comparison of MLSDO to other competing algorithms based on cooperative techniques.

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3 Proposed algorithm In the following subsections, we first describe how the distances are computed in the search space. Then, we describe the overall scheme of the proposed MLSDO algorithm and the initialization procedure. 3.1 Distance handling In this paper, we propose to define the search space as a d-dimensional Euclidean space, since it is the simplest and most commonly used space. The inner product is given by the usual dot product, denoted by h·, ·i, and the Euclidean norm is denoted by k·k. Then, as the search space may not have the same bounds on each dimension, we use a “normalized” basis. We denote by ∆i the size of each interval that defines the search space, where i ∈ {1, ..., d}. Then, the unit vectors (ei ) in the direction of each axis of the Cartesian coordinates system are scaled, in order to produce modified basis vectors (ui ), defined as {u1 = ∆1 e1 , u2 = ∆2 e2 , ..., ud = ∆d ed }. This change in basis transforms a hyperrectangular search space into a hyper-square search space, according to (2), where x′i and xi are the ith coordinates of a solution expressed in the hyper-square space, and in the hyper-rectangular space, respectively. x′i =

xi for i = 1, 2, ..., d ∆i

(2)

3.2 Overall scheme MLSDO is a multi-agent algorithm that makes use of a population of agents to explore the search space. Agents are nearsighted: they only have a local view of the search space. More precisely, agents are only performing local searches; they jump from their current position to a better one, in their neighborhood, until they cannot improve their current solution, reaching thus a local optimum. A selection of these optima are saved in order to accelerate the convergence of the algorithm. The overall scheme of MLSDO consists of the following two modules (Figure 1): 1. Memory manager: in case of a multimodal environment, a dynamic optimization algorithm needs to keep track of many of the found local optima, since one of them can become the new global optimum after a change occurs in the objective function. Thus, we propose to save the found optima in memory. The memory manager maintains the archive of local optima that are provided by the coordinator. 2. Coordinator: it supervises the whole search, and manages the interactions between the memory manager and the agents. It compensates for the nearsightedness of the agents, and it is able to prevent them from searching in unpromising zones of the search space. The coordinator is informed about the found local optima, and manages the creation, destruction and relocation of the agents.


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!

Fig. 1 Overall scheme of MLSDO. N is the number of currently existing agents. Each agent performs a search procedure described in subsection 3.4.

The initialization step in Figure 1 mainly consists in computing the initial set of agents, as described in the next subsection. The stopping criterion depends on the optimization problem.

3.3 Computation of the initial set of agents The coordinator starts by creating the agents in the initialization phase of MLSDO. The number of agents to be created is fixed by the parameter na . The initial positions of these agents are not randomly generated, but are computed in order to prevent several agents from being placed close to each other. This is done by sequentially placing the agents at the locations generated by a heuristic. The implementation details of this heuristic are given in subsection 4.3. Thus, at the end of this heuristic, we get a set of initial positions for the set of agents that is widely covering the search space.

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Fig. 2 Flowchart of the main procedure of a MLSDO agent.

3.4 The flowchart of an agent Agents proceed by running their local search independently of each other. The flowchart of the search procedure of an agent is illustrated in Figure 2. One can see that two special states, named “CHECK POINT A” and “CHECK POINT B”, appear in this flowchart. These states mark the end of one step of the procedure of an agent. Hence, if one of these states has been reached, then the agent halts its execution until all other agents have reached one of these states. Afterwards, the execution of the agents is resumed; i.e., if an agent halts on CHECK POINT A (CHECK POINT B, respectively), then it resumes its execution on CHECK POINT A (CHECK POINT B, respectively). This special state allows the parallel execution of the agents.

4 The optimization strategies used in MLSDO In the following subsections, the strategies used in MLSDO are described in detail.

4.1 The exploration strategy of the agents MLSDO agents explore the search space step-by-step, moving from their current solution Sc to a better one S′c in their neighborhood, until they reach a local optimum.


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initializeAgent Inputs: Snew ,rnew local variables: ∅ Sc ← Snew

Evaluate Sc D←0

R ← rnew

U ← 0.0

return {Sc ,D,R,U}

Fig. 3 Procedure that initializes the local search of an agent. Snew and rnew are the initial solution and the initial step size of the agent, respectively. Sc is the current solution of the agent. D is the direction vector of the last displacement of the agent. R is the step size of the agent. U is the value of the cumulative dot product (see equation 3). The function Evaluate computes the value of the objective function of a given solution and assigns this value to the solution. 0 is a vector of d zeros.

An agent can be created for two reasons: to explore the search space or to track an archived optimum, when a change in the objective function is detected. An agent has a step size R, adapted during its local search, and initialized to rnew (a real parameter in the interval (0, 1]). However, an agent created to explore the search space requires a greater initial step size than a “tracking” agent. Thus, when initializing the local search of a new agent, the value of rnew depends on the condition of the creation of the agent. The parameter rnew is equal to rl for a “tracking” agent, and to re otherwise, where rl and re are two parameters to be fixed. If the local search of an agent has to be reinitialized, then rnew must be equal to re . The initialization of the local search of an agent is described in Figure 3. This procedure is called at the creation, and at the relocation of an agent. At the initialization of the local search of an agent, in addition to the step size R, the current solution Sc of the agent is set to Snew , where Snew is the new starting solution given by the coordinator. The direction vector D and the cumulative dot product U, used to adapt R, are set to the null vector and zero, respectively. We focus now on the local search of a single agent, summarized in Figure 4 and described in detail below. The procedure in Figure 4 provides the details of the state indicated in Figure 2 by the description “perform one step of the local search of the agent”. Thus, it is repeated each time the agent is in this “local search” state, so as to enable the convergence of the local search. This is the main procedure of the local search performed by an agent, and it calls the subprocedures selectCandidate, stoppingCriterion and updateStepSize that are defined below. At each step of its local search, the agent moves from its current solution Sc to the new candidate solution S′c according to the mechanism in Figure 5. As we can see, two candidate solutions are evaluated per dimension of the search space, denoted by Sprev and Snext . They stand in opposite directions from S′c along an axis of the search

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localSearch Inputs: Sc ,d,R,U,D,rl , δ ph , δ pl local variables: stop {S′c ,Sw ,D′ } ← selectCandidate(Sc ,d,R)

stop ← stoppingCriterion S′c ,Sw ,Sc ,R,rl , δ ph , δ pl

{U,R} ←

updateStepSize(S′c ,Sc ,U,D,D′ ,R)

if S′c 6= Sc then Sc ← S′c D ← D′

end if stop = true then Stopping criterion (of the local search) satisfied: Sc is the local optimum found end return {Sc ,R,U,D} Fig. 4 Procedure that performs one step of the local search of an agent. Sc is the current solution of the agent. d is the dimension of the search space. R is the step size of the agent. U is the value of the cumulative dot product (see equation 3). D is the direction vector of the last displacement of the agent. rl , δ ph and δ pl are parameters of the algorithm. The procedures selectCandidate, stoppingCriterion and updateStepSize are described in subsection 4.1.

space, at equal distance R from S′c . For each axis of the search space, the best solution among Sprev , Snext and S′c becomes the new candidate solution S′c . Other mechanisms can be used as that described in [11]. At the end of this procedure, the normalized direction vector D′ from Sc to S′c , and the worst candidate solution Sw , are also returned. D′ is used later to update the step size, R, of the current agent, and Sw is used in the stopping criterion of the agent. To illustrate this process, the successive displacement vectors of an agent, after its initialization, are presented in Figure 6 (a). Once normalized, these vectors are the direction vectors that are used to adapt the step size of the agent (denoted by D and D′ ). As we can see, in Figure 6 (b), these displacement vectors are provided by the steps performed by the agent along each axis of the search space. The adaptation of the step size R is performed through the procedure described in Figure 7. Depending on the situation, the step size is doubled or halved: – if the procedure in Figure 5 cannot find a better candidate solution in the neighborhood of Sc , i.e., if S′c = Sc , then R is halved; – if the agent appears to be moving in a forward direction, according to the “cumulative dot product” described below, then R is doubled to accelerate the convergence of the agent. The cumulative dot product makes use of trajectory information gathered along the steps of the agent. It is computed according to the equation (3) :


11

selectCandidate Inputs: Sc ,d,R local variables: i,Sprev ,Snext ,Si S′c ← Sc

Sw ← Sc

for i = 1 to d do Sprev ← S′c − R × ui Snext ← S′c + R × ui

if Sprev is outside the search space then Sprev ← the closest point to Sprev inside the search space

end

if Snext is outside the search space then Snext ← the closest point to Snext inside the search space end

Evaluate Sprev and Snext Si ← the best solution among Sprev and Snext if Si is strictly better than S′c then S′c ← Si end

Si ← the worst solution among Sprev and Snext if Si is worse or equal to Sw then Sw ← Si end end D′ ← S′c − Sc

D′ ← kD1′ k × D′

return {S′c ,Sw ,D′ } Fig. 5 Selection mechanism: selects the candidate solution S′c to replace the current solution Sc of an agent. d is the dimension of the search space. R is the step size of the agent. ui is the basis vector of the ith axis of the search space. Sw is the worst tested candidate solution. D′ is the direction vector of the current displacement of the agent. The function Evaluate computes the value of the objective function of a given solution and assigns this value to the solution.

Un =

1  2 × Un−1 + hDn−1 , Dn i if n > 0 

0

(3)

otherwise

where Un is the cumulative dot product of the successive direction vectors Dn of the displacements of the agent, at the step n since its initialization, and hDn−1 , Dn i is the dot product of the last two direction vectors. In the procedure updateStepSize (Figure 7), the sequence Un corresponds to U, Dn−1 corresponds to D and Dn corresponds to D′ (the direction vectors of the previous and the current steps of the agent, respectively).

12


1

1

u2

u2

0

u1

1 (a)

0

u1

1 (b)

Fig. 6 Illustration of the local search performed by an agent on an example of a two-dimensional objective function. u1 and u2 are the normalized basis vectors of the search space. (a) shows the successive displacement vectors of the agent. Each of them is performed during one step of the local search procedure of the agent. (b) shows the path followed by the agent. This path is made of the best candidate solutions evaluated along each axis of the search space. These solutions are depicted by black-filled circles.

updateStepSize Inputs: S′c ,Sc ,U,R,D,D′ ,re local variables: ∅ if S′c = Sc then U ← 12 ×U R←

1 2

×R

U←

1 2

×U + hD, D′ i

else if U < −re then U ← −re

else if U > re then R ← 2×R if R > 1 then R←1 end

U ← 0.0 end end return {U,R} Fig. 7 Procedure to update the step size R of an agent. S′c is the candidate solution found to replace the current solution Sc of the agent. U is the value of the cumulative dot product (see equation 3). D and D′ are the direction vectors of the previous and the current displacements of the agent, respectively. re is a parameter of the algorithm.

In the case of U greater than re , R is doubled and U is reset to 0, where re is a parameter of the algorithm. To prevent U from having high negative values, at the end of the procedure updateStepSize, U is constrained to be higher or equal to −re .


13

stoppingCriterion Inputs: S′c ,Sw ,Sc ,R,rl , δ ph , δ pl local variables: p if R < rl then if S′c 6= Sc then p ← |fitness (S′c ) − fitness (Sc )| else

end

p ← |fitness (Sw ) − fitness (Sc )|

if Sc is the best solution found since the last change in the objective function then if p ≤ δ ph then return true end else if p ≤ δ pl then return true end end end return f alse

Fig. 8 Procedure that tests the stopping criterion of an agent. fitness is a function that returns the value of the objective function of a given solution. S′c is the candidate solution found to replace the current solution Sc of the agent. Sw is the worst tested candidate solution. R is the step size of the agent. rl is the initial step size of “tracking” agents. δ ph and δ pl are the highest and the lowest precision parameters of the stagnation criterion, respectively.

The stopping criterion of the local search is presented in Figure 8, where δ ph and δ pl are two parameters of MLSDO. If the stopping criterion is satisfied, then the procedure returns true, otherwise, it returns f alse. As we can see, if the current solution of an agent is the best solution found by MLSDO since the last change in the objective function, then we use a higher precision δ ph in the stagnation criterion of its local search, otherwise we use a lower precision δ pl . We choose δ ph to be not larger than δ pl . In this way, we prevent the fine-tuning of low quality solutions, which could lead to a waste of fitness function evaluations; only the best solution found by the algorithm is fine-tuned.

4.2 The diversity maintaining strategy If an agent has found a local optimum, then it sends it to the coordinator that transmits it to the memory manager. Afterwards, the coordinator gives the agent its new position (the procedure used to generate an initial solution is detailed in subsection 4.3), in order to perform a new local search. This relocating is done by calling the procedure initializeAgent (see Figure 3) with the new position of the agent (and R is initialized to re ).

14


relocateOrDestroyAgent Inputs: N,na ,d,Am ,Ai ,re local variables: d0 ,Snew if N > na then destroy the agent end {d0 ,Snew } ← newInitialSolution(d,Am ,Ai ) if (N > 1) and (d0 ≤ re ) then destroy the agent else relocate the agent at Snew with an initial step size of re end

Fig. 9 Procedure that destroys or relocates an agent. N is the number of currently existing agents. na is the maximum number of “exploring” agents. d is the dimension of the search space. Am is the archive of the local optima found by the agents. Ai is the archive of the last initial positions of the agents. re is the initial step size of “exploring” agents.

To prevent several agents from exploring the same zone of the search space, and to prevent them from converging to the same local optimum, an exclusion radius is attributed to each agent. This exclusion radius is the parameter re . Hence, if an agent detects one or several other agents at a distance lower than re , then only the agent with the best fitness, among the detected agents including the agent having detected them, is allowed to continue its search. All the other agents have to be relocated. 4.3 The relocation of the agents If an agent has found an optimum, then this optimum is transmitted to memory through the coordinator. Afterwards, the coordinator can either destroy the agent, or let the agent start a new local search at a given position. This decision is also taken for an agent that has been found too close to other agents (see Figure 2). This procedure is summarized in Figure 9. We can note that this procedure makes use of two archives: Am and Ai . The archive Am contains the saved optima, and its capacity is equal to a fixed value nm . We will see how this value is computed in subsection 4.5. The archive Ai saves the last nm initial positions of agents to be created or relocated. Each time a change in the objective function is detected, the archive Ai is cleared. This procedure produces a new position for the agent which has to be relocated, which is far from all the other agents, and from the solutions stored in Am and Ai , by using the newInitialSolution procedure (see Figure 10). This heuristic generates several random locations uniformly distributed in the search space, selects one of them and returns it. The selection mechanism of this heuristic is as follows. For each generated location, the distance to the closest location in the set of current solutions of the agents, and of solutions stored in Am and Ai , is calculated. Then, the generated location that has the greatest calculated


15

newInitialSolution Inputs: d,Am ,Ai local variables: S,d1 ,d2 ,ai ,Sc ,Si d0 ← −∞

repeat 10 × d times S ← a random solution uniformly chosen in the search space d1 ← +∞

for each agent ai do Sc ← the current solution of ai

d2 ← the distance between S and Sc if d2 < d1 then d1 ← d2 end

end for each Si ∈ (Am ∪ Ai ) do d2 ← the distance between S and Si if d2 < d1 then d1 ← d2 end end if d1 > d0 then d0 ← d1 Snew ← S

end end return {d0 ,Snew } Fig. 10 Procedure to generate an initial solution Snew for an agent. It also returns the distance d0 between this solution, and the closest one in the set of current solutions of the agents, and of solutions stored in Am (the archive of the found local optima) and Ai (the archive of the last initial positions of the agents). d is the dimension of the search space.

distance is selected. The number of generated locations has been empirically set to 10 × d, where d is the dimension of the search space. It is a compromise between the computational cost, and the accuracy of the heuristic. If the new position Snew for the agent is in an unexplored zone of the search space (if it is not too close to another agent, to an archived optimum or to an archived initial position), then the agent is relocated at Snew . Otherwise, the search space is considered saturated and the coordinator destroys the agent. The decision of destroying the agent is also taken if more than na agents exist. It happens if agents are created to track archived optima after the detection of a change in the objective function, as described in the next subsection.

16


addAgents Inputs: na ,nc ,N,m,Am ,rl ,re local variables: nt ,nn ,Obest nt ← max(0,min(na + nc − N,nc ,m))

nn ← max(0,min(na − N − nt ,m − nt ))

repeat nt times Obest ← the best optimum in Am Am ← Am − {Obest }

create an agent with initial solution Obest and initial step size rl end repeat nn times Obest ← the best optimum in Am Am ← Am − {Obest }

create an agent with initial solution Obest and initial step size re end

Fig. 11 Procedure to create additional agents after a change, to track the best archived optima and to make the number N of existing agents at least equal to na (the maximum number of “exploring” agents). max and min are functions that return the maximum and the minimum value among several given values, respectively. nc is the maximum number of “tracking” agents. m is the number of local optima currently stored in the archive Am . rl and re are the initial step sizes of “tracking” and “exploring” agents, respectively.

4.4 The change detection and the tracking of the optima The coordinator detects the changes in the environment. This detection is performed when all the agents have completed one step of their search procedure, i.e., when all the agents are in a CHECK POINT state (see subsection 3.4). Changes in the environment are detected by reevaluating the fitness of a randomly chosen agent or archived optimum, and comparing it to its previous value. If these values are different, a change is supposed to have occurred, and the following actions are taken: the fitnesses of all agents and archived optima are reevaluated; then, the procedure of the creation of additional agents (Figure 11) is executed. These additional agents are initialized using the best optima in Am as initial solutions. Each time an agent is created, the optimum used to initialize it is removed from Am . The maximum number of “tracking” agents to create (to track optima when a change is detected) is nc . After creating the “tracking” agents, if the number N of currently existing agents is lower than na , then at most na − N “exploring” agents are created.

4.5 Archive management The memory manager maintains the archive Am of local optima found by the agents. This archive must be bounded, its size is fixed to a number nm of entries. We propose the expression (4) to calculate the value of nm :


nm = round

d re

17

(4)

where the round function rounds a number to the nearest integer, re is the exclusion radius of the agents and d is the dimension of the search space. This expression was defined empirically. We introduce a flag isNotUpToDate that indicates if a change in the objective function occurred since the detection of a given stored optimum: if a change occurred, it returns true; otherwise, it returns f alse. If the archive is full, then we use the following conditions to update the archive: 1. If the new optimum, denoted by Oc , is better than the worst optimum in the archive, or its value is at least equal to the one of this worst optimum, then: (a) If there is one or several optima in the archive where isNotUpToDate returns true, then the worst of them is replaced by Oc ; (b) otherwise, the worst optimum of the archive is replaced by Oc . 2. If there is one or several optima in the archive that are “too close” to Oc (an archived optimum is considered “too close” to Oc if it lies at a distance from Oc lower or equal to the geometric average of rl and re ), then all these optima close to each other are considered to be dominated by the best of them. Thus, this subset of solutions is replaced by only their best one. The different steps of this replacement are in Figure 12.

replaceDominatedOptima Inputs: Oc ,Am ,rl ,re local variables: Asub ,Obest ,Oi Asub ← ∅

Obest ← Oc

for each Oi ∈ Am do√ if kOc − Oi k ≤ rl × re then Asub ← Asub ∪ {Oi }

if Oi is strictly better than Obest then Obest ← Oi end

end end Am ← Am − Asub

Am ← Am ∪ {Obest } return Am

Fig. 12 Procedure that replaces the dominated optima. Oc is the newly found optimum. Am is the archive of the local optima found by the agents. rl and re are the initial step sizes of “tracking” and “exploring” agents, respectively.

18


5 Benchmark sets It is common that real-world problems have time costly evaluations of fitness functions. Hence, the computational cost of a tested algorithm in most testbeds is expressed in terms of number of evaluations. It is the case for the benchmarks used in this paper: the time corresponds to the number of evaluations since the beginning of a benchmark execution. In dynamic environments, the goal is to quickly find and follow the global optimum over the time. Then, the measures used to evaluate the performance of dynamic optimization algorithms take into account this fact. In this section, the main two benchmarks in dynamic environments and their measures of performance are described.

5.1 The Moving Peaks Benchmark To date, the most commonly used benchmark for dynamic optimization is the Moving Peaks Benchmark (MPB) [4, 5]. MPB is a maximization problem that consists of a number of peaks that randomly vary their shape, position and height upon time. At any time, one of the local optima can become the new global optimum. The peaks change position every α evaluations, and α is called “time span”. They move by a fixed amount s (the change severity). More details about MPB are available in [5]. In order to evaluate the performance, the “offline error” is used. The offline error (oe) is defined in equation 5: ! 1 Nc 1 Ne ( j) ∗ ∗ oe = (5) ∑ Ne ( j) ∑ f j − f ji Nc j=1 i=1 where Nc is the total number of fitness landscape changes within a single experiment, Ne ( j) is the number of evaluations performed for the jth state of the landscape, f j∗ is the value of the optimal solution for the jth landscape, and f ji∗ is the current best fitness value found for the jth landscape. In [5], three sets of parameters, called scenarios, were proposed. It appears that the most commonly used set of parameters for MPB is scenario 2 (see Table 1), hence, it will be used in this paper.

5.2 The Generalized Dynamic Benchmark Generator The Generalized Dynamic Benchmark Generator (GDBG) is the second benchmark used in this paper, it is described in [18, 20]. It was provided for the CEC’2009 Special Session on Evolutionary Computation in Dynamic and Uncertain Environments. It is based on the Sphere, Rastrigin, Weierstrass, Griewank and Ackley test functions

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Parameter Number of peaks Dimension d Peak heights Peak widths Change cycle α Change severity s Height severity Width severity Correlation coefficient Number of changes Nc

19

Scenario 2 10 5 [30,70] [1,12] 5000 1 7 1 0 100

Table 1 MPB parameters in scenario 2. Parameter Dimension d (fixed) Dimension d (changed) Change cycle α Number of changes Nc

Value 10 [5,15] 10000 × d 60

Table 2 GDBG parameters used during the CEC’2009 competition.

that are commonly used in the literature. They are presented in detail in [20]. These functions were rotated, composed and combined to form six problems with different degrees of difficulty: F1 : rotation peak function (with 10 and 50 peaks) F2 : composition of Sphere’s function F3 : composition of Rastrigin’s function F4 : composition of Griewank’s function F5 : composition of Ackley’s function F6 : hybrid composition function A total of seven dynamic scenarios with different degrees of difficulty was proposed: T1 : small step change (a small displacement) T2 : large step change (a large displacement) T3 : random change (Gaussian displacement) T4 : chaotic change (logistic function) T5 : recurrent change (a periodic displacement) T6 : recurrent with noise (the same as above, but the optimum never returns exactly to the same point) T7 : changing the dimension of the problem The basic parameters of the benchmark are given in Table 2, where the change cycle corresponds, as for MPB, to the number of evaluations that makes a time span. There are 49 test cases that correspond to the combinations of the six problems with the seven change scenarios (indeed, function F1 is used twice, with 10 and 50

20


peaks respectively). For every test case, the tested algorithm is run several times. The number of runs of the tested algorithm is equal to 20 in our experiments. As defined in [20], the convergence graphs, showing the relative error ri (t) of the run with median performance for each problem, are also computed. For the maximization problem F1 , the formula used for ri (t) is defined in equation 6, and for the minimization problems F2 to F6 , it is defined in equation 7: ri (t) =

fi (t) fi∗ (t)

(6)

ri (t) =

fi∗ (t) fi (t)

(7)

where fi (t) is the value of the best found solution at time t since the last occurrence of a change, during the ith run of the tested algorithm, and fi∗ (t) is the value of the global optimum at time t. The marking scheme proposed by the authors of GDBG works as follows: a mark is calculated for each run of a test case, and the average value of this mark is denoted by mark pct . The sum of all marks mark pct gives a score that corresponds to the overall performance of the tested algorithm, denoted by op. The percentage of the mark of each test case in this final score is defined by a coefficient markmax . It is also the maximum value of mark pct that can be obtained by the tested algorithm on each test case. The authors of GDBG have set the values of the coefficients markmax such that the maximum value of op is equal to 100. This score is a measure of the performance of an algorithm in terms of both convergence speed and solution quality. It is based on the value of an approximated offline error, and on the value of the best relative error, calculated for each time span. More details about this marking scheme are given in [20].

6 Results and discussion In the following subsections, the parameter setting of the proposed algorithm is discussed. Then, an analysis of the computational complexity of MLSDO is presented. Afterwards, its sensitivity analysis is given, followed by an empirical analysis of its components. A statistical analysis is performed to determine the strategies of MLSDO that are useful to improve its performance. Then, we show how the balance between exploring and tracking agents is dynamically adapted. Finally, a convergence analysis of MLSDO and a comparison with competing algorithms are presented.

6.1 Parameter setting of MLSDO Table 3 summarizes the six parameters of MLSDO that the user has to define. In Table 3, the values given in the “MPB” column are suitable for the “Moving Peaks


21

Name re

Type real

Interval (0,1]

MPB 0.1

GDBG 0.1

Short description exclusion radius of the agents, and initial step size of “exploring” agents

rl

real

(0,re )

0.005

0.005

initial step size of “tracking” agents

δ ph

real

[0, δ pl ]

0.001

0.001

highest precision parameter of the stagnation criterion of the agents local searches

δ pl

real

[δ ph ,+∞]

1.5

1.5

lowest precision parameter of the stagnation criterion of the agents local searches

na

integer

[1,10]

1

5

maximum number of “exploring” agents

nc

integer

[0,20]

10

0

maximum number of “tracking” agents created after the detection of a change

Table 3 MLSDO parameter setting for MPB and GDBG.

Benchmark” (see subsection 5.1), and the ones given in the “GDBG” column are suitable for the “Generalized Dynamic Benchmark Generator” (see subsection 5.2). These values were fixed empirically, and used to perform all our experiments. Among several sets of values for the parameters, we selected the one that leads to the best performance. The first parameters in Table 3 (re , rl , δ ph and δ pl ) that share the same values for the two benchmarks are suitable to induce a good performance both for MPB and GDBG. However, we do not have the proof that they are optimal for other dynamic problems. The best performance on MPB (scenario 2) is obtained with na = 1. This can be explained by the characteristics of this scenario: the heights of the peaks are close during the first time span. Hence, there is no need to use several exploring agents in order to accelerate the discovery of a promising peak. Several agents would converge to local optima having a similar fitness, and the global convergence of the algorithm would be slowed down. However, having more than one exploring agent can lead to better performances for other problems, as it is the case for GDBG and for the problem used in subsection 6.3, based on the Rosenbrock function. Indeed, rather than performing sequentially a local search initialized in different areas of the search space, the use of several local searches performed in parallel can lead to a faster convergence to a good solution. For instance, if three agents are used in MLSDO, then at each iteration of the algorithm, each agent performs one step of its local search, i.e. one displacement from its current solution to a better one in its neighborhood (Figure 13 (c)). By contrast, if only one agent is used, sequentially, to perform these local searches, using the same initial solutions, then it has to complete one of these local searches before starting the next one (Figure 13 (a)). Hence, the parallel execution of the local searches can lead to a different convergence curve for MLSDO than the sequential one (Figures 13 (b) and (d)). If the fitness function is multimodal and composed of several peaks having different heights, then the average number of evaluations required to reach a good solution by the sequential execution of the local searches could be higher than by their parallel execution. It is especially the case if

22


3 Fitnness

Fitnness

3 2 1

2 1

0

500

1000 Evaluations

0

500

(a)

(b)

3 Fitnness

3 Fitnness

1000 Evaluations

2 1

2 1

0

500

1000 Evaluations

(c)

0

500

1000 Evaluations

(d)

Fig. 13 Effects of the parallel and sequential execution of the local searches on the convergence of MLSDO. (a) illustrates a possible evolution of the fitness of the current solution over the evaluations, for three local searches performed sequentially. (b) illustrates the convergence curve that corresponds to this sequential execution. (c) illustrates the evolution of the fitness of the current solution for a parallel execution of these local searches. The same initial solutions are used, and the successive iterations of the local searches are just interleaved. (d) illustrates the convergence curve that corresponds to this parallel execution.

significant improvements are made early on in an agent’s local search, with small improvements during final fine-tuning. This case is illustrated in Figure 13. In this simple maximization example, we assume that: – an agent needs 500 evaluations to complete its local search ; – it needs 50 evaluations to perform one step of its local search ; – the initial solutions of the local searches performed by the three agents are located in the attraction zone of different peaks having different heights ; – the most significant improvements are made at the beginning of the local searches of the agents, with small improvements afterwards. The offline performance is a performance measure defined as the average of f ∗ (t), where f ∗ (t) is the value of the best solution found at the t th evaluation since the last change in the objective function [5]. The offline performance after 1500 evaluations, for each possible order by which the local searches are initialized, is presented in Table 4. As it can be seen, the parallel strategy leads to a better offline performance for 4 initialization orders among the 6 ones, and to a better average offline performance. Then, we can say that depending on the width and height of the peaks, so depending on the nature of the problem, the parallel strategy may be better than the sequential one. For a problem with unknown characteristics, the parameter na can be fitted by testing its possible values from 1 to 10 and by keeping the best one.


Sequential Parallel

1,2,3 2.24 3.14

1,3,2 2.59 3.18

Initialization order 2,1,3 2,3,1 3,1,2 2.59 2.92 3.27 3.17 3.21 3.25

3,2,1 3.27 3.25

23 Average 2.81 3.20

Table 4 Offline performance for each possible order by which the initial solutions are taken to initialize the local searches (in the example illustrated in Figure 13, the initialization order is 2,1,3). Its average value over all initialization orders is also given.

The lowest precision parameter δ pl needs to be correctly adapted to the objective function and to its change severity. A low value for δ pl prevents the agents from widely exploring the search space in a fast changing environment, since they will spend too many evaluations on fine-tuning their current solution, and too few ones on exploring other zones of the search space. Hence, the lower the number of evaluations between two changes is, the higher the value of δ pl should be. This means that the compromise between a high precision of the found optima (intensification) and a wide exploration of the search space (diversification) needs to be in favor of diversification in fast changing environments. As we can see in the procedure in Figure 11, the initial step size of “tracking” agents is equal to rl , whereas the initial step size of “non-tracking” agents (the “exploring” ones) is equal to the exclusion radius re . The exclusion radius re should match the radius of the attraction zone of an optimum and rl has to be lower than re . In this case, a low initial step size for “tracking” agents is needed to track an optimum because a larger initial step size will allow the agent to leave the attraction zone of the tracked optimum, and begin exploring the search space elsewhere. The parameter rl has also to be well suited to the severity of the changes, since a too low value of rl in a fast changing environment requires a significant number of adaptations of the step size, and a waste of many fitness function evaluations. The parameter nc corresponds to the number of archived optima that need to be tracked at every change of the objective function. If the objective function changes strongly enough, and the positions of the optima can move to any random location in the search space, then nc tends to 0. On the contrary, if the changes are smooth enough, the tracking of the optima is possible, then, the value of nc corresponds to the number of promising optima to track. For GDBG, we fixed nc = 0. Thus, the archived local optima are not tracked but rather redetected using exploring agents. It means that the changes in the objective function are too strong for the use of tracking agents to be effective. Then, in order to achieve good performance for this benchmark, MLSDO has to quickly detect the new positions of the local optima, using exploring agents. If the objective function is completely and randomly modified after a change, no information can be used from the past states of the function. Hence, it is not possible to do better than a random restart of the search process. Then, to be efficient in such environment, we need indeed a fast convergence to the global optimum, or at least, to a good solution.

24


Process relocation of agents which are exploring the same zone of the search space

Complexity O(d 2 (na + nc )(na + nc + nm )) O(d(na + nc ))

execution of one step of the local search of each agent archiving the local optima found by the agents

O(d nm (na + nc ))

detection of a change in the objective function, and reevaluation of the solutions of the agents and of the archives, if a change is detected

O(na + nc + nm ) O(na + nc )

creation of tracking agents, if a change is detected relocation of the agents having their stopping criterion satisfied Total MLSDO complexity

O(d 2 (na + nc )(na + nc + nm )) O(d 2 (na + nc )(na + nc + nm ))

Table 5 Computational complexity analysis of MLSDO.

6.2 Computational complexity analysis of MLSDO All the procedures used in MLSDO are repeated between two changes in the objective function. Hence, the computational complexity of the algorithm is studied between two detections of a change. The processes performed between them are summarized in Table 5, with their computational complexity. The complexity of MLSDO, calculated by summing the complexities of these processes, is also given. Therefore, for a fixed set of parameters of MLSDO, and by substituting the expression of nm (equation (4)) into the expression of the complexity of MLSDO (in Table 5), we get a worst case computational complexity of O(d 3 ) for MLSDO, where d is the dimension of the problem. 6.3 Sensitivity analysis of MLSDO The sensitivity of MLSDO is studied by varying the value of one of its parameters, while the others are left unchanged. Each parameter is studied this way, by applying the algorithm on MPB (scenario 2) and on the Rosenbrock function in five dimensions (as defined in equation 8). The Rosenbrock function is used as a static minimization problem that admits only one global optimum equal to 0. We use this commonly used static test function in the analysis because the user may not know in advance if the problem to solve is static or dynamic. Hence, MLSDO should also be able to solve static functions, and the sensitivity of its parameters should also be studied for them.

f (x) =

d−1

∑

i=1

(1 − xi )2 + 100(xi+1 − x2i )2

(8)

where d is the number of dimensions, the search space is [−10, 10]d , x is a solution to be evaluated and xi is its ith coordinate. The values used for the unchanged parameters are the ones defined in Table 3 for MPB, and the ones defined in Table 6 for the Rosenbrock function, i.e. the

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Parameter Value

na 2

nc 0

rl 0.005

re 0.03

δ ph 1E−10

25

δ pl 1E−04

Table 6 MLSDO parameter setting for the Rosenbrock function (used for the unchanged parameters in the sensitivity analysis).

ones fixed empirically that lead to the best performances. The ranges of values for the varying parameters have to be sufficiently wide for the analysis. We choose the following ones: na ∈ [1, 10]; nc ∈ [0, 20]; rl ∈ [5E−4, 1E−2]; re ∈ [0.01, 0.23] for both problems, δ ph ∈ [3E−8, 5E−1]; δ pl ∈ [1E−3, 4E+0] for MPB, and δ ph ∈ [5E−11, 3E−6]; δ pl ∈ [3E−6, 3E−2] for the Rosenbrock function. Using MPB, the algorithm is stopped when 5 × 105 evaluations have been performed, and its performance is measured using the offline error (see subsection 5.1). Using the Rosenbrock function, it is stopped when the value of the best solution found by the algorithm falls below 0.01, and its performance is measured using the number of evaluations needed to satisfy this stopping criterion. We use this performance measurement on the Rosenbrock function because it is more important for a dynamic optimization algorithm to find the global optimum quickly than to find it accurately. For both test problems, the results are averaged over 100 runs, and illustrated in Figure 14 and Figure 15. As we can see on both problems, when the number of agents na becomes too high, the performance of MLSDO decreases. Best results are obtained using na = 1 on MPB and using na = 2 on the Rosenbrock function. Varying the value of nc does not perturb the performance of MLSDO on the Rosenbrock function, since it is a static function and thus, no tracking agent is used. However, on MPB, the performance of MLSDO increases with the value of nc . Thus, nc has to be high enough to let the algorithm track a sufficient number of local optima. It appears that a value greater or equal to 10 is fitted for MPB. On MPB, we can see that rl , re and δ pl have similar behavior, and that they can perturb the performance of MLSDO significantly if they are not correctly fitted. On the contrary, it appears that rl has no impact on the performance of MLSDO on the Rosenbrock function, and that re and δ pl do not highly perturb it. The behavior of δ ph is similar in both test cases, and it needs to be sufficiently low to obtain accurate results. The optimal values of the MLSDO parameters are problem dependent, but their behaviors appear to be smooth and unimodal. Hence, the MLSDO parameters do not need to be accurately fitted in order to obtain good results. Besides, in these figures, we can see that the parameter values in Tables 3 and 6 correspond to the lowest values of offline error (Figure 14) and to the lowest numbers of evaluations (Figure 14), i.e. the values that correspond to the best performances for these benchmarks, according to these figures. 6.4 Empirical analysis of the strategies used in MLSDO An evaluation of the components of the MLSDO algorithm is made, in order to justify their requirement for obtaining high quality results. This evaluation is performed on

26


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Fig. 14 Sensitivity on MPB. (a) represents the evolution of the offline error for several values of the parameter na . (b) represents it for nc . (c) represents it for rl . (d) represents it for re . (e) represents it for δ ph . (f) represents it for δ pl .

MPB (scenario 2, because it is the most used one among the three scenarios proposed in [5], see subsection 5.1) and on the problems F2 and F3 of the GDBG benchmark (based on the unimodal Sphere function and on the multimodal Rastrigin function, respectively). Using MPB, the maximum number of iterations is fixed to 5 × 105. It corresponds to 100 changes occurred in the objective function during each run. The resulting offline errors and standard deviations averaged over 100 runs are summa-


27

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Fig. 15 Sensitivity on the Rosenbrock function. (a) represents the evolution of the number of evaluations needed to find the global optimum (with a precision of 0.01) for several values of the parameter na . (b) represents it for nc . (c) represents it for rl . (d) represents it for re . (e) represents it for δ ph . (f) represents it for δ pl .

rized in the second column of Table 7, for several variants of MLSDO. Using F2 and F3 , the sum of the marks markrun , obtained for the seven change scenarios of GDBG, gives a score for each variant of MLSDO. These scores and their standard deviations averaged over 20 runs are summarized in the third and fourth columns of Table 7. In

28


Variant MLSDOnoCDPA

oe for MPB 0.35 ± 0.09

Score for F2 14.30 ± 0.19

Score for F3 3.41 ± 0.20

Short description no cumulative dot product adaptation of R

MLSDO

0.36 ± 0.08

13.93 ± 0.21

6.74 ± 0.20

the unmodified MLSDO algorithm

MLSDOnoExcl

0.42 ± 0.11

13.51 ± 0.24

6.74 ± 0.21

no exclusion radius for the agents

MLSDOno δ pl

0.59 ± 0.26

13.93 ± 0.22

5.88 ± 0.21

using only δ ph in the stopping criterion

MLSDOrandInit

0.61 ± 0.24

13.89 ± 0.23

6.74 ± 0.29

with uniform random initial solutions for the local searches of the agents

MLSDOno rl

0.85 ± 0.09

13.89 ± 0.23

6.68 ± 0.23

using re instead of rl as initial step size of “tracking” agents

MLSDOno δ ph

1.08 ± 0.07

12.46 ± 0.26

6.01 ± 0.21

using only δ pl in the stopping criterion

MLSDOnoDom

1.34 ± 0.44

13.83 ± 0.22

6.70 ± 0.17

without removing dominated optima from the archive of local optima

MLSDOnoTrack

4.08 ± 0.25

13.93 ± 0.25

6.73 ± 0.21

without agents

MLSDOnoArch

7.22 ± 0.34

13.84 ± 0.23

6.76 ± 0.25

without archiving local optima

creating

“tracking”

Table 7 Performance of each simplified variant of MLSDO for MPB and the problems F2 and F3 of GDBG.

this table, the variants are sorted from the best to the worst according to the offline error (oe) obtained for MPB. The Kruskal-Wallis statistical test has been applied on the results obtained by the variants of MLSDO. For MPB, as well as for the problems F2 and F3 , this test indicates at 99% confidence level that there is a significant difference between the performances of at least two variants. Then, the Tukey-Kramer multiple comparisons procedure has been used to determine which variants differ in terms of offline error for MPB, and in terms of score for F2 and F3 . The results obtained by the variants that perform significantly differently from the unmodified MLSDO algorithm, at 99% confidence level, appear in bold in this table. Among the simpler variants of MLSDO, we can note that MLSDOnoExcl does not perform a detection of other agents inside the exclusion radius re of an agent, i.e., an agent has not to start a new local search elsewhere when it is too close to another agent. Thus, all agents can explore the same zone of the search space. In MLSDOrandInit , the heuristic in Figure 10 is not used. This heuristic generates an initial solution for an agent that is far from the already explored zones of the search space. Hence, in MLSDOrandInit , the initial solutions of the local searches of the agents are randomly generated uniformly in the search space. In MLSDOnoCDPA , the cumulative dot product adaptation of the step size is not used, and the only adaptation process used is the reduction of R (when no better candidate solution can be found in the local landscape of an agent). In MLSDOno rl , the initial step size of a “tracking”


29

agent, created by the coordinator when a change is detected in the objective function (on the location of a previously found optimum), is not equal to rl but to re . Hence, the initial step size of a “tracking” agent is the same as the one of an “exploring” agent, in this variant. In MLSDOnoArch , the local optima found by the agents are not stored (Am is always empty). In MLSDOnoDom , the procedure in Figure 12 is not used. Thus, the local optima that are dominated by another one are not removed from Am . In MLSDOno δ pl , δ pl is replaced by δ ph , so that the precision of the stopping criterion is always equal to δ ph (the parameter δ pl is not used in this variant). The opposite replacement is done in MLSDOno δ ph , where the parameter δ ph is not used. Finally, we can also note that in MLSDOnoTrack , no “tracking” agent is created when a change is detected in the objective function (the parameter nc is not used).

In Table 7, the results obtained by MLSDOnoArch and MLSDOnoTrack are far worse than the ones of the other variants on MPB. Hence, we can conclude that the most important components of the MLSDO algorithm, in order to achieve a better result, are the archiving of found local optima and their tracking using dedicated agents. Besides, we can see that MLSDOnoArch performs significantly worse than MLSDOnoTrack . Indeed, in MLSDOnoArch , the found local optima are not archived. Thus, they can neither be tracked, nor be used by the heuristic in Figure 10 to generate initial solutions for the agents that are far from these found local optima. However, in MLSDOnoTrack , the local optima are not tracked, but they are used in the generation of the initial solutions of the agents. Thus, we can also conclude that the use of Am in this generation process is important to achieve a good performance. By comparing the results obtained by MLSDOno δ pl and MLSDOno δ ph , we can conclude that the use of the lowest precision parameter is less critical than the highest one, in order to achieve a better result. Hence, if we have only one of these parameters in the stopping criterion, it is better to set it to a low value in order to locate all the local optima with a high precision. However, the combined use of these two levels of precision is required to significantly improve the performance of MLSDO. Looking at the results obtained by MLSDOrandInit , MLSDOno rl and MLSDOnoDom , we can also conclude that an intelligent generation of the initial solutions for the agents, as well as the use of a dedicated initial step size rl for the “tracking” agents, and the removal of dominated optima from the archive of local optima, are important strategies in MLSDO. Finally, the statistical analysis shows that a significant difference exists between the unmodified MLSDO algorithm and all its variants except MLSDOnoCDPA and MLSDOnoExcl on MPB, at 99% confidence level as well as at 95%. Hence, we can conclude that all the tested components of MLSDO, except the cumulative dot product adaptation of the step size of an agent and the use of an exclusion radius, are useful to get good results on MPB. However, MLSDO performs significantly better than MLSDOnoCDPA for the problem F3 of GDBG, and significantly better than MLSDOnoExcl for the problem F2 . Hence, even if these strategies are not helpful for MPB, they can greatly improve the performance of MLSDO for other dynamic problems.

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Agents 6 4 2 0

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Fig. 16 Evolution of the number of exploring and tracking agents during the first 40000 evaluations of a run of MPB.

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t Fig. 17 Evolution of the number of solutions stored in the archive of the found local optima, during the first 40000 evaluations of a run of MPB.

6.5 Analysis of the balance between exploring and tracking agents The evolution of the number of exploring and tracking agents over the time is shown in Figure 16, for the first 40000 evaluations of a run of MLSDO on MPB (scenario 2). The evolution of the number of archived local optima is also shown in Figure 17, as it is linked to the number of tracking agents created after a change, i.e. the highest number of tracking agents created after a change is equal to the number of archived optima. As we can see, during the first time span, five local optima are found among ten. Then, the remaining ones are progressively detected. The tracking agents require few evaluations to locate the new position of the detected local optima. Hence, most evaluations of a time span of MPB are used to explore the search space, in order to find the local optima that remain to be detected. For the GDBG benchmark, the number of exploring agents stays equal to five, and no tracking agent is created during a run. Indeed, we only make use of exploring agents to detect the new position of the local optima.

6.6 Convergence analysis and comparison on MPB The convergence of MLSDO is studied in Figure 18, using the run with median performance among 100 runs on MPB. As one can see, the convergence of MLSDO on each time span is fast. The first time spans show the highest values, because MLSDO


31

Fig. 18 Relative error of the fitness on MPB. The axis y corresponds to the first 40 time spans, the axis x corresponds to the first 800 evaluations of a time span with a granularity of 20 evaluations, and the axis z is equal to the relative error scale for the axis z.

10

∗ fy∗ − fyx fy∗

, using the notations of equation 5. For more clarity, we used a logarithmic

-1

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Fig. 19 Average relative error of the fitness on MPB. x corresponds to the evaluations of a time span, and ∗ for j = 1,...,N . For more clarity, we used a logarithmic scale for f¯∗ axis. f¯x∗ is the average value of f jx c x

has not yet recorded the locations of each local optimum of the landscape. Once the local optima are found, the algorithm converges faster by tracking them, rather than redetecting them. The average evolution of the relative error of the fitness among all time spans is given in figure 19. As illustrated in this figure, it takes only 20 evaluations to get a relative error lower than 10−1 , and 431 evaluations to get a relative error lower than 10−2 . The comparison, on MPB, of MLSDO with the other leading optimization algorithms in dynamic environments is summarized in Table 8. These competing algorithms are the only ones that we found suitable for comparison in the literature, i.e., they are tested by their authors using the most commonly used set of MPB parameters (see Table 1). The offline errors and the standard deviations are given, and the algorithms are sorted in increasing order of offline error. Results are averaged over 50 runs of the tested algorithms. Results of competing algorithms are given in the

32

Julien Lepagnot et al. Algorithm Moser and Chiong, 2010 [27] MLSDO Novoa et al., 2009 [29] Lepagnot et al., 2009 [16, 17] Moser and Hendtlass, 2007 [27, 28] Yang and Li, 2010 [36] Liu et al., 2010 [23] Lung and Dumitrescu, 2007 [24] Bird and Li, 2007 [1] Lung and Dumitrescu, 2008 [25] Blackwell and Branke, 2006 [3] Mendes and Mohais, 2005 [26] Li et al., 2006 [22] Blackwell and Branke, 2004 [2] Parrott and Li, 2006 [31] Du and Li, 2008 [9]

Offline error 0.25 ± 0.08 0.36 ± 0.08 0.40 ± 0.04 0.59 ± 0.10 0.66 ± 0.20 1.06 ± 0.24 1.31 ± 0.06 1.38 ± 0.02 1.50 ± 0.08 1.53 ± 0.01 1.72 ± 0.06 1.75 ± 0.03 1.93 ± 0.06 2.16 ± 0.06 2.51 ± 0.09 4.02 ± 0.56

Table 8 Comparison with competing algorithms on MPB using standard settings (see Table 1). Algorithm MLSDO Moser and Chiong, 2010 [27] Lung and Dumitrescu, 2007 [24] Lepagnot et al., 2009 [16, 17] Moser and Hendtlass, 2007 [27, 28]

Offline error 14.00 ± 2.33 16.50 ± 5.40 24.60 ± 0.25 34.64 ± 2.72 480.5 ± 70.1

Table 9 Comparison with competing algorithms on MPB using d = 100.

references listed in the first column. As we can see, MLSDO is the second ranked algorithm in terms of offline error. The first ranked algorithm on MPB is the one proposed by Moser and Chiong in 2010, called Hybridised EO [27]. Though they differ in several points, both Hybridised EO and MLSDO restart local searches repeatedly in the search space, and store the found local optima in order to track them after a change. Compared to MLSDO, Hybridised EO is a “multi-phase” algorithm that does not perform several local searches in parallel. An interesting difference between them is that, in order to generate an initial solution for a local search in a promising area of the search space, Hybridised EO evaluates several candidate solutions and uses the best one as the initial solution of the local search. More precisely, it samples every dimension of the search space in equal distances. Thus, in order to start a local search in a d-dimensional space, it has first to evaluate 10 × d candidate solutions. Then, the question of the scalability of the algorithm regarding the number of dimensions of the problem arises. In comparison, MLSDO uses only the archived solutions to produce an initial solution for a local search, i.e. it does not evaluate any additional one. Then, to compare the performance of MLSDO with the other competing algorithms in the high dimensional case, the same comparison (on MPB, scenario 2) is made using d = 100, as summarized in Table 9. However, the numerical results using d = 100 of several algorithms of Table 8 are not available (they are not included in this comparison). The results of the algorithm proposed by Lung and Dumitrescu [24] are

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Algorithm Moser and Chiong, 2010 [27] MLSDO Lepagnot et al., 2009 [16, 17] Moser and Hendtlass, 2007 [27, 28] Yang and Li, 2010 [36] Liu et al., 2010 [23] Lung and Dumitrescu, 2007 [24] Bird and Li, 2007 [1] Lung and Dumitrescu, 2008 [25] Blackwell and Branke, 2006 [3] Li et al., 2006 [22] Parrott and Li, 2006 [31]

s=1 0.25 ± 0.08 0.36 ± 0.08 0.59 ± 0.10 0.66 ± 0.20 1.06 ± 0.24 1.31 ± 0.06 1.38 ± 0.02 1.50 ± 0.08 1.53 ± 0.01 1.75 ± 0.06 1.93 ± 0.08 2.51 ± 0.09

Offline error s=2 0.47 ± 0.12 0.60 ± 0.07 0.87 ± 0.12 0.86 ± 0.21 1.17 ± 0.22 1.98 ± 0.06 1.78 ± 0.02 1.87 ± 0.05 1.57 ± 0.01 2.40 ± 0.06 2.25 ± 0.09 3.78 ± 0.09

33

s=3 0.49 ± 0.12 0.92 ± 0.10 1.18 ± 0.13 0.94 ± 0.22 1.36 ± 0.28 2.21 ± 0.06 2.03 ± 0.03 2.40 ± 0.08 1.67 ± 0.01 3.00 ± 0.06 2.74 ± 0.09 4.96 ± 0.12

Table 10 Comparison with competing algorithms on MPB using s = 1,2,3.

taken from [27]. The results of the remaining competing algorithms are taken from their corresponding published papers. As we can see, MLSDO is the best ranked algorithm in terms of offline error, in this high dimensional case, whereas it was ranked at the second place using d = 5, i.e., MLSDO obtains a worse offline error than the algorithm proposed by Moser and Chiong [27] using d = 5, whereas it is the opposite using d = 100. It means that the performance of MLSDO scales well regarding the number of dimensions of the problem, compared to competing algorithms.

Finally, we compare the performance on MPB (scenario 2) of MLSDO with the competing algorithms that provide comparable numerical results, using higher change severities. It is made using s = 1, 2, ..., 6, as summarized in Tables 10 and 11. The results of the algorithm proposed by Li et al. [22] are from [23], and the results of the algorithm proposed by Parrott and Li [31] are from [1]. The results of the remaining competing algorithms are taken from their corresponding papers. It is known that the optima are more and more difficult to track with the increase of the change severity. Therefore, the performance of all algorithms degrades when the change severity increases. However, we can see that the performance of the following four algorithms does not degrade as much as the one of MLSDO: Moser and Chiong, 2010 [27]; Moser and Hendtlass, 2007 [27, 28]; Yang and Li, 2010 [36] and Lung and Dumitrescu, 2008 [25]. Hence, though MLSDO is ranked at the second place using s = 1, 2, 3, it is ranked at the third, fourth and fifth place for s = 4, 5, 6, respectively. This decrease in the performance of MLSDO is explained by the fact that the initial step size rl of the tracking agents has to be adapted to the severity of the changes, as stated in subsection 6.1. Therefore, better results can be obtained by adapting the value of rl to the change severity s, as shown in Table 12.

34

Julien Lepagnot et al. Algorithm Moser and Chiong, 2010 [27] MLSDO Lepagnot et al., 2009 [16, 17] Moser and Hendtlass, 2007 [27, 28] Yang and Li, 2010 [36] Liu et al., 2010 [23] Lung and Dumitrescu, 2007 [24] Bird and Li, 2007 [1] Lung and Dumitrescu, 2008 [25] Blackwell and Branke, 2006 [3] Li et al., 2006 [22] Parrott and Li, 2006 [31]

s=4 0.53 ± 0.13 1.22 ± 0.12 1.49 ± 0.13 0.97 ± 0.21 1.38 ± 0.29 2.61 ± 0.11 2.23 ± 0.05 2.90 ± 0.08 1.72 ± 0.03 3.59 ± 0.10 3.05 ± 0.10 5.56 ± 0.13

Offline error s=5 0.65 ± 0.19 1.62 ± 0.13 1.86 ± 0.17 1.05 ± 0.21 1.58 ± 0.32 3.20 ± 0.13 2.52 ± 0.06 3.25 ± 0.09 1.78 ± 0.06 4.24 ± 0.10 3.24 ± 0.11 6.76 ± 0.15

s=6 0.77 ± 0.24 2.01 ± 0.19 2.32 ± 0.18 1.09 ± 0.22 1.53 ± 0.29 3.93 ± 0.14 2.74 ± 0.10 3.86 ± 0.11 1.79 ± 0.03 4.79 ± 0.10 4.95 ± 0.13 7.68 ± 0.16

Table 11 Comparison with competing algorithms on MPB using s = 4,5,6. Value of s Value of rl Offline error

1 0.005 0.36 ± 0.08

2 0.010 0.54 ± 0.09

3 0.015 0.67 ± 0.09

4 0.019 0.84 ± 0.08

5 0.023 1.00 ± 0.12

6 0.027 1.38 ± 0.14

Table 12 Performance of MLSDO on MPB using s = 1,2,...,6. The parameter rl of MLSDO is empirically adapted to the value of s used. The other parameters of MLSDO are left unchanged to the values defined in subsection 6.1.

6.7 Convergence analysis and comparison on GDBG The experimental results on the GDBG benchmark are gathered in Table 13. An example of the convergence behavior of MLSDO on the problem F1 with 10 peaks is depicted in Figure 20, where t is the number of evaluations since the beginning of the run. In this figure, the relative error ri (t) is given for each change scenario Tk as ri (t) + k − 1, where 0 ≤ ri (t) ≤ 1, i is the run with median performance and k = 1, 2, ..., 7. As we can see, the curves of the relative error for all change scenarios are sharp. It means that MLSDO needs only few evaluations to converge to a better local optimum. To illustrate this fast convergence, a zoom on the first change in the objective function of this problem, for all change scenarios, is made on Figure 20. On the change scenarios T1 , T4 and T6 , the relative error reaches a value close to 1 during each time span. It means that the global optimum is found for all the time spans of these three scenarios. The average number of evaluations required to reach a relative error of 0.99 during a time span is equal to 2511 on T1 , 1596 on T4 and 5523 on T6 . Thus, MLSDO provides a fast convergence to the global optimum on these change scenarios. On the other scenarios, we can see that the global optimum is also found quickly for most time spans. As we can see in Table 13, MLSDO is able to produce reasonable solutions for all problems. Only for problem F3 , the proposed MLSDO algorithm could not find the optimum quick enough during dynamic changes. This is however a hard problem, and all competing algorithms have difficulties to solve it. The best mark obtained by

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Function Change F1 (10 peaks) T1 F1 (10 peaks) T2 F1 (10 peaks) T3 F1 (10 peaks) T4 F1 (10 peaks) T5 F1 (10 peaks) T6 F1 (10 peaks) T7 Sum for F1 (10 peaks) F1 (50 peaks) T1 F1 (50 peaks) T2 F1 (50 peaks) T3 F1 (50 peaks) T4 F1 (50 peaks) T5 F1 (50 peaks) T6 F1 (50 peaks) T7 Sum for F1 (50 peaks) F2 T1 F2 T2 F2 T3 F2 T4 F2 T5 F2 T6 F2 T7 Sum for F2 F3 T1 F3 T2 F3 T3 F3 T4 F3 T5 F3 T6 F3 T7 Sum for F3

markmax 1.5 1.5 1.5 1.5 1.5 1.5 1.0 10.0 1.5 1.5 1.5 1.5 1.5 1.5 1.0 10.0 2.4 2.4 2.4 2.4 2.4 2.4 1.6 16.0 2.4 2.4 2.4 2.4 2.4 2.4 1.6 16.0

markpct 1.493 1.455 1.428 1.496 1.464 1.481 0.966 9.78 1.490 1.433 1.386 1.481 1.469 1.473 0.934 9.67 2.277 1.876 1.940 2.348 1.762 2.258 1.469 13.93 1.913 0.461 0.821 1.250 0.652 0.920 0.724 6.74

Function Change F4 T1 F4 T2 F4 T3 F4 T4 F4 T5 F4 T6 F4 T7 Sum for F4 F5 T1 F5 T2 F5 T3 F5 T4 F5 T5 F5 T6 F5 T7 Sum for F5 F6 T1 F6 T2 F6 T3 F6 T4 F6 T5 F6 T6 F6 T7 Sum for F6 Overall performance op

35 markmax 2.4 2.4 2.4 2.4 2.4 2.4 1.6 16.0 2.4 2.4 2.4 2.4 2.4 2.4 1.6 16.0 2.4 2.4 2.4 2.4 2.4 2.4 1.6 16.0 100.0

markpct 2.261 1.808 1.863 2.273 1.681 2.226 1.427 13.54 2.280 2.276 2.278 2.288 2.268 2.272 1.520 15.18 1.945 1.792 1.763 1.891 1.935 1.832 1.277 12.44 81.28

Table 13 Performance measurement on GDBG.

competing algorithms on the problem F3 is 5.15 [7], and MLSDO obtains a better score of 6.74. Among the 49 test cases summarized in Table 13, MLSDO performs worst for the problem F3 using the change scenario T2 . It can be explained by the large displacements of the optima produced by this change scenario, inducing a major modification of the landscape of this difficult problem. MLSDO needs only few evaluations per time span to find a good solution. This is a major advantage in a fast changing environment. The results obtained by the competing algorithms on GDBG are summarized in Tables 14 and 15. For each problem of GDBG, the sum of mark pct obtained for the seven change scenarios is given in Table 14. To make the results obtained by the algorithms comparable, we use the same values of markmax for the problem F1 as for the other problems, i.e. for each problem, markmax = 1.6 for the change scenario T7 and markmax = 2.4 for the other change scenarios. Conversely, for each change scenario, the sum of mark pct obtained for all problems is given in Table 15. We use the same values of markmax for the change scenario

36

Julien Lepagnot et al. 7 6 5

ri (t)

4 3 2 1 0 0

1E6

2E6

3 3E6

4E6

5E6

6E6

t

7 6 5

ri ((t)

4 3 2 1 0 99000

99500

100000

100500

101 1000

101500

102000

102500

103000

t Fig. 20 Convergence graph for the GDBG problem F1 (10 peaks).

T7 as for the other scenarios, i.e. for each change scenario, markmax = 1.5 for the problem F1 (with 10 and 50 peaks) and markmax = 2.4 for the other problems. From Table 14, we can see that every algorithm performs best for the function F1 , and worst for the functions F3 and F6 , though the results obtained for F6 are better than for F3 . For the other functions, results are mitigated and we cannot conclude. Compared to the other algorithms, MLSDO obtains the best results for all functions except F2 and F4 . From Table 15, we can see that the algorithms perform best for the change scenarios T1 and T4 . For the other scenarios, results are mitigated and we cannot conclude. Compared to the other algorithms, MLSDO obtains the best results for all change scenarios.


37

Performance Algorithm MLSDO Lepagnot et al., 2009 [16, 17] Brest et al., 2009 [7] Korosec and Silc, 2009 [15] Yu and Suganthan, 2009 [38] Li and Yang, 2009 [19] França and Zuben, 2009 [10]

F1 (10 peaks) 15.65 15.64 15.07 14.66 13.13 14.45 12.22

F1 (50 peaks) 15.47 15.44 15.00 14.52 13.30 14.36 12.63

F2

F3

F4

F5

F6

13.93 14.65 10.91 11.25 10.64 10.64 8.55

6.74 0.20 5.15 3.36 4.37 1.70 0.16

13.54 14.12 10.78 9.90 10.31 9.47 6.72

15.18 13.26 14.16 13.50 8.95 10.30 4.17

12.44 9.10 9.94 8.96 7.31 7.46 3.15

Table 14 Performances of the competing algorithms for each problem of GDBG. In this table, the performance of an algorithm for a problem is calculated as the sum of mark pct obtained for the seven change scenarios, using markmax = 1.6 for T7 and markmax = 2.4 for the other change scenarios.

Algorithm MLSDO Lepagnot et al., 2009 [16, 17] Brest et al., 2009 [7] Korosec and Silc, 2009 [15] Yu and Suganthan, 2009 [38] Li and Yang, 2009 [19] França and Zuben, 2009 [10]

T1 13.66 11.28 12.57 11.46 8.58 9.78 8.12

T2 11.10 10.42 9.18 8.61 8.11 7.75 5.45

Performance T3 T4 T5 11.48 13.03 11.23 10.19 10.97 10.20 9.19 12.06 9.43 8.97 10.65 8.95 8.42 10.10 8.53 7.70 11.49 7.71 5.57 6.89 3.55

T6 12.46 10.66 10.58 10.02 8.89 7.78 4.96

T7 12.47 10.56 10.08 9.80 8.19 8.06 5.63

Table 15 Performances of the competing algorithms for each change scenario of GDBG. In this table, the performance of an algorithm for a change scenario is calculated as the sum of mark pct obtained for all problems, using markmax = 1.5 for F1 and markmax = 2.4 for the other problems.

100 81.28 80

70.76

69.73

65.21 58.09

op

60

57.57 38.29

40 20 0 MLSDO

Lepagnot Brest et al., Korosec Yu and Li and Yang, França and et al., 2009 2009 [6] and Silc, Suganthan, 2009 [18] Zuben, [15, 16] 2009 [14] 2009 [37] 2009 [9]

Fig. 21 Comparison with competing algorithms on GDBG.

The comparison, on GDBG, of MLSDO with the other leading optimization algorithms in dynamic environments is summarized in Figure 21. The algorithms are ranked according to their overall performance. As we can see, MLSDO is the first ranked algorithm on this benchmark.

38


7 Conclusion A new algorithm for DOPs has been proposed, called MLSDO. It has been developed in order to solve a wide range of DOPs. It is a cooperative search algorithm based on several coordinated local search agents, and on the archiving of the found local optima, in order to track them after a change in the objective function. It makes use of the main strategies and cooperative techniques proposed for DOPs in the literature, and implements them using inovative heuristics. The architecture and the main concepts of MLSDO have been presented, then its algorithms and their goals have been described in detail. Afterwards, the setting and the sensitivity of its parameters, its computational complexity, the efficiency of its strategies, the dynamic adaptation of the number of its exploring and tracking agents, and its convergence have been studied through an extensive experimental analysis and discussion. Comparisons with competing algorithms on the well known Moving Peaks Benchmark, as well as on the Generalized Dynamic Benchmark Generator, show the efficiency of the proposed algorithm. In works in progress, the MLSDO algorithm is applied to several real-world problems. Currently, we are working on its application to the segmentation, and to the registration of sequences of biomedical images. We plan to adapt MLSDO to dynamic combinatorial optimization. We would like also to make the critical parameters of MLSDO adaptive, i.e., such that they will be automatically adjusted. Finally, as many real-world DOPs are multiobjective, or have many constraints that can be dynamic, the proposed algorithm may also be adapted to the dynamic multiobjective optimization. We are pleased to share the source code of MLSDO and will make it publicly available at http://lissi.fr.

Appendix: Nomenclature of all the variables used Name α Ai Am ∆i δ ph

δ pl d D D′ Dn ei f (x) f (x,t) f ∗ (t)

Short description number of evaluations that makes a time span in the benchmarks archive of the last nm initial positions of agents that are created or relocated, in MLSDO archive of the local optima found by the agents, in MLSDO size of the interval that defines the search space on the ith axis in the “non-normalized” basis parameter of MLSDO that defines the highest precision parameter of the stagnation criterion of the agents local searches parameter of MLSDO that defines the lowest precision parameter of the stagnation criterion of the agents local searches dimension of the search space direction vector of the preceding displacement of an agent, in MLSDO direction vector of the current displacement of an agent, in MLSDO direction vector of the displacement of an agent at the nth step of its local search, in MLSDO ith unit vector of the “non-normalized” basis of the search space the objective function of a static optimization problem the objective function of a dynamic optimization problem value of the best solution found at the t th evaluation since the last change in the objective function

A Multiple Local Search Algorithm for Continuous Dynamic Optimization Name fi (t) fi∗ (t) f j∗ f ji∗ Fk f¯x∗ f itness gk (x,t) h j (x,t) isNotU pToDate m markmax mark pct max min N na nc Nc Ne ( j) nm Oc oe op R re ri (t) rl rnew round s Sc S′c Sprev Snew Snext Sw t Tk u U ui Un v x xi

39

Short description value of the best solution found at the t th evaluation of GDBG value of the global optimum at the t th evaluation of GDBG value of the global optimum for the jth time span in the benchmarks value of the best solution found at the ith evaluation of the jth time span of MPB kth problem of GDBG average relative error of the best fitness found at the xth evaluation of a time span of MPB function that returns the value of the objective function of a given solution, in MLSDO the kth inequality constraint of a dynamic optimization problem the jth equality constraint of a dynamic optimization problem flag that indicates if a change in the objective function occurred since the detection of a given stored optimum number of optima currently stored in the archive Am maximal mark that can be obtained on the considered test case of GDBG mark obtained on the considered test case of GDBG function that returns the maximum value among several given values function that returns the minimum value among several given values number of agents currently existing during the execution of MLSDO parameter of MLSDO that defines the maximum number of “exploring” agents parameter of MLSDO that defines the maximum number of “tracking” agents created after the detection of a change number of changes in the benchmarks evaluations performed on the jth time span of MPB capacity of the archives Ai and Am a newly found optimum offline error used in MPB overall performance used in GDBG step size of an agent of MLSDO parameter of MLSDO that defines the exclusion radius of the agents, and the initial step size of “exploring” agents relative error of the best fitness found at the t th evaluation of the ith run of GDBG parameter of MLSDO that defines the initial step size of “tracking” agents initial step size of an agent, in MLSDO (can be equal to either re or rl ) function that rounds a given number to the nearest integer change severity used in MPB current solution of an agent, in MLSDO best candidate solution of an agent at the current step of its local search, in MLSDO a candidate solution generated with Snext in the local search of an agent, in MLSDO the initial solution of the local search of an agent, in MLSDO a candidate solution generated with Sprev in the local search of an agent, in MLSDO worst candidate solution of an agent at the current step of its local search, in MLSDO number of evaluations performed since the beginning of the tested algorithm kth change scenario of GDBG number of equality constraints value of the cumulative dot product used to adapt the step size of an agent, in MLSDO ith vector of the “normalized” basis of the search space value of the cumulative dot product of an agent at the nth step of its local search, in MLSDO number of inequality constraints a solution in the search space of an optimization problem ith coordinate of the solution vector x

40


References 1. Bird S, Li X (2007) Using regression to improve local convergence. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Singapore, pp 592–599 2. Blackwell T, Branke J (2004) Multi-swarm optimization in dynamic environments. Lecture Notes in Computer Science 3005:489–500 3. Blackwell T, Branke J (2006) Multi-swarms, exclusion and anti-convergence in dynamic environments. IEEE Transactions on Evolutionary Computation 10(4):459–472 4. Branke J (1999) Memory enhanced evolutionary algorithms for changing optimization problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Washington, DC, USA, pp 1875–1882 5. Branke J (1999) The Moving Peaks Benchmark website. http://people.aifb.kit.edu/jbr/MovPeaks 6. Branke J, Kaußler T, Schmidt C, Schmeck H (2000) A multi-population approach to dynamic optimization problems. In: Proceedings of Adaptive Computing in Design and Manufacturing, Springer, Berlin, Germany, pp 299–308 7. Brest J, Zamuda A, Boskovic B, Maucec MS, Zumer V (2009) Dynamic optimization using selfadaptive differential evolution. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Trondheim, Norway, pp 415–422 8. Dréo J, Siarry P (2006) An ant colony algorithm aimed at dynamic continuous optimization. Applied Mathematics and Computation 181(1):457–467 9. Du W, Li B (2008) Multi-strategy ensemble particle swarm optimization for dynamic optimization. Information Sciences 178(15):3096–3109 10. de França FO, Zuben FJV (2009) A dynamic artificial immune algorithm applied to challenging benchmarking problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Trondheim, Norway, pp 423–430 11. Gardeux V, Chelouah R, Siarry P, Glover F (2009) Unidimensional search for solving continuous high-dimensional optimization problems. In: Proceedings of the IEEE International Conference on Intelligent Systems Design and Applications, IEEE, Pisa, Italy, pp 1096–1101 12. Gonzalez JR, Masegosa AD, Garcia IJ (2010) A cooperative strategy for solving dynamic optimization problems. Memetic Computing 3(1):3–14 13. Jin Y, Branke J (2005) Evolutionary optimization in uncertain environments – a survey. IEEE Transactions on Evolutionary Computation 9(3):303–317 14. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks IV, IEEE, Perth, Australia, pp 1942–1948 15. Korosec P, Silc J (2009) The differential ant-stigmergy algorithm applied to dynamic optimization problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Trondheim, Norway, pp 407–414 16. Lepagnot J, Nakib A, Oulhadj H, Siarry P (2009) Performance analysis of MADO dynamic optimization algorithm. In: Proceedings of the IEEE International Conference on Intelligent Systems Design and Applications, IEEE, Pisa, Italy, pp 37–42 17. Lepagnot J, Nakib A, Oulhadj H, Siarry P (2010) A new multiagent algorithm for dynamic continuous optimization. International Journal of Applied Metaheuristic Computing 1(1):16–38 18. Li C, Yang S (2008) A generalized approach to construct benchmark problems for dynamic optimization. In: Proceedings of the 7th International Conference on Simulated Evolution and Learning, Springer, Melbourne, Australia, pp 391–400 19. Li C, Yang S (2009) A clustering particle swarm optimizer for dynamic optimization. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Trondheim, Norway, pp 439–446 20. Li C, Yang S, Nguyen TT, Yu EL, Yao X, Jin Y, Beyer HG, Suganthan PN (2008) Benchmark generator for CEC 2009 competition on dynamic optimization. Tech. rep., University of Leicester, University of Birmingham, Nanyang Technological University 21. Li X (2004) Adaptively choosing neighbourhood bests using species in a particle swarm optimizer for multimodal function optimization. In: Proceedings of the Genetic and Evolutionary Computation Conference, Springer, Seattle, Washington, USA, pp 105–116 22. Li X, Branke J, Blackwell T (2006) Particle swarm with speciation and adaptation in a dynamic environment. In: Proceedings of the Genetic and Evolutionary Computation Conference, ACM, Seattle, Washington, USA, pp 51–58


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23. Liu L, Yang S, Wang D (2010) Particle swarm optimization with composite particles in dynamic environments. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 40(6):1634– 1648 24. Lung RI, Dumitrescu D (2007) Collaborative evolutionary swarm optimization with a Gauss chaotic sequence generator. Innovations in Hybrid Intelligent Systems 44:207–214 25. Lung RI, Dumitrescu D (2008) ESCA: A new evolutionary-swarm cooperative algorithm. Studies in Computational Intelligence 129:105–114 26. Mendes R, Mohais A (2005) DynDE: A differential evolution for dynamic optimization problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Edinburgh, Scotland, pp 2808–2815 27. Moser I, Chiong R (2010) Dynamic function optimisation with hybridised extremal dynamics. Memetic Computing 2(2):137–148 28. Moser I, Hendtlass T (2007) A simple and efficient multi-component algorithm for solving dynamic function optimisation problems. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Singapore, pp 252–259 29. Novoa P, Pelta DA, Cruz C, del Amo IG (2009) Controlling particle trajectories in a multi-swarm approach for dynamic optimization problems. In: Proceedings of the International Work-conference on the Interplay between Natural and Artificial Computation, Springer, Santiago de Compostela, Spain, pp 285–294 30. Parrott D, Li X (2004) A particle swarm model for tracking multiple peaks in a dynamic environment using speciation. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, San Diego, California, USA, pp 98–103 31. Parrott D, Li X (2006) Locating and tracking multiple dynamic optima by a particle swarm model using speciation. IEEE Transactions on Evolutionary Computation 10(4):440–458 32. Pelta D, Cruz C, Gonzalez JR (2009) A study on diversity and cooperation in a multiagent strategy for dynamic optimization problems. International Journal of Intelligent Systems 24(7):844–861 33. Pelta D, Cruz C, Verdegay JL (2009) Simple control rules in a cooperative system for dynamic optimisation problems. International Journal of General Systems 38(7):701–717 34. Tfaili W, Siarry P (2008) A new charged ant colony algorithm for continuous dynamic optimization. Applied Mathematics and Computation 197(2):604–613 35. Wang H, Wang D, Yang S (2009) A memetic algorithm with adaptive hill climbing strategy for dynamic optimization problems. Soft Computing 13(8-9):763–780 36. Yang S, Li C (2010) A clustering particle swarm optimizer for locating and tracking multiple optima in dynamic environments. IEEE Transactions on Evolutionary Computation 14(6):959–974 37. Yang S, Yao X (2008) Population-based incremental learning with associative memory for dynamic environments. IEEE Transactions on Evolutionary Computation 12(5):542–562 38. Yu EL, Suganthan P (2009) Evolutionary programming with ensemble of external memories for dynamic optimization. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE, Trondheim, Norway, pp 431–438 39. Zeng S, Shi H, Kang L, Ding L (2007) Orthogonal Dynamic Hill Climbing Algorithm: ODHC. In: Yang S, Ong YS, Jin Y (eds) Evolutionary Computation in Dynamic and Uncertain Environments, Springer, pp 79–104

Appendix B

Prediction based on fractional integration This article was published in IEEE Mobile Computing @articleDBLP:journals/tmc/NakibDDS14, author = Amir Nakib and Boubaker Daachi and Mustapha Dakkak and Patrick Siarry, title = Mobile Tracking Based on Fractional Integration, journal = IEEE Trans. Mob. Comput., volume = 13, number = 10, pages = 2306–2319, year = 2014,

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IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 13, NO. 10, OCTOBER 2014

Mobile Tracking Based on Fractional Integration Amir Nakib, Boubaker Daachi, Mustapha Dakkak, and Patrick Siarry Abstract—While the static indoor geo-location of mobile terminals (MTs) has been extensively studied in the last decade, the prediction of the trajectory of an MT is still a major problem when designing mobile location (tracking) systems (TSs). In fact, Global Positioning System (GPS) works quite well in outdoor conditions and relatively unobstructed spaces, but falls short in many urban conditions and other realistic use cases. It is important to augment mobile geo-location architectures with a prediction dimension to deal with distortions caused by obstacles, and ultimately produce a more accurate positioning system. Different prediction approaches have been proposed in the literature, the most common is based on prediction filters such as linear predictors (LPs), Kalman filters (KFs), and particle filters (PFs). In this paper, we take the prediction one step further by using digital fractional integration (DFI) to predict the actual trajectory of MTs. We evaluate the performance of our proposed DFI prediction in two indoor trajectory scenarios inspired by typical user mobility patterns in typical indoor conditions (museum visit and hospital doctor walk). To illustrate the efficiency of the proposed method in particularly noisy environments, we consider two other MT trajectory scenarios, namely spiral and sinusoidal trajectories. Experimental results show a significant performance improvement over most common predictors in the relevant literature, particularly in noisy cases. Extensive study of short-archive principle using 5, 10, and 25 previous estimated positions, showed the benefit of using DFI operator with only the most recent locations of an MT. Index Terms—Indoor location, path tracking, digital fractional integration, prediction filters, mobile location, Kalman filter

1

I NTRODUCTION

T

HE

popularity of location-based services and embedded micro-sensing technologies has greatly promoted the development of several services. The availability of accurate users’ geo-location information has indeed many promising commercial applications [12], [50]. Despite the fact that there has been a substantial amount of research in indoor location architectures [8], [12], [16], [28], there are still some well-known issues related to the accuracy in a noisy environment. At the core of indoor positioning systems is the tracking systems (TSs), which allows for various applications in individual navigation, social networking, asset management, traffic management, or mobile resource management, etc. There are different tracking techniques are applied to construct TSs in indoor environments where the signal is noisy, weak or non-existing [24], [38], [53]. In this paper, we propose a new technique to enhance the performance of the classical MT tracking in an indoor environment. We particularly take advantage of properties of the digital fractional integration (DFI) to enhance the performance of the classical techniques such as linear predictors and Kalman filters. Indeed, these methods are not efficient when environment is closed with many obstacles and, when multiple targets need to be tracked. Those cases are usual in indoor geo-location problems. Then, the geo-location process becomes less accurate, • The authors are with Laboratoire Images, Signaux et Systèmes Intelligents (LISSI, E. A. 3956), Université de Paris-Est Créteil, Créteil 94010, France. E-mail: {nakib, daachi, siarry}@u-pec.fr; [email protected]. Manuscript received 16 Aug. 2012; revised 15 Jan. 2013; accepted 14 Feb. 2013. Date of publication 10 Mar. 2013; date of current version 26 Aug. 2014. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference the Digital Object Identifier below. Digital Object Identifier 10.1109/TMC.2013.37

and the achieved position may be insufficient for many applications. The outline of this paper is as follows. In the next section, we state the problem dealt in this paper and present the related work. In Section 3, the formalism of fractional integration is presented. Section 4 is addressed to explain the properties of a fractionally integrated trajectory. Proposed method is exposed in Section 5, while the experimental results, discussion, and comparison study are presented in Section 6. We conclude this paper in Section 7.

2

P ROBLEM S TATEMENT AND R ELATED W ORK

Tracking scenario consists of sensors, which produce noisy measurements, such as azimuth angle. The purpose of a tracking algorithm is to determine the target trajectory using sensor measurements. There is additional prior information on the dynamics of targets, which narrows the prediction domain of target trajectories to keep only those that are possible when the laws of physics are taken into account. Additionally, the actual tracking sensors may have different accuracies, which will lead to different types of measurements. Obviously, a larger number of sensors will improve the estimation process as more information on the same target trajectory can be relied on during trajectory predictions. In case of multiple targets there is an additional difficulty. Without additional information one cannot determine which measurements corresponds to which target. This is known in the literature as the problem of data association. The same problem applies to false alarm or clutter measurements as it is difficult to associate a given measurement to a target MT. One can use attribute measurements for inferring properties of the targets. In this paper, we do not deal with this problem and assume that the data association

c 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 1536-1233 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

NAKIB ET AL.: MOBILE TRACKING BASED ON FRACTIONAL INTEGRATION

problem is solved. For more details on this issue, please refer to [7], [34], [56]. Besides, predicting or filtering in this context is equivalent to optimal state estimation in dynamic environment. The aim is to use indirect noisy measurements to estimate hidden signal consisting of sequence of states. The prediction is optimal in the sense that it minimizes the expected estimation error. In case of multiple targets, minimizing the estimation error also requires solving the problem of data association with or without possible help of attribute measurements. Estimation is done on-line, which means that updated state estimate is available as soon as new measurements are obtained. In Bayesian sense filtering, this is equivalent to a recursive estimation of posterior distribution of states. Clearly, the tracking problem is a form of filtering where the hidden signal consists of states of moving or stationary targets (position, velocity, etc.). Therefore, the purpose of tracking is to estimate the position of these targets as accurately as possible using only indirect and noisy measurements such as azimuths, distances and elevations. In addition to these kinematic measurements, there are also attribute measurements that are most useful in the process of MT-measurement association by providing information on types of targets in area. In order to achieve a high-level of accuracy, the prediction algorithm should be able to track all existing targets, and clutter insensitive such that there are no extra targets in addition to existing ones. In practice, the location systems can be classified into two main categories: static or memory-less positioning systems and dynamic positioning (tracking) systems (DTSs). In the former approach, it is assumed that the MT remains stationary [1], [6], [25], [31], [45], [47], [57], [58], while the latter approach relies on the correlation of MT positions over the time [13], [30], [33], [46], [55]. On the static MT positioning, there are several approaches have been used to estimate using the current measurement. A first class of approaches contains geometric (rangebased) approaches which depend on the propagation time, as Time Of Arrival (TOA), Time Difference Of Arrival (TDOA), or Round Trip Time (RTT) and/or Angle Of Arrival (AOA), to calculate the distance between MTs and the WLAN access points (APs). Afterwards, the MT position is estimated by applying triangulation or trilateration method [59]. The second class of approaches to MT positioning, called range-free approaches, is based on the nearest neighbor [1], or probabilistic [58], and pattern recognition techniques [4], [5], [14]. On the other hand, Kalman Filters and Particle Filters are commonly used in the domain for localization [10], [11], [17], [19], [29] as real-time linear predictors. A summarized version of this work was presented in [9]. The particle filters can be seen as the best alternative rather than the classical approach using the model-based Kalman filter techniques [18], [26]. Linear predictor and Kalman filtering methods underperform when the state domain is restricted by the presence of obstacles and when multiple hypotheses need to be tracked. Both of these situations are common in indoor localization problems. In this case, the prediction process becomes more approximate, and the achieved accuracy may be insufficient for critical

2307

applications. In [32], authors introduce the Nonparametric Information (NI) filter, a novel state-space Bayesian filter for RSS-based tracking in indoor WLANs. The filter provides a recursive position estimate over time by fusing information from RSS measurements and a motion model. Authors propose an intelligent tracking system that mimics the characteristics of a cognitive dynamic system. Cognitive dynamic systems learn rules of behavior through interactions with the environment to deal with uncertainties in a robust and reliable manner [22]. Advantages of cognitive systems are illustrated by the cognitive radio [20] and radar systems [21]. In this paper, we propose to enhance the performance of the linear prediction filter and Kalman filters to allow their use in an indoor environment. In order to have an idea about the degree of predictability of a path, it is common to use the autocorrelation function as metric to measure the redundancy and the relationship between samples of a signal. The autocorrelation refers to the correlation of a path with its own past and future values. Moreover, autocorrelated path are probabilistically predictable because future values depend on current and past values. Then, a long memory path is characterized by a correlation function which decreases slowly when the lag increases, and viceversa. From a geometric point of view, the regularity can be also used to characterize the predictability of a path. So, it is easy to see that a regular path that present a smooth curvature can be predicted with a good accuracy. Consequently, to enhance the performance of a predictor, one has just to increase its autocorrelation and the regularity of the path at its input. To do so, we propose to use the Digital Fractional Integration based filter. We propose to use the Digital Fractional Integration based filter to enhance the performance of a predictor, and to increase its autocorrelation. A remarkable merit of fractional differentiation operators is that they may still be applied to functions that are not differentiable in the classical sense. Unlike the integer order derivative, the fractional order derivative at point x is not determined by an arbitrary small neighborhood of x. In other words, the fractional derivative is not a local property of the function. There are several well-known approaches to unification of differentiation and integration notions, and their extension to non-integer orders [48]. A general survey on the different approaches is given in [37], [41], [51]. The theory of fractional integrals was primarily developed as a theoretical field of mathematics. More recently, fractional integration has found applications in various areas: in control theory it is used for path planning [2], [36], [40], [42]–[44]; it is also used to solve the inverse heat conduction problem [3]; other applications are reported for instance in modeling using fractal networks [49], in image processing for edge detection [35], in biomedical signal processing [15], [52], in thermal systems [54] and in biological tissues [23], [52].

3

F RACTIONAL I NTEGRATION

3.1 Definition of Fractional Calculus We consider a fractional integral operator that has the function y(t) as output and x(t) as the input function. Then, the

2308


equation defining this operator is: α

y(t) = J x(t),

(1)

where, J is the fractional integration operator and α is the fractional integration order. The word fractional is used for historical reasons and means any real, imaginary, or complex number. In the case of fractional integration the real part of α is negative. We denote the α th integral operator by Jα . In this paper, we recall the definitions introduced in [39] to make this paper self-consistent. Then, the operator Jα proposed by Riemann-Liouville for fractional differentiation is defined by the formula: 1 J x(t) = (α) α

t

(t − ξ )α−1 x(ξ ) dξ ,

(2)

where: x(t) is a real function, t > 0, Re(α) > 0, c the integral reference and the Euler-gamma function. Whereas the fractional differentiation Dα (the left sided fractional integration or the inverse operation) is given as: Dα = Dm Jm−α , m − 1 < α ≤ m, m ∈ N,

(3)

where Dm denotes the ordinary derivative of order m. We also define: J0 = D0 = I,

(4)

where I is the identity operator. In particular, we have: t x(ξ ) dξ.

We note the semi-group property: (6)

(7)

Jα Dα x(t) = x(t).

(8)

that means

3.2 Discrete Form: Grünwald Approach The first derivative of a function x(t) is given by: x(t) − x(t − h) . h

(9)

Using a finite sampling step h of the time t, i.e. t = Kh, leads to: x(Kh) − x((K − 1)h) D1 x(t) = . (10) h Introducing the delay operator q−1 , defined by: q−1 x(Kh), we can write: 1 − q−1 x(Kh). h The same calculus is achieved at the order 2: 2 1 − q−1 D2 x(t) = x(Kh). h2 D1 x(t) =

Dα x(t) =

K α(α − 1) . . . (α − k + 1) 1 (−1)k x(t − kh), (16) α h k! k=0

where K is as defined before. It is obvious to see that when a negative real part for the fractional order of differentiation α is chosen, the fractional integral (2) can be computed. Definition (16) shows that the fractional integral of a function takes into account the past of the function x(n). For more details about the definition of the fractional integration, we suggest to the reader to see [39], [48].

3.3 Computation of the Coefficients To implement the fractional differentiation, the computation of its coefficients is necessary: α(α − 1)....(α − k + 1) . (17) k! One of the possible approaches is to use the recurrence relationships: (α) ω0 = 1

(18) (α) (α) ωk = 1 − α+1 ωk−1 , k = 1, 2, 3, . . . , K. k (α)

Jα Dα = I

h→0

As x(t) = 0 fot t < 0, one has x(t − kh) = 0 for t − kh < 0, the finite sum in the previous equation can be reduced to:

ωk

and the expression for α ≥ 0:

D1 x(t) = lim

= x((K − k)h) = x(t − kh), another represenSince tation of the fractional derivative is: ∞ α(α − 1) . . . (α − k + 1) 1 (−1)k Dα x(t) = α x(t − kh). (15) h k!

(5)

c

Jα Jβ = Jα+β for α ≥ 0, β ≥ 0

k=0

q−k x(Kh)

k=0

c

Jx(t) =

The generalization to any order (real or complex) is immediate: α 1 − q−1 Dα x(t) = x(Kh), (13) hα where α is real or complex. Developping (1 − q−1 )α from the Newton binomial formula gives: ∞ 1 α k α(α − 1) . . . (α − k + 1) D x(t) = α (−1) x(Kh). (14) h k!

(11)

= (−1)k

In the case of a fixed value of α, this approach allows to create an array of coefficients that can be used to differentiate (integrate) various functions. However, in some problems the most suited value of α must be found: various values of α are considered, and for each particular value of α the (α) coefficients ωk must be computed separately. In such case the fast Fourier transform method can be used [39], [48]. (α) The coefficients ωk can be considered as the coefficients of the power series for the function (1 − z)α : ∞ ∞ α k (α) k (−1)k ωk z , (19) z = (1 − z)α = k k=0

(12)

k=0

where z = q−1 is the delay operator. Substituting z = e−iϕ , we have: 1 − e−iϕ

α

=

∞ k=0

(α)

ωk e−ikϕ

(20)


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Fig. 1. Illustration of the application of the DFI to X coordinates of the museum path. (a) Original X coordinates X (n). (b) DFI of X (n) path with α = −0.1. (c) DFI of X (n) path with α = −0.3. (d) DFI of X (n) path with α = −0.5. Where, n is the number of the samples and the distance between the points is expressed in meter. (α)

and the coefficients ωk Fourier transform:

are expressed in terms of the

2π 1 fα (ϕ)eikϕ dϕ, (21) 2πi 0 α where fα (ϕ) = 1 − e−iϕ . Then, any technical implementation of fast Fourier transform can be used to compute the (α) coefficients ωk . (α)

ωk

4

=

P ROPERTIES OF A F RACTIONALLY I NTEGRATED PATH

In this section the properties of the fractionally integrated path are studied. We start by the geometric properties in terms of regularity, then the statistical properties are presented.

4.1 Regularity Indeed, the regularity of the path provides information on how the path is varying over the time: a path that has low regularity means that the mobile is changing its direction frequently and it would be hard to predict its position. On the contrary, a mobile path that is highly regular means that the mobile does not often change its direction, then, its path would be easier to track (predict). In this paragraph, we show that the application of the DFI with a negative order allows enhancing the regularity of a mobile path and, consequently, enhancing its predictability. To illustrate our proposition, the modifications of the regularity at the application of the fractional integration on a positive function (a path) with negative DFI order, we consider an example of a given path (Fig. 1(a)), where the different coordinates (positions or measures) are positive. Now let us see what happens when we differentiate fractionally the path. So,

three different orders: −0.1, −0.3, and −0.5 are considered and, the corresponding results are presented in Fig. 1(b), (c), and (d), respectively. One can see that the path becomes smooth when the absolute value of the DFI order increases. In Fig. 1(b) the path is very smooth (regular) and tends to be linear. However, in Fig. 1(c) and d we can remark that it is smoother (regular). Then, we can say that for a given path an optimal DFI order exists where the path becomes regular and can be predicted more accurately. In practice, the path is a two dimensional function, then, we consider each dimension separately (two separated functions): the X coordinates function, called X(n) and the Y coordinates function, called Y(n). The example of Fig. 2(a) illustrates a museum visit path and the Fig. 2(b) presents the final result of applying the digital fractional integration (DFI) to both dimensions (XandY) of considered path with α = −0.5 is presented in Fig. 1(b). One can see that the path is totally modified and smoother than the original path. The application of the fractional integration with negative order (differentiation) provides a compressed or regular path, where the amplitude range is decreasing with the increase of the absolute value of α. On the contrary, as it can be derived from the property of the differentiated path given in expression (8) that, when α is positive, the amplitude range of the integrated function increases. This property can also be explained from the frequency properties of the fractional differentiation filter point of view.

4.2

Statistical Analysis of the Fractionally Integrated Path In this subsection, we present the statistical effect of the application of the fractional integration on a given

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Fig. 2. Illustration of the application of the DFI to a path. (a) Orginal museum visit path. (b) DFI of the museum path with α = −0.5. Where, n is the number of samples and the distance between the points is expressed in meter.

path. Indeed, we know that theoretically the fractional integration is reversible but in the discrete case, some errors can appear due to the approximation. To do so, we studied the average value of the fractionally differentiated path. Then, as pointed out before, we analyze the autocorrelation coefficients of the fractionally differentiated path to show their increase with the application of the DFI.

4.2.1 Average Value of the Integrated Function The Z− transform function of the discrete filter given in (16) can be written as: G(z) = (1 − z)α . H(z) = F(z)

(22)

From (22), we can easily show that the average value of the output path, i.e. the integrated function, is given by:

position of an MT. We define the predictor at hand by:

X(k) =

N

ai X(k − i),

where ai are predictor coefficients, X(i) is the position at the ˆ time i of the MT, and X(i) is the estimated position. To predict the position X(k), the predictor uses a linear combination of the N previous positions. The parameters ai , are called the prediction coefficients and 1 ≤ i ≤ N. Then, the prediction error is defined by: e(k) = X(k) − X(k).

(23)

where μf is the average value of the function f . The value of μg is infinite when the DFI order is positive. Consquently, to keep constant the average value of the path, the original trajectory is centered before applying the DFI.

4.2.2 Autocorrelation Proposition 1. The application to a function of the DFI with a negative order increases the autocorrelation of a function and consequently reduces the prediction error. The opposite case happens when a DFI with a positive order is applied to this function. As the performance of the Linear predictor is directly related to the correlation coefficient below we show the proof of our proposition in that case. However, in the case of KF and non-linear filters their principles are different, then, we proof that the DFI allows to increase the autocorrelation, that means increasing the relationship between the different past measures (positions) of given path. We took in example a Gaussian white noise as path. Proof in the case of Linear predictor: In order to proof this proposition we consider the case of a linear predictor of order N that we use to predict the

(25)

The variance of the prediction error e(k) is defined by: σe2 =

N

ai aj X (i − j)+

i,j=0

μg = H(0) × μf ,

(24)

i=1

N

ai σ 2ˆ ,

i=1

X

(26)

where X (k) is the autocorrelation function of the path X(k). It is obvious that if the original path P(k) is not totally random and the coefficients ai are correctly chosen, the variance of the error e(k) is lower than that of the original trajectory. The coefficients ai minimizing the mean square error σe2 = E (X − X)2 , are given by the equation system: δσe2 = aj X (i − j) + σ 2ˆ = 0, X δai N

(27)

j=0

where i = 1, 2, . . . , N and the variance σ 2ˆ is unknown at X the beginning, because it depends on the prediction error 2 variance σe . In practice this equation system will be solved by successive approximations. Using (27) in (26), the mini2 mal value of the residual variance σe2 , denoted by σe,min , is given by the following equation: 2 σe,min = σX2 −

N

ai X (i).

(28)

i=1

Then, it is easy to find the following expression (29): N 2 = σX2 1 − ai ρX (i) , (29) σe,min i=1


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Fig. 3. Principle of applying DFI to classical prediction filters.

where σX2 is the variance of the path X(n) and ρX (i) are its autocorrelation coefficients. The ratio of the variances is expressed by the relation: −1 N σ2 GX = 2 X = 1 − ai ρX (i) . (30) σe,min i=1 The prediction gain expresses the achieved variance reduction by linear prediction of order N. Then, to prove that the DFI of the path X(n) is more correlated than the original path, one must prove that the prediction gain GDα X of DFI of X(n) is greater than GX . To do so, it is sufficient to demonstrate that the autocorrelation coefficients of the fractionally integrated path, with negative order, are greater than those of X(n). We define the DFI of X(n) by: Dα X(k) =

N

ωjα X(k − j)

(31)

j=0

and its autocorrelation coefficients by: ρDα X(k) = Dα X(k) × Dαk+j X(k + j).

where E {.} is expected value operator given by: E {x(n)} =

X(k) +

N

where N is the total number of measures. Autocorrelation corresponds to the special case x(n) = y(n), then: Rxx (m) = E {x(n + m) · x(n)} .

ωlα X(k − l) ≥ X(k)

(33)

and for k + l: Dα X(k + l) = X(k + l) +

N

ωjα X(k + l − j)

j=1

≥ X(k + l).

(34)

From (33) and (34), and as X(k) is a positive function, we can write the following relation: Dα X(k) × Dα X(k + l) ≥ X(k) × X(k + l) Then,

Dα X(k) × Dα X(k + l) ≥

k

X(k) × X(k + l).

(35)

(36)

k

Finally, we have: ρDα X(k) ≥ ρX(k) ⇒ GDα X ≥ GX .

(37)

Autocorelation increase in the case of the Gaussian white noise: In this paragraph, we show that the DFI with a negative order allows to increase the autocorrelation of any path at the input. The cross-correlation expression of two functions x(n) and y(n) is given by: Rxy (m) = E x(n + m) y(n) = E x(n) y(n − m) , (38)

(40)

The estimation of the raw autocorrelations is given by: ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ N−m−1 f (n + m) · f (n) m ≥ 0 , (41) Rˆ xx (m) = n=0 ⎪ ⎪ ⎩ Rˆ xx (−m) m < 0⎭ where N is the length of the sequence f and m is the lag: m = 0, 1, . . . , N − 1. In the case of zero mean white noise signal, the normalized autocorrelation coefficient of the differentiated function is given by:

(32)

l=1

(39)

i=1

ρ(m) =

k

From (31) it is obvious that:

N 1 x(i), N

Rxx (m) (1 + α)(m − α) = . Rxx (0) (−α) (m + α + 1)

(42)

The expression of the correlation coefficients given in (42) shows that: the autocorrelation of a path increases when the DFI is applied with a negative order α. Indeed, the increase of the autocorrelation can be seen through the increase of the autocorrelation coefficients values that correspond to the lag values. It is well known that the variations of the autocorrelation for different values of lag indicates that when the lag is equal to zero, the autocorrelation is equal to 1 and it decreases when the lag increases. In our case, when α varies from 0 to −1, this decrease is more and more smooth, thus indicating the increase of the dependency between the different samples.

5

P ROPOSED M ETHOD

In this section, we present the proposed method to enhance the path prediction procedure. It consists in adding on downstream the digital fractional integration operator to any classical prediction filter. Fig. 3 illustrates the principle of this procedure. The main idea takes profit from the properties of the DFI, and the first step consists in applying the DFI with a negative order α to the current path, which forms the entry of any classical prediction filter. Reciprocally, we apply the DFI with inversed order (−α) to the output (estimated trajectory), which consists of the predicted locations of the MT. Our proposed method is based on two assumptions: the first is that the data association problem is solved, and the second is that the path of the mobile terminal (MT) can

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be seen as a positive function P(i). Then, locations P(i) = (X(i), Y(i)) of an MT are separated into two positive coordinates trajectories X(i) and Y(i), where X(i) expresses the abscissa trajectory, and Y(i) is the ordinate one. Hereinafter, we apply the DFI to X(i) and to Y(i) separately. Let X(i) = {X(1), X(2), . . . , X(N)} be abscissa of the MT discrete trajectory locations. Then, the DFI of this trajectory, for a given DFI order α (negative), is defined by the formula (16). Then, we obtain Dα X(i) = {Dα X(1), Dα X(2), . . . , Dα X(N)}. Afterwards, we use Dα X(i) as the input of any classical prediction filter to estimate the future differenX(i). Then, to have MT positions, we tiated position Dα X(i), then differentiate with the opposite order (−α): D−α finally, we get the estimated positions of the MT. The same operations are applied to the ordinate trajectory Y(i), to track the mobile.

6

E XPERIMENTAL R ESULTS AND D ISCUSSION

In this section we present the obtained results using the proposed method. In order to provide to the reader a global vision of the advantage of using DFI, we show the results of two different experiments: the goal of the first is to show the provided enhancement of the performance when DFI is used before the predictor. The second experiment shows a consequence of using the DFI that is the reduction of the archive (saved past measurements) to predict the future position of the MT. For all our experiments, we simulated different scenarios to have an idea about the behaviour of our proposed method in all situations. The considered scenarios are: two scenarios of indoor trajectories inspired from daily life promenades (museum visit (57 points or positions) and hospital doctor walk (102 points)) (Fig. 4(a) and (b)), and two other scenarios (spiral (58 points) and sinusoidal (45 points)) (Fig. 4(c) and (d)). The performances of the proposed method are compared to those obtained using LP alone and KF alone.

6.1 Enhancement of the Path Prediction Using DFI Based on the properties of the DFI seen in Section 4.2.2, the application to a function of the DFI with α negative increases the autocorrelation of the considered function. Fig. 5 illustrates the use of the linear prediction filter. One can see the small difference between a DFI path and its prediction (low prediction error)(see Fig. 5(b)). However, the prediction error is significant between the path and its prediction (see Fig. 5(a)). The prediction errors (defined in (25)) between the original trajectory and its estimated one using only LP (linear predictor) of order 3, and the prediction error values with the estimated trajectory using DFI-LP are presented in Fig. 5(c). One can notice that the linear predictor estimates better the transformed trajectory using the DFI than the original trajectory. This can also be seen when looking to the Fig. 5(c) where the amplitude of the prediction error on each point using the DFI is less than that of LP alone. In Fig. 5(d) we present the variation of root mean square error (RMS) for different values of α. As it can be seen, the decrease of the RMS is linear with the increase of the absolute value of the DFI order |α|. It can

be seen that best DFI order is −0.9 for this path. From this, one can conclude that the best order is always for any path, unfortunately, as we will see later, the best DFI order must be optimized for each path. As stated before, instead of the original path, the input of prediction filters is the fractionally differentiated trajectory, therefore the prediction error value decreases. The classical prediction filters, as Kalman filter (KF), and LP have two main computational steps, the first step consists of the update time, while the second one is the measurement update. More analysis and experiments will be presented in the following. In our experimentations we considered two cases: noiseless and noisy. Of course the more important case is the noisy case that was obtained by adding a Gaussian white noise of variance 2 to the orignal measurements (observations). In Fig. 4(a), (b), (c) and (d) the corresponding noisy paths are depicted using dots. In the following, we first present the obtained results on the linear predictor (LP), then those on Kalman filter (KF). In each future MT location, we exploited the last half observations of the traveled path. In Table 1, we summarize the obtained results of applying DFI to the linear filter (LP). In the column Acc(m) we show the obtained accuracy, in meter, using LP, while αx column is the best DFI order on the abscissa axis (OX), and αy column is the best DFI order on the axis OY. The Var% column shows the provided enhancement in terms of the variance of the prediction error expressed in percent. The DFIAcc(m) shows the new accuracy when the DFI is used and the "Enhancement" column presents the global enhancement using DFI on the corresponding path. For each path we presented the two cases: noiseless and noisy. Regarding the noiseless case, one can see that enhancement of the accuracy can reach 57.02% (in the case of Museum path presented in Fig. 5(a)) and for worst case 14.04% in the sinusoidal case (see Fig. 5(d)). Looking to the optimal values of α: they are close to 1 when the path is not regular that means the MT changes its orientation after few observations (positions or measures). On the contrary, these values are close to 0 when the path is regular. Indeed, the museum and hospital scenarios have corner points, where the direction changes drastically, and the difference is not significant between the variation of the abscissa and the ordinates (see Fig. 4(a) and (c)). While in the noisy case, the DFI allows also to enhance the prediction of the mobile path with at least 3.8%. In the museum visit scenario, the proposed method achieves an average prediction error of 30cm, while in the case of the LP alone it is of 56cm, thus the enhancement is around 49%. Besides the enhancement in terms of the variance of the prediction error using DFI is 72.28%. As noisy measurements degrade the accuracy, the performance of the LP decreases drastically: the accuracy becomes 1.52m. Despite that the DFI allows to have an accuracy of 1.38m that corresponds to 9.2%. In the other scenarios (hospital and sinusoidal noisy measurements), the accuracy achieved when the DFI is added is better than that achieved by the LP alone. Thus, in all cases, the proposed method has achieved the best performance. Looking to these results the optimal values of α for the proposed method is sensitive to variations of the DFI order of 1/100.


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Fig. 4. Four simulated paths scenarios: MT real trajectories (solid lines) and the observations (dots). Scenarios of daily promenades and noisy measurements. (a) Museum visit. (b) Hospital doctor walk. (c) Spiral trajectory. (d) Sinusoidal trajectory.

TABLE 1 Illustration of the Enhancement Using DFI in the Case of LP

The Cases Noiseless and Noisy Measurements are Considered.

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Fig. 5. Illustration of using LP and DFI-LP on given path. (a) Original trajectory (solid line) and its estimation using LP (dashed line). (b) DFI of the path and its estimation with α = −0.5. (c) Prediction error values using LP (solid line) and prediction error using DFI-LP (dashed line). (d) Variation of the root mean squared error with the increase of the DFI order (α).

Fig. 6. Illustration of the performance of KF and DFI-KF predictors for different KF parameters in the case of museum. (a) Q = 0.01, R = 5. (b) Q = 0.01, R = 10. (c) Q = 0.01, R = 100. (d) Q = 0.1, R = 5. (e) Q = 0.1, R = 10. (f) Q = 0.1, R = 100.


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TABLE 2 Q and R, Impact: Comparison of the Kalman Filter with the DFI-Kalman Filter, Q = 0.01, R ∈ {5, 10, 100}

Fig. 7. Archive size reducing. Comparison between the LP and the DFILP using noiseless and noisy measurements.

TABLE 3 Comparison of the Kalman Filter with the DFI-Kalman Filter, Q = 0.1, R ∈ {5, 10, 100}

The second predictor considered in our experiments is the classical Kalman filter (KF). It is known that the performance of the KF depends on its two parameters: the estimation of system covariance (Q) and the estimation of the observation covariance (R), see Appendix A, which is available in the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/37, for more details. In this case we considered only the noisy case, because in the noiseless case KF allows to predict perfectly the path, then there is no need to differentiate the original path (α = 0). So, in the noisy case, we compared results obtained by applying KF and DFI-KF, to the considered scenarios (Fig. 4) for predicting the MT path (the museum visit, the spiral, and the sinusoidal path). Tables 2 and 3 present the experimental results where different values of the parameters of KF are considered: Q ∈ {0.01, 0.1} and R ∈ {5, 10, 100}, this choice is arbitrary, and one can choose other values. The proposed method provides a good performance in most scenarios with different parameters values of Q and R. Looking to the Table 2 the best enhancement for museum path is around 21% and it starts from 0.45% with R = 5 and Q = 0.01. These results show that the DFI allows to increase performance even of a robust predictor as KF. The same remark can be made about the second case where Q = 0.1 (Table 3), where we can notice that in the test of museum path with R = 10 and Q = 0.1 the DFI does

not allow to enhance the performance because the KF was already optimal. Fig. 6 presents the predicted MT trajectory in the museum visit scenario, using DFI-KF (KF preceded by the DFI) and KF alone. From these experiments, we can say that the obtained results confirm what we expect by increasing the autocorrelation by the DFI. Consequently, the proposed method enhances the performance of the classical predictors for all scenarios.

6.2 Decreasing of the Archive Size In this subsection, we show empirically another consequence of using DFI when predicting a terminal path: the reduction of the archive used to predict the path of a mobile terminal. It is obvious that the benefit of using a shortarchive when predicting is the decrease of the number of locations used for the estimation of the path. Moreover, it allows to eliminate the influence of the accumulated approximation errors during long-time tracking. The use of small archive leads to use a lower number of positions that will be taken into account to predict the location of the TM: therefore, in our experiments, we considered the last 5, 10, 15 and 25 positions. To illustrate the decrease of the archive size to predict the position of an MT path, a performance analysis has been done using the four simulated scenarios exposed in Fig 4. The obtained results for LP and DFI-LP, are presented in Fig. 7 to Fig. 10. While the comparison to KF is depicted in Fig. 9. 6.2.1 The Linear Predictor (LP) Case To compare the proposed predictor (DFI-LP) to the linear predictor (LP), different results for the noisy and noiseless measurements cases are presented in this paragraph. In the noisy case: the provided enhancement varies from 0% in the case of the sinusoidal path (with LP order equal to 5) to 13.26% in the case of the spiral path (with the LP order equal to 25). In the case of the hospital doctor promenade scenario, the best enhancement is when the LP order is equal to 25 and is 5.64%. The obtained results are summarized at Fig. 7. So, from these experiments we can see that in most cases the enhancement increases with the increase of the archive size. However, this increase is not linear and tends to be constant when the archive size is greater than

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Fig. 8. Illustration of the performance of the LP of order 25 preceded with DFI. (a) Museum scenario. (b) Hospital scenario. (c) Sinusoidal scenario. (d) Spiral scenario.

Fig. 9. Illustration of the archive reducing using DFI and comparison between DFI-KF and KF for different parameters of KF. (a) Museum and Hospital scenarios where Q = 0.01. (b) Sinusoidal and Spiral scenarios, where Q = 0.01. (c) Museum and Hospital scenarios where Q = 0.1. (d) Sinusoidal and Spiral scenarios, where Q = 0.1.


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Fig. 10. Enhancements of the prediction for different scenarios. (a) Enhancements for Museum scenario. (b) Prediction error on OX with Q = 0.01, R = 100 and m = 5. (c) Prediction error on OY with Q = 0.1, R = 100 and m = 5. (d) Enhancements for Hospital doctor walk. (e) Prediction error on OX with Q = 0.01, R = 100 and m = 5. (f) Prediction error on OY with Q = 0.1, R = 100 and m = 5. (g) Enhancements for Spiral scenario. (h) Prediction error on OX with Q = 0.01, R = 100 and m = 5. (i) Prediction error on OY with Q = 0.1, R = 100 and m = 5. (j) Enhancements for Sinusoidal scenario. (k) Prediction error on OX with Q = 0.01, R = 100 and m = 5. (l) Prediction error on OY with Q = 0.1, R = 100 and m = 5.

15 in most of the considered scenarios. This proposition is not verified in the case of the museum scenario, where the enhancement increases linearly. Then, we can say that depending on the accuracy required for the application, an optimal archive size can be found. In the noiseless case, one can remark that the best performance in the first scenario is obtained for an archive size of 15 where the enhancement is equal to 51.9%. For the hospital scenario the best score is 25.87% and for Spiral scenario it is of 56.28%, all these scores are obtained with an LP of order 25 (see Fig. 7). Then, in the case of LP the addition of DFI allows to have a good enhancement with the decrease of the archive size to predict the position of the mobile. The obtained paths are presented in Fig. 8.

6.2.2 The Kalman Filter Case To illustrate the performance of the short archive on the Kalman filter, we compare DFI-KF to the KF alone by varying the R and Q values. In first, we considered only the noisy case and as in previous experiments, we present the obtained results for Q ∈ {0.1, 0.01}, R ∈ {5, 10, 100}, and the archive size m ∈ {5, 10, 25}. Fig. 9 depicts the obtained results. One can notice that, for Q = 0.01 and archive size m = 5, the performance increases with the increase of R in most cases. For example, in the museum scenario, the enhancement is 17%, 24%, and 36% for R = 5, R = 10, and R = 100 respectively (see the Fig. 9). The results in Fig. 9 show the usefulness of the DFI before the Kalman filter. It can be seen also that whatever the values of the parameters of the KF, DFI allows to enhance its performance. Indeed,

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in Fig. 10(a), (d), (g) and (j) one can see a pseudo linear dependency between the increase of the archive and the enhancement. So, the enhancement increases with the decrease of the archive size. The Fig. 10(b), (c), (e), (f), (h), (i), (k) and (l) show the prediction error on OX axis and OY axis where the archive size is equal to 5. One can notice that in all cases the amplitude of the prediction error in the case of DFI-KF is lower than that in the case of KF alone, this is clear in the case of the museum scenario. However, in some cases, the DFI does not provide a significant enhancement as for example in the case of Q = 0.1, R = 5 and m = 10. This can be explained by the fact that the parameters of KF were well estimated.

7

C ONCLUSION

In this paper, we showed that, under some assumptions, the digital fractional integration allows increasing the autocorrelation of a path. We then exploited this property in order to track a mobile terminal and predict its trajectory with relatively reasonable confidence intervals. Particularly, we used the paradigm of digital fractional integration (DFI) to significantly enhance the accuracy of MT mobility pattern predictions in indoor environments. Our approach to MT trajectory prediction outperforms considerably other existing linear predictors and the Kalman filter demonstrates the performance of fractional integrals to enhance the performance of the classical predictors. We also found that our DFI-based approach allows reducing the archive size that leads in many cases to a decrease in the spiraling effect of accumulated rounding error that builds up over a long tracking time, due to smaller number of addends. Our future works will focus on designing an algorithm to further optimize the performance of our DFI-based trajectory estimation algorithm. Several applications of our proposed approach are under development such as indoor navigation of vision-impaired persons within the university campus. The proposed technique can be added as a layer in existing devices where only the prediction engine would be updated. For instance, the position estimation engine in [27] can be easily changed.

R EFERENCES [1] P. Bahl and V. N. Padmanabhan, “Radar: An in-building RF-based user location and tracking system,” in Proc. INFOCOM, 2000, pp. 775–784. [2] A. Banos, J. Cervera, P. Lanusse, and J. Sabatier, “Bode optimal loop shaping with crone compensator,” J. Vibration Control, vol. 17, no. 13, pp. 1964–1974, Nov. 2011. [3] J. L. Battaglia, O. Cois, L. Puigsegus, and A. Oustaloup, “Solving an inverse heat conduction problem using a non-integer identified model,” Int. J. Heat Mass Trans., vol. 44, no. 14, pp. 2671–2680, Jul. 2001. [4] M. Borenovic, A. Neskovic, D. Budimir, and L. Zezelj, “Utilizing artificial neural networks for WLAN positioning,” in Proc. PIMRC, Cannes, France, 2008, pp. 1–5. [5] M. Brunato and R. Battiti, “Statistical learning theory for location fingerprinting in wireless lans,” Comput. Netw., vol. 47, no. 6, pp. 825–845, 2005. [6] Y. Chen, Q. Yang, J. Yin, and X. Chai, “Power-efficient access-point selection for indoor location estimation,” IEEE Trans. Knowl. Data Eng., vol. 18, no. 7, pp. 877–888, Jul. 2006. [7] H. Dai, Z. Zhu, and X. Gu, “Multi-target indoor localization and tracking on video monitoring system in a wireless sensor network,” J. Netw. Comput. Appl., vol. 36, no. 1, pp. 228–234, Jan. 2013.

[8] M. Dakkak, A. Nakib, B. Daachi, P. Siarry, and J. Lemoine, “Indoor localization method based on RTT and AOA using coordinates clustering,” Comput. Netw., vol. 55, no. 8, pp. 1794–1803, 2011. [9] M. Dakkak, A. Nakib, B. Daachi, P. Siarry, and J. Lemoine, “Mobile indoor location based on fractional differentiation,” in Proc. IEEE WCNC, Apr. 2012, pp. 2003–2008. [10] A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice. New York, NY, USA: Springer Verlag, 2001. [11] A. Doucet, S. J. Godsill, and C. Andrieu, “On sequential simulation-based methods for Bayesian filtering,” Statist. Comput., vol. 10, no. 3, pp. 197–208, 2000. [12] M. Ebling and R. Cáceres, “Gaming and augmented reality come to location-based services,” IEEE Pervasive Comput., vol. 9, no. 1, pp. 5–6, Jan./Mar. 2010. [13] F. Evennou, F. Marx, and E. Novakov, “Map-aided indoor mobile positioning system using particle filter,” in Proc. IEEE WCNC, 2005, pp. 2490–2494. [14] S. H. Fang and T. N. Lin, “Indoor location system based on discriminant-adaptive neural network in IEEE 802.11 environments,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1973–1978, Nov. 2008. [15] Y. Ferdi, “Fractional order calculus-based filters for biomedical signal processing,” in Proc. MECBME, 2011, pp. 73–76. [16] B. Ferris, K. Watkins, and A. Borning, “Location-aware tools for improving public transit usability,” IEEE Pervasive Comput., vol. 9, no. 1, pp. 13–19, Jan./Mar. 2010. [17] W. Gilks, S. Richardson, and D. Spiegelhalter, Markov Chain Monte Carlo in Practice. Boca Raton, FL, USA, Chapman & Hall/CRC, 1996. [18] F. Gustafsson, Adaptive Filtering and Change Detection. Chichester, U.K.: Wiley, 2000. [19] I. Guvence, C. T. Abdallaf, R. Jordan, and O. Dedeoglu, “Enhancements to RSS based indoor tracking systems using Kalman filters,” in Proc. GSPx ISPC, Dallas, TX, USA, 2003. [20] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Communications, vol. 23, no. 2, pp. 201–220, Feb. 2005. [21] S. Haykin, “Cognitive radar: A way of the future,” IEEE Signal Process. Mag., vol. 23, no. 1, pp. 30–40, Jan. 2006. [22] S. Haykin, “Cognitive dynamic systems,” in Proc. IEEE ICASSP, vol. 4. Honolulu, HI, USA, 2007, pp. 1369–1372. [23] C. M. Ionescu, J. A. T. Machado, and R. De Keyser, “Modeling of the lung impedance using a fractional-order ladder network with constant phase elements,” IEEE Trans. Biomed. Circuits Syst., vol. 5, no. 1, pp. 83–89, Feb. 2011. [24] C. S. Jensen, H. Lu, and B. Yang, “Graph model based indoor tracking,” in Proc. Mobile Data Manag., Taipei, Taiwan, 2009, pp. 122–131. [25] K. Kaemarungsi and P. Krishnamurthy, “Properties of indoor received signal strength for WLAN location fingerprinting,” in Proc. MobiQuitous, Boston, MA, USA, 2004, pp. 14–23. [26] T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation (Information and System Sciences), Upper Saddle River, NJ, USA: Prentice-Hall, 2000. [27] T. Kha, P. Dinh, A. Brett, and V. Svetha, “Indoor location prediction using multiple wireless received signal strengths,” in Proc. 7th AusDM CRPIT, Glenelg, South Australia, 2008, vol. 87, pp. 187–192. [28] S. A. Khan, B. Daachi, and K. Djouani, “Overcoming localization errors due to node power drooping in a wireless sensor network,” Int. J. Electron. Telecommun., vol. 57, no. 3, pp. 341–346, 2011. [29] R. Kumar, Y. A. Powar, and V. Apte, “Improving the accuracy of wireless lan based location determination systems using Kalman filter and multiple observation,” in Proc. IEEE Wireless Commun. Netw. Conf., 2006, pp. 463–468. [30] A. Kushki, K. Plataniotis, and A. N. Venetsanopoulos, “Location tracking in wireless local area networks with adaptive radio maps,” in Proc. IEEE ICASSP, Toulouse, France, 2006, pp. 741–744. [31] A. Kushki, K. N. Plataniotis, and A. N. Venetsanopoulos, “Kernelbased positioning in wireless local area networks,” IEEE Trans. Mobile Comput., vol. 6, no. 6, pp. 689–705, Jun. 2007. [32] A. Kushki, K. N. Plataniotis, and A. N. Venetsanopoulos, “Intelligent dynamic radio tracking in indoor wireless local area networks,” IEEE Trans. Mobile Comput., vol. 9, no. 3, pp. 405–419, Mar. 2010.


[33] A. M. Ladd, K. E. Bekris, A. Rudys, D. S. Wallach, and L. E. Kavraki, “On the feasibility of using wireless ethernet for indoor localization,” IEEE Trans. Robot., vol. 20, no. 3, pp. 555–559, Jun. 2004. [34] M. Liua, H. Lia, Y. Shenb, J. Fana, and S. Huanga, “Elastic neural network method for multi-target tracking task allocation in wireless sensor network,” Comput. Math. Appl., vol. 57, no. 11–12, pp. 1822–1828, 2009. [35] B. Mathieu, P. Melchior, A. Oustaloup, and C. Ceyral, “Fractional differentiation for edge detection,” Signal Process., vol. 83, no. 11, pp. 2421–2432, 2003. [36] R. Matli, M. Aoun, F. Levron, and A. Oustaloup, “Analytical computation of the H2-norom of fractional commensurate transfer functions,” Automatica, vol. 47, no. 11, pp. 2425–2432, 2011. [37] K. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations. New York, NY, USA: Wiley, 1993. [38] W. Muttitanon, N. K. Kumar, and M. Souris, “An indoor positioning system (IPS) using grid model,” J. Comput. Sci., vol. 3, no. 12, pp. 907–913, 2007. [39] K. Oldham and J. Spanier, The Fractional Calculus. New York, NY, USA: Academic, 1974. [40] A. Oustaloup et al., “The CRONE aproach: Theoretical developments and major applications fractional differentiation and its applications,” in Proc. IFAC FDA, Porto, Portugal, 2006, pp. 324–354. [41] A. Oustaloup, F. Levron, B. Mathieu, and F. Nanot, “Frequency band complex non integer differentiator: Characterization and synthesis,” IEEE Trans. Circuits Syst., vol. 47, no. 1, pp. 25–40, Jan. 2000. [42] A. Oustaloup and H. Linares, “The CRONE path planning,” Math. Comput. Simul., vol. 41, no. 3–4, pp. 209–217, 1996. [43] A. Oustaloup, B. Mathieu, and P. Lanusse, “The CRONE control of resonant plants: Application to a flexible transmission,” Eur. J. Control, vol. 1, no. 2, pp. 113–121, 1995. [44] A. Oustaloup, J. Sabatier, and P. Lanusse, “From fractal robustness to the CRONE control, fractional calculus and applied analysis,” Int. J. Theory Appl., vol. 2, no. 1, pp. 1–30, 1999. [45] J. J. Pan, J. T. Kwok, Q. Yang, and Y. Chen, “Multidimensional vector regression for accurate and low-cost location estimation in pervasive computing,” IEEE Trans. Knowl. Data Eng., vol. 18, no. 9, pp. 1181–1193, Sep. 2006. [46] A. Paul and E. Wan, “WIFI based indoor localization and tracking using sigma-point Kalman filtering methods,” in Proc. IEEE/ION Position, Location Navig. Symp., Monterey, CA, USA, 2008, pp. 646–659. [47] P. Prasithsangaree, P. Krishnamurthy, and P. Chrysanthis, “On indoor position location with wireless lans,” in Proc. 13th IEEE Int. Symp. Personal, Indoor, Mobile Radio Commun., 2002, pp. 720–724. [48] I. Prodlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fract. Calcul. Appl. Anal., vol. 5, no. 4, pp. 367–386, 2002. [49] C. Ramus-Serment, X. Moreau, M. Nouillant, A. Oustaloup, and F. Levron, “Generalised approach on fractional response of fractal networks,” Chaos Solitons Fractal., vol. 14, no. 3, pp. 479–488, Aug. 2002. [50] A. Roy, S. K. Das, and K. Basu, “A predictive framework for location-aware resource management in smart homes,” IEEE Trans. Mobile Comput., vol. 6, no. 11, pp. 1270–1283, Nov. 2007. [51] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Yverdon, Switzerland: Gordon and Breach Science, 1993. [52] L. Sommacal et al., “Improvement of the muscle fractional multimodel for low-rate stimulation,” Biomed. Signal Process. Control, vol. 2, no. 3, pp. 226–233, 2007. [53] K. Tran, D. Q. Phung, B. Adams, and S. Venkatesh, “Indoor location prediction using multiple wireless received signal strengths,” in Proc. AusDM, 2008, pp. 187–192. [54] S. Victor, P. Melchior, and A. Oustaloup, “Robust path tracking using flatness for fractional linear mimo systems: A thermal application,” Comput. Math. Applicat., vol. 59, no. 5, pp. 1667–1678, 2010. [55] H. Wang, A. Szabo, J. Bamberger, D. Brunn, and U. Hanebeck, “Performance comparison of nonlinear filters for indoor WLAN positioning,” in Proc. Int. Conf. Inf. Fusion, Cologne, Germany, 2008, pp. 1–7.

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[56] X. Wang and S. Wang, “Collaborative signal processing for target tracking in distributed wireless sensor networks,” J. Parallel Distrib. Comput., vol. 67, no. 5, pp. 501–515, 2007. [57] J. Yin, Q. Yang, and L. M. Ni, “Learning adaptive temporal radio maps for signal-strength-based location estimation,” IEEE Trans. Mobile Comput., vol. 7, no. 7, pp. 869–883, Jul. 2008. [58] M. Youssef and A. K. Agrawala, “The Horus WLAN location determination system,” in Proc. 3rd Int. Conf. MobiSys, Seattle, WA, USA, Jun. 2005, pp. 205–218. [59] J. Zhen and S. Zhang, “Adaptive AR model based robust mobile location estimation approach in NLOS environment,” in Proc. IEEE Veh. Technol. Conf., Milan, Italy, May 2004, vol. 5, pp. 2682–2685.

Amir Nakib received the M.D. degree in electronics and image processing in 2004 from the University of Paris 6 and in December 2007 he received the PhD degree in computer sciences from the University of Paris Est Créteil. His current research interests include stochastic global optimization heuristics and their applications (image processing, transport, drinking water networks, geo-localization).

Boubaker Daachi is an Associate Professor of robotics and computer science since 2003. He received his PhD degree in robotics from the University of Versailles in 2000, the M.S. degree in robotics from the University of Paris 6 in 1998, and the B.S. degree in computer science from the University of Setif (Algeria) in 1995. His current research interests include ubiquitous robotics, wireless sensor networks and adaptive control of robotic systems. He has published articles on robotics and wireless networks.

Mustapha Dakkak received the PhD degree from the University of Paris Est in 2012, the master’s degree from the Institut National Polytechnique de Grenoble in 2001, and studied in higher education at the Higher Institute of Applied Science and Technology (HIAST), Syria, in 1996. He has been an Engineer in the Graphics and Image Processing Laboratory, HIAST, since 1996. His current research interests include application of the multiobjective optimization for the geo-localization using multiple technologies of transmission and positioning.

Patrick Siarry received the PhD degree from the University Paris 6, in 1986 and the Doctorate of Sciences (Habilitation) from the University Paris 11, in 1994. He was first involved in the development of analog and digital models of nuclear power plants at Electricité de France (E.D.F.). Since 1995, he has been a Professor of automatics and informatics. His current research interests include adaptation of new stochastic global optimization heuristics to continuous variable problems and their application to various engineering fields. He is also interested in the fitting of process models to experimental data and the learning of fuzzy rule bases and neural networks. For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.

Appendix C

A framework for analysis of brain cine MR sequences This article was published in Computerized medical imaging and graphics. @articleNaki12, author = Amir Nakib and Patrick Siarry and Philippe Decq, title = A framework for analysis of brain cine MR sequences, journal = Comp. Med. Imag. and Graph., volume = 36, number = 2, pages = 152–168, year = 2012,

143

Computerized Medical Imaging and Graphics 36 (2012) 152–168

Contents lists available at SciVerse ScienceDirect

Computerized Medical Imaging and Graphics journal homepage: www.elsevier.com/locate/compmedimag

A framework for analysis of brain cine MR sequences Amir Nakib a,∗ , Patrick Siarry a , Philippe Decq b a

Laboratoire Images, Signaux et Systèmes Intelligents (LISSI, E.A. 3956), 61 avenue du Général de Gaulle, 94010 Créteil, France Service de Neurochirurgie, Centre Hospitalier Universitaire Henri Mondor, Unité d’Analyse et de Restauration du Mouvement (LBM ENSAM-CNRS 8005), 51 avenue du Maréchal de Lattre de Tassigny, 94000 Créteil, France b

a r t i c l e

i n f o

Article history: Received 19 February 2011 Received in revised form 27 September 2011 Accepted 28 September 2011 Keywords: Image registration Image segmentation Fractional differentiation Deformation model Dynamic optimization Evolutionary algorithm

a b s t r a c t In this paper, we propose a framework to automate the assessment of the movements of a third cerebral ventricle in a cine MR sequence. Indeed, the goal of this assessment is to build an atlas of the movements of the healthy ventricles in the context of the hydrocephalus pathology. This approach is composed of two phases: a contour extraction, using fractional integration and a registration method, based on dynamic evolutionary optimization. The first phase of the framework is based on the fractional integration thresholding, that allows delineating the contours of the area of interest. In order to track over time each point of the primitive and achieve the assessment of the deformation, a matching method, based on a new dynamic optimization algorithm, called Dynamic Covariance Matrix Adaptation Evolution Strategy (D-CMAES), is used. The obtained results for quantification have been clinically validated by an expert and compared to those presented in the literature. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Recently, a new technique for obtaining brain images of cineMR (Magnetic Resonance) type has been developed by Hodel et al. [1]. The principle of this technique is to synchronize the MRI signal with the ECG (Electrocardiographic) signal. The MRI signal provides three-dimensional images and cuts of high anatomical precision, and the ECG signal is obtained from the heart activity. An image of brain cine-MRI is therefore built by making the average of the MRI signals acquired during the R–R period of the ECG signal. This technique allows to have a good visualization of the movements of the walls of the third ventricle during the cardiac cycle. For more details about this method see [1]. In this paper, we are interested in the automation of the assessment of the movements of the walls, which allows a better understanding of physiological brain functioning and the provision of aid to diagnosis and therapeutic decision. Here, we will not be interested in the cerebrospinal fluid (CSF), the reader can have more details about analysis of the motion using CSF in [2,3]. This problem was already presented in a short paper [4] and in [1] a manual method was used to evaluate the movements. Several methods for the movement quantification have been proposed in the literature for myocardium images. This image processing application requires the partitioning of the image into

∗ Corresponding author. Tel.: +33 145171491. E-mail address: [email protected] (A. Nakib). 0895-6111/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2011.09.003

homogeneous regions. Then, the fundamental process used is called image segmentation, that plays an important role in entire image analysis system [5]. It is generally performed before the analysis and the decision-making process in many image analyses, such as the quantification of tissue volumes, diagnosis, and localization of disease, the study of anatomical structures, the matching and motion tracking. Since the segmentation is considered by many authors to be an essential component of any image analysis system, this problem has received a great deal of attention; thus any attempt to completely survey the literature in this paper would be too space-consuming. However, surveys of most segmentation methods may be found in [6] and [7]; another review of thresholding methods is in [8]. In [7,9–11] the authors proposed level set based methods to assess cardiac movements. These methods cannot be applied directly to our brain sequences. Indeed, firstly, the amplitude of movements in the heart is much greater than that of the walls of the third ventricle. Secondly, due to the presence of cerebrospinal fluid, one cannot properly segment the entire ventricle and quantify its movements in the sequence, because the stopping criteria are adapted to our images and the algorithmic complexity is high. For our application, we were looking for a low computation complexity (quick) method to assess and quantify these movements. Then, we proposed a framework that consists of two phases: in the first phase, the extraction of contour of the walls of the 3rd ventricle is performed. In the second phase, a contour registration is done to achieve the deformation model. In the segmentation step (phase 1), images are segmented by a proposed method based on

A. Nakib et al. / Computerized Medical Imaging and Graphics 36 (2012) 152–168

153

Fig. 1. Illustration of the first image of the original sequence. (a) Pathologic case (hydrocephalus) and (b) sane case.

fractional differentiation, inspired from those proposed in [12,13]. It is followed by a technique aimed at finding the best differentiation order. This phase allows to enhance the contrast and to extract the contours of a selected region of interest (ROI) at different times of the cardiac cycle. In the second step, the information provided by these contours is combined through their mapping. This registration procedure allows us to track the movements of each point of the contour of the ROI over time, and to obtain a better mathematical modeling of the movement. The parameters of this model are calculated over all the sequence. As we perform optimization several times for every couple of images, then the landscape of the function to be optimized (objective function) changes. Thus, we talk about dynamic optimization. In recent times, optimization in dynamic environments has attracted a growing interest, due to its practical relevance. For dynamic environments, the goal is not only to locate the optimum, but also to follow it as closely as possible. A dynamic optimization problem can be expressed by: Minimize f (x , t)

(1)

where f (x , t) is the objective function of the problem that may change over time, denoted by t. The problem can be seen as a dynamic optimization problem with constant time constraints. Then, we propose to adapt the algorithm proposed by Hansen and Ostermeier [14] to solve the dynamic problem at hand. The proposed algorithm is called Dynamic Covariance Matrix Adaptation Evolution Strategy (DCMAES), that includes an archive module (memory) and a new mechanism to manage the diversity of the stored solutions. This paper is organized as follows: Section 2 is for the description of the problem at hand. In Section 3, the characteristics of the MRI acquisitions are given. The proposed method will be detailed in Section 4. In Section 5, we present the results and their clinical validation. We end with a conclusion and perspectives (Section 6).

interest (ROI). In Fig. 1(a), we present an example of an ROI: Lamina Terminalis (LT). It is the first image of a brain cine-MRI sequence in a pathologic case (hydrocephalus). The sane case is presented in Fig. 1(b). Hydrocephalus is usually due to the blockage of CSF outflow in the ventricles. Patients with hydrocephalus have abnormal accumulation of CSF in the ventricles. This may be source of the increase of the intracranial pressure inside the skull and progressive enlargement of the head, convulsion, and mental disability. The goal of developing this framework is to contribute to the study of the diagnosis of endoscopic third ventriculostomy (ETV) patency. To validate the proposed method, 16 age-matched healthy volunteers were explored with the same MR protocol (12 women, 4 men; average age 38.5 years, interquartile range: 25.5–54). This study was approved by the local ethics committee; written informed consent was obtained for all the patients. 3. MRI data acquisition Data were obtained using 1.5-T MR (Siemens, Avanto, Erlangen, Germany). The protocol included a retrospectively gated cine true

2. Problem at hand The goal behind this work is to build an atlas of the movements of the healthy ventricles in the context of the hydrocephalus pathology. Then, in this paper, our objective is to segment the walls of the third ventricle, and quantify their movements. We have tested several image segmentation methods based on edge detection approach: the method of Canny, derivative methods and more robust methods such as level set methods [7]. All these methods give similar unconvincing results because they do not reproduce precisely the contours of the third ventricle on the images of the used sequences. This is due to the fact that they are very noisy, due to the presence of CSF, and variation of their contrasts on the same subject (third ventricle). For our application, the segmentation of the entire walls is not possible (differences between the pathologic and sane cases) and is not necessary. Consequently, we decided to work only on a region of

Fig. 2. Overall scheme of the proposed framework.

154


Fig. 3. Illustration of the modifications on the dynamic of the image with DFD-2D order. (a) Original image, (b) variation of the original dynamic, (c) integrated image with the order −0.1, (d) variation of the dynamic with the order −0.1, (e) integrated image with the order −0.2, and (f) variation of the dynamic with ˛ = −0.2.

Fast Imaging with Steady-state Precession Magnetic Resonance (cine True FISP MR) sequence: mid-sagittal plane was defined on a transverse slice from the center of the Lamina Terminalis to the cerebral aqueduct [1]. The parameters of the acquisition are given in Table 1. This acquisition technique provides only a sequence of 2D image, then a 3D image segmentation technique cannot be applied.

4. Proposed framework In this section we present the proposed framework that consists in two phases: the first consists in the extraction of contours of the image ROI with a segmentation technique, based on twodimensional-digital-fractional-integration (2D-DFI). In the second


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Table 1 Parameters of the MRI acquisition. Parameters

Values

TR/TE Number of excitations Flip angle Field of view Matrix size Slice thickness Acquisition time Phases during the R to R interval

78 ms/3 ms 7 82◦ 160 mm 256 × 256 2.5 mm From 2 to 4 min 20

phase, we combine the information provided by the contours of the image sequence through a registration. The steps of our framework are illustrated in Fig. 2. 4.1. Segmentation algorithm In order to segment every image of the sequence, we propose a new method based on two-dimensional-digital-fractionalintegration (2D-DFI). In our previous work [13], the proposed algorithm used a positive fractional order that is equivalent to a fractional derivative. The original image was segmented looking to the histogram of the differentiated image. In this paper, we propose the 2D-DFI, where the fractional order is negative (in the interval (−1, 0]). The fractional integration (also called fractional differentiation (FD)) is based on the works of Leibniz and Hospital in 1965. The applications of this method are numerous, it is used in automatics [15,16], in signal processing [17] and in image processing [12,13,18]. The FD of Riemann-Liouville is defined as follows: D−˛ f (x) =

1 (˛)

x

(x − )

˛−1

f ()d

(2)

c

where f(x) is a real and causal function, x > 0, ˛ the order of the FD, c the interval of the integral and the function of Euler–Lagrange. In the case of integration, the order ˛ is negative. In the discrete domain, the approximation of the DFI is given by: M

g ˛ (x) = D−˛ f (x) ≈

1 ˇk (˛)f (x − kh) h˛

(3)

k=0

where h is the sampling step, M the number of samples, x = M · h and ˇk (˛) are defined by:

Fig. 4. Segmentation result of the image in Fig. 3: (a) with the order −0.1 and (b) superposition of the original image and the segmentation results.

In order to adapt the 2D-DFI to the segmentation problem, we propose to take only the values of the DFI in the interval [0, L]. Then, from Eq. (5), the integrated image is given by:

g ˛ (x, y) =

⎧ ⎪ ⎨L ⎪ ⎩

if D−˛ f (x, y) ≥ L

0

if D−˛ f (x, y) ≤ 0

D−˛ f (x, y)

elsewhere

(6)

where L is the total number of gray levels (generally, 256) and g˛ (x, y) the fractionally integrated pixel of coordinates x and y. In order to segment

the image, we only used a threshold empirically fixed: t = 0.8 × max(g ˛ ) applied on the fractionally differentiated image, where g˛ is the maximum gray-level value in the original image. The example of Fig. 3 illustrates the way the DFD-2D allows to increase the image contrast without increasing the noise. Indeed, in Fig. 3(a), an example of a Lamina Terminalis is used to illustrate the advantage of applying this 2D filter, we present the variations of the gray-levels that are meshing as 2D function (Fig. 3(b), (d), (f) and (h)). In Fig. 4, we present the segmentation result using the proposed segmentation method.

4.1.1. Optimal 2D-DFD order As one can remark, the optimal segmentation of an image corre(k + 1) − ˛ − 1 sponds to finding the optimal 2D-DFI order (˛). In order to find the ˇk (˛), k = 0, 1, 2, . . . , M − 1(4) ˇ0 (0) = 1, ˇk+1 (˛) = (k + 1) best value, we propose to use a criterion to characterize the best segmentation, then the best value is that providing a segmentation Eq. (3) is equivalent to the Riemann–Liouville equation for h tending that optimizes the defined criterion. Indeed, a good segmentation to zero. The 2D-DFI is given, for a real and bounded function f(x, y), means that all regions (connected components) are homogenous by: and their number will not be under 2 and greater than 3. Less than ˛ ˛ 2 (equal to 1), it means that there is only one region; in this case, ∂ ∂ D−˛ f (x, y) = f (x, y) the segmentation result is bad. Greater than 3, it means that there ∂x ∂y are too many regions and the image is over segmented, thus the M/2 N/2 segmentation result is bad. 1 In order to obtain the optimal order for the segmentation prop(k, l)f (x − hk, y − lh) ≈ 2˛ (5) h cess, the well known uniformity criterion is used, it is defined by k=−M/2l=−N/2 [19]: M and N are the number of elements of f taken into account p 2 2p (fi − i ) for calculating the differential image, M × N represents the size u=1− (7) 2 N of the mask, p(k, l) = ˇk (˛) × ˇl (˛) are the elements of the matrix (fmax − fmin ) j=0 j ∈ Cj (˛) PM,N (p(k, l)), calculated from Eq. (3), which corresponds to the horizontal and vertical components, respectively. x denotes the where p is the number of thresholds (in our case 1), Cj is the jth connected component, N the image size, fi the gray level of pixel i, integer part of x.

156


i the mean gray level of pixels in jth connected component, fmax and fmin the maximum and the minimum gray levels of pixels in the image, respectively. When u is close to 1, the uniformity is very good and vice versa. Algorithm 1 summarizes the different steps of the segmentation algorithm. Algorithm 1 (Search for the optimal ˛opt .). Set ˛ = −0.9 // initialization Step 1: while˛ ≤ 0 a. Apply 2D-DFI at the order ˛ to the original Image (I) b. Compute t that corresponds to the order ˛. c. Calculate the segmentation quality measure criterion corresponding to t d. Increment ˛ by 0.1 and go to Step 1: Step 2: Find topt that optimizes the segmentation quality measure criterion. Step3: Print results: optimal order (˛opt ), optimal threshold (topt ) and segmented image.

4.2. Geometric matching of the contours The obtained contours in the segmentation phase will be used to assess the movement of the contours of the ROI over time. This step requires matching these contours. However, several false matches appear. To eliminate this problem, we have separated the obtained contours after segmentation and indexing and we have only kept the contour corresponding to the third ventricle (in the case of the Lamina Terminalis, this is in the right side). To evaluate the ventricle deformation, a contour matching operation is required after the segmentation stage in order to track the position of points belonging to the contours of the region of interest (ROI) over time. This operation is carried out in two steps: first, a rigid registration, called alignment, takes into account the displacement of the global membrane. Then, a morphing process performs accurate elastic matching of the successively aligned contours. 4.2.1. Contours’ alignment This step consists in looking, for each point in the curve of a reference image, at the nearest point in the curve in the destination image, based on a predefined minimum distance. The different steps of that phase matching are summarized as follows: For each point of the source curve: 1. Calculate the distance between this point and all points of the destination curve. 2. Match this point with the nearest point of the destination curve. In this step, the goal is to associate each point of the initial contour to one point of the second contour. We assume that each point of the first contour can be associated with at least one point of the second contour. Let C1 = {p1 (j), j = 1, . . ., L1 } and C2 = {p2 (k), k = 1, . . ., L2 } be two curves corresponding to successive contours to be paired. The alignment procedure realizes a registration between these two curves through a geometrical transformation controlled by a dissimilarity criterion based on 3 parameters. We considered also a non linear deformation model, however, due to the small movements of the walls, the provided results did not show a significant difference. – The Euclidean distance between two points: dj,k =

2

(x1 (j) − x2 (k)) + (y1 (j) − y2 (k))

2

(8)

– The difference of local curvature of these two points:

K1 (j) − K2 (k) K (j) + K (k)

Kj,k = 1 + log 1 +

1

2

(9)

where K1 (j) and K2 (k) correspond to the two values of the curvature. – The difference of orientation of the normals to the curves at these two points: nj,k = 2 − nT1 (j) · n2 (k)

(10)

where n1 (j) and n2 (k) are the values of the normal. Finally, the measurement of dissimilarity between two points C1 (j) and C2 (k) belonging to two successive contours C1 and C2 is given by: DC2 (k) (C1 (j)) ≡ Dj,k = dj,k · (Kj,k )K · (nj,k )n

(11)

where Dj,k > 1, K and n are positive weighting factors, fixed empirically to 4. Taking the curve C1 as a reference, one has to find, for each point C2 (k) = (x2 (k), y2 (k)) of C2 , a single point C1 (j) = (x1 (j), y1 (j)) of C1 that minimizes the dissimilarity measurement defined by (4). So, C1 (j) is defined as follows: C2 = arg min DC2 (C1 )

(12)

C1

where C2 is a set of sorted points of the contour C2 that correspond to contour C1 , respectively. 4.2.1.1. False matches. In the matching phase, a point is mapped to the nearest point, but the direct application of the matching method to the ROI contour without indexing produces false matches. Other false matches are due to the presence of several equidistant points from the point to match. Zhang [20] proposed a method using an adaptive statistical threshold, which is based on the statistical characteristics of the distances between matched points, such as the average and the standard deviation. We used this method to eliminate false matches and to assess the quality of the registration obtained. False matches of this type appear only if an alignment takes the first curve as a reference, i.e. by matching the contour of each sequence with the first points of the contour of the first image. 4.2.2. The deformation model In order to approximate the deformation model, we use a registration technique. The idea of registration is to combine information from two (or more) images of the same scene, but obtained at different moments, with different views or different acquisition methods. Then, the aim of a registration system is to determine the mapping information (positions, grayscale, structures, etc.) representing a physical reality on these different images. The goal of image registration is to find the best transformation T˜ among all the transformations T applied to an image I that looks at best like the image J. It is quite difficult to review all the image registration methods that have been proposed in the literature, a survey of many methods is presented in [21]. According to the used primitives (attributes), in our work we considered the geometric approach of image registration. In this approach an extract of the geometric primitives (points, curves or surfaces) is performed, and then a geometric similarity criterion is used to find the best correspondence. In the case where the primitives are points, the Euclidean distance is usually the most used [22]. In the case of surfaces and curves, the most widely used algorithm is the Iterative Closest Point (ICP) algorithm [20,23–25]. Other criteria are based on geometric calculations cards, Chamfer distances or Hausdorff distance [22,26]. Geometric approaches have the advantage of holding high-level information, but remain vague regarding the extraction of primitives. The optimization is a very important step in image registration. It aims at determining an optimum processing according to a similarity criterion. The optimization process is iterative. It must find


157

Fig. 5. Overall scheme of D-CMAES.

at each stage the parameters T of the processing that ensure the best match between the two images until the convergence to the optimal solution. Broadly speaking, the optimization problem is formulated by: Topt = arg max(J, T (I))

Fig. 6. Example of movement assessment.

they own a high robustness against changes of the problem instance over a certain range. For more details about the description of the historical development of evolutionary algorithms, see [27]. In order to assess the deformation model, we have matched the contours of the indexed images of the sequence by taking at each time two contours of two successive images. The similarity criterion minimizes the distance function described by Eq. (12). Considering the distortion models that exist, we assume that the movements of the third ventricle are governed by an affine transformation. This model is characterized by a rotation , two translations (tx , ty ) and two scaling factors (s1 , s2 ) according to x and y:

(13)

T ∈˝

where I is the original image, J is the image to register, ˝ is the search space for possible transformations, is the similarity criterion chosen, and Topt is the optimum processing. Among the optimization methods, we find numerical methods (without use of the gradient), such as Simplex and Powell [9,26] methods, and gradient based methods, such as gradient descent, conjugate gradient, the method of Newton and Levenberg–Marquardt [22,26]. In this paper, we propose to use an evolution strategy based method (class of evolutionary algorithms) to solve this optimization problem. Evolutionary algorithms (EA) are a class of nature inspired problem solvers. They use mechanisms known from natural evolution and have in common the transfer of their biological background into optimization. EA have proven their potentials in many real world applications. It is known that, in static environments, evolutionary algorithms find good or nearly optimal solutions, even for difficult problems, in a short time. In addition,

⎛

x2

⎞

⎛

s1 · cos

⎜ ⎟ ⎜ ⎝ y2 ⎠ = ⎝ s1 · sin 1

−s2 · sin s2 · cos

0

0

tx

⎞ ⎛ ⎟ ⎜

x1

⎞ ⎟

ty ⎠ · ⎝ y1 ⎠ 1

1

p1 (x1 , y1 ) is a point of the reference primitive and p2 (x2 , y2 ) is the point obtained with the geometric model.

Start

Sequence of M ROI’s contours

n ≤ MovSize

Cn

No

yes n=n+1

matching the contours

C1

et

Cn

Calculang the maximum shiing of each

(14)

C 1 point

Calculang the maximum shiing of each

C 1 point

Show the amplitudes of the movement

End Fig. 7. Procedure of the quantification of the movements (MovSize is the total number of the images in the sequence; in our experiments, it is equal to 20).

158


Fig. 8. Original sequence.

The parameters of this transformation are defined by minimizing the squared error among all points of the curve and those obtained with the model given by this function: SE(x) =

N 1 i=1

[C 2 (i) − T

(x)

(C1 (i))]

2

(15)

N1 is the cardinal of all the points of the contour C1 and x ≡ (s1 , s2 , , tx , ty ) is the vector of parameters. Several authors such as [9,20] use an iterative local search algorithm to solve this problem. To avoid local minima, the optimization criteria are modified by adding weight terms inversely proportional to the distances between matched points. This makes the optimization algorithm more complex. 4.3. Dynamic optimization algorithm (D-CMAES) In this section, we present the dynamic optimization algorithm that we use to optimize the criterion defined by the expression (15). The proposed algorithm consists in an adaptation of the

well-known Covariance Matrix Adaptation Evolution Strategy (CMAES) algorithm to the dynamic case, which we denote by D-CMAES. In Fig. 5, we present an overview of the D-CMAES algorithm, one can see that our contribution consists in the addition of a memory block and a mechanism to manage it. CMAES generates new population members by sampling from a probability distribution that is constructed during the optimization process. One of the key concepts of this algorithm involves the learning of correlations between parameters and the use of these correlations to accelerate the convergence of the algorithm. The adaptation mechanism of CMAES consists of two parts, (1) the adaptation of the covariance matrix and (2) the adaptation of the global step size. The covariance matrix is adapted through the evolution path and difference vectors between some of the best individuals in the current and previous generations. The detailed description of the algorithm is given in [28]. The application of memory schemes has proved to be able to enhance the performance of the evolution strategy algorithms in dynamic environments, especially when the environment changes cyclically in the search space. In these environments, with time


159

Fig. 9. Illustration of the segmentation result for the sequence in Fig. 8.

going, an old environment can reappear and the associated solution in the memory, which exactly remembers the old environment, will instantaneously move the search to the reappeared environment. The basic principle of the proposed memory scheme consists in storing the useful information from the current environment and reusing it later in new environments. To manage this memory, there are three major technical considerations: What to store in the memory? How to update the memory? and How to retrieve the memory? For the first aspect, our choice is to store good solutions and reuse them at every change of the environment. In our problem, at every start of the registration process, the algorithm starts the optimization procedure with the stored solutions. The memory space is usually limited (fixed at 5 individuals, in our case) for the efficiency of computation and search. This leads to the second consideration of memory updating mechanisms. To update the memory, we select one individual of the memory to be removed or updated using the best individual from the population. Thus, the stored individuals are of above average fitness, not too old, and distributed across several promising areas of the search space. This operation is conditioned by the increase of the diversity. To estimate the fitness diversity of the memory individuals when occurs an update of the memory, we propose the following coefficient:

F avg ı = min 1, − 1 F +ε

(16)

best

where Fbest and Favg are, respectively, the best and average values among the fitness values of all the individuals already stored in the memory. If ı ≈ 1, it means that there is a high diversity (in terms of fitness values) among individuals of the memory and, on the contrary, if ı ≈ 0, it means that the diversity is low. 4.3.1. Assessment of the movements The matching phase allows us to locate each point of a curve chosen as a reference curve in a chosen destination. Thus, we can calculate the movement of each point over time. Consequently, the first image of the sequence is taken as reference. The positions of

the points on the ROI’s contour in this image are considered as its initial positions. In this case, the matching is carried out between the contour of the first image (taken as a reference) and the outline of the sequence (Fig. 6). 4.3.2. Amplitudes of the movement To calculate the values of the amplitudes of the movement of every point belonging to a contour, we used the Euclidean distance (given by Eq. (8)) as a similarity criterion. The distance is calculated between the contours of the first image of the sequence, and all the contours of the other images of the sequence. The motivation behind the use of this criterion allows to obtain significant measures of movement, and it does not create false matches, which is noticeable when using the distance function based on the Euclidean distances, and the normal curves at each point to match. The calculation of horizontal and vertical displacements is relative to the x-axis and y-axis images. The whole procedure proposed to quantify the movements of the selected ROI from the walls of the third ventricle is presented in Fig. 7.

5. Results and discussion In this section, we first present the results of the preprocessing and extraction of the ROI’s contours (for the Lamina Terminalis), followed by the results for the registration and the quantification of the movement. Finally, we will present the results obtained for the quantification of other ROIs of the sequences. The results of the proposed quantification method have been clinically validated by an expert and compared to those in [1] obtained using a manual segmentation. The first phase of the framework consists in the extraction of the ROI. It is carried out in two steps: the segmentation and the indexation. The 2D-DFI algorithm is used to segment the different images of the sequence. Then, only perimeters of the segmentation results are considered. In order to illustrate this phase, we consider the original sequence in Fig. 8. The obtained segmentation results are presented in Fig. 9.

160


Fig. 10. Segmentation and matching of the Lamina Terminalis borders for the cine-MRI sequence.

Table 2 Obtained set of parameters of the deformation model. Images

s

tx

ty

img1–2 img2–3 img3–4 img4–5 img5–6 img6–7 img7–8 img8–9 img9–10 img10–11 img11–12 img12–13 img13–14 img14–15 img15–16 img16–17 img17–18 img18–19 img19–20

0.9917 0.9999 1.0044 0.9985 0.9977 1.0018 0.9984 0.9985 0.9967 0.9959 1.0018 1.0004 1.0087 0.9843 1.0033 1.0086 1.0047 1.0028 1.0073

0.0023 0.0015 0.0042 0.0025 0.0028 0.0117 0.0076 −0.0016 −0.0011 −0.0015 −0.0004 −0.001 0.0085 0.007 −0.0033 −0.0086 0.0047 0.0028 −0.0073

0.615 0.2151 0.2884 −0.2611 0.0736 0.529 0.333 0.1376 0.1136 0.2835 −0.2613 0.1992 0.1123 0.4257 −0.5332 0.371 0.1351 −0.1303 0.2596

1.4813 −0.639 0.4406 −0.515 0.1055 0.1088 −0.19 −0.2963 −0.1026 −0.3654 −0.567 −0.4103 −0.6622 0.78 −0.3712 0.1734 −0.1494 0.9025 0.9034

From the segmentation results and after the matching process, the representation of movement of the ROI can be shown. In Fig. 10, we present the case of the example in Fig. 9. The registration process allows to have all parameters of the deformation for all the different couples of images (i.e. deformation of the contour in Image 1 towards that of Image 2, etc.). In Table 2, we illustrate the obtained set of parameters. 5.1. Quantification of the ROI’s movements To quantify the movement of the ROI, we have aligned all the contours of the sequence with the contour of the first image. Fig. 11 shows the amplitudes of displacements of each point over time in the case of the Lamina Terminalis (Fig. 8).

Fig. 11. Representation of the displacements (mm).

The movements that we are interested in this work are those that have maximum amplitude. In the sequence used, the maximum movement is of 2.57 mm. This has been clinically validated by an expert in this study and is in the same range as the results published in [1]. The horizontal and vertical displacements are given in Table 3. 5.2. Performance analysis In this section we analyze the performances of the different phases of the framework: segmentation procedure, registration. Then, we compare the obtained results to those from classical, manual (by an expert) and recent methods. Let us summarize the different steps of the framework: (1) segmentation, (2) contours matching, and (3) contours’ registration. One can remark that the critical phase is the segmentation process, in other terms, if this phase fails, all results will be false. Herein, we

Table 3 Example 1 of obtained results: the points that have maximum of displacement. Position of the point in the contour C1

Time

Horizontal displacement (pixels)

Vertical displacement (pixels)

Horizontal displacement (mm)

Vertical displacement (mm)

Total displacement (mm)

18 20

5 5

4 4

−1 −1

2.5 2.5

−0.625 −0.625

2.57 2.57


b

3

Phase (degrees)

Magnitude (dB)

a

2 1 0 -1 -2 0

0.1

0.2

0.3

0.4

0.5

0.6

Magnitude (dB)

d

0.8

0.9

5

10 0

4 2 0 -2 0.2

0.3

0.4

0.5

0.1

0.6

0.7

0.8

0.9

h Phase (degrees)

Magnitude (dB)

5 0 -5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Phase (degrees)

Magnitude (dB)

10

0

-10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00 0.30 0.19 0.30 0.09 0.06

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20

0.8

i

1.00 0.60 0.48 0.60 0.36 0.29 0.48 0.29 0.23

l

1.00 0.90 0.85 0.90 0.810 0.77 0.85 0.77 0.73

0 -20 -40 -60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Frequency (´ rad/sample)

k

0.1

0.7

0.19 0.06 0.04

-80 0

1

20

-20 0

0.6

-30


j

0.5


10

0.1

0.4

-20

-40

1

15

-10 0

0.3

-10


g

0.2


e

0.1

0

rad/sample)

6

0

f

-10

8

-4

1.00 0.10 0.05 0.10 0.01 0.005 0.05 0.005 0.003

-5

-15

1

c

0

Phase (degrees )

Normalized Frequency

0.7

(´

161

0.9

1

50

0

-50

-100

-150 0

0.1


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fig. 12. Frequency responses (amplitude and phase) and impulse responses of 2D-DFI filters (size 3 × 3) for different ˛ values. (a), (b) and (c): ˛ = −0.1, (d), (e) and (f): ˛ = −0.3, (g), (h) and (i): ˛ = −0.6, (j), (k) and (l): ˛ = −0.9.

assume that the second procedure is perfect and focus only on the first and the last procedures. 5.2.1. Fractional filter analysis From the expression of the filtered image g(x, y) = D−˛ f(x, y) (5), where f is the original image, the 2D Discrete Fourier transform of the impulse response is: M−1 N−1

H(u, v) =

G(u, v) 1 p(k, l) · exp−h˛((uk/M)+(vl/N)) = F(u, v) M · N · h2˛ k=0 l=0

(17)

From (8), we can easily show (using the properties of the twodimensional Discrete Fourier transform) that the average value of the output image, or the differentiated function, is given by: M−1 N−1

g = f · H(0, 0) = f ·

1 p(k, l) 2˛ M·N·h

(18)

k=0 l=0

where f and g are the average values of the functions f and g, respectively, H is the two-dimensional Discrete Fourier transform of h(x, y). This assertion can also be interpreted from the amplitude

Fig. 13. Original sequence and ROI used for the segmentation performance analysis.

162


Fig. 14. Segmentation result of the sequence in Fig. 13 using Otsu method.

Fig. 15. Segmentation result of the sequence in Fig. 13 using 2DE method.


Fig. 16. Segmentation result of the sequence in Fig. 13 using 2DRE method.

Fig. 17. Segmentation result of the sequence in Fig. 13 using 2DLE method.

163

164


Fig. 18. Segmentation result of the sequence in Fig. 13 using K-means method.

Fig. 19. Segmentation result of the sequence in Fig. 13 using the proposed method (2D-DFI).


165

Fig. 20. Example 2 of the ROI selected.

Fig. 21. Example 3 of the ROI selected.

frequency responses in Fig. 12. From this analysis, we can explain the pixels’ gray-level increase over all the images [29].

These methods are: the classical Otsu method [30], the Twodimensional global entropy based method (2DE) [31], the 2D relative entropy (2DRE) method [32], the 2D local entropy based method (2DLE) [29], and the K-means based method [33]. The obtained results are illustrated in Figs. 14–20, respectively. One can see that in Fig. 14 all images of the sequence are not well segmented: the pixels of contours between the regions are connected. The use of this method to evaluate the movement implies the use of many post-processing techniques. The same remark can be done in Fig. 15, while in Fig. 16 9/20 images are well

5.2.2. Segmentation results In this subsection, we present a comparison of the performances of the segmentation procedure to those of five other methods from the literature. These methods are non-supervised and use the same thresholding approach to segment images. To do so, we consider the original sequence and ROI in Fig. 13.

Table 4 Transformation parameters provided by D-CMAES, and their corresponding SE, for the registration of each couple of images. Couple of images

s1

s2

tx

ty

SE × 105


1.02 1.002 0.993 0.972 1.008 0.999 1.02 1.01 0.96 0.99 1 1.014 1.018 0.975 1.005 1.034 1.004 1.013 1.009

1 0.977 0.986 1.008 0.996 0.995 0.992 0.995 1 0.997 1 0.994 0.993 1.007 0.999 0.986 0.999 0.998 0.997

−0.011 0.001 −0.003 0.018 0.002 −0.007 −0.016 −0.01 0.001 −0.007 0 −0.014 −0.016 0.015 −0.003 −0.026 −0.002 −0.008 −0.005

−0.005 −0.022 0.042 −0.022 −0.098 −0.181 −0.266 −0.15 −0.173 −0.132 0 −0.012 −0.04 −0.019 0 0.008 −0.001 −0.002 −0.002

0.148 0.334 0.213 0.123 0.115 0.036 0.007 0.002 0 0 0 0.023 −0.002 0.077 0.024 0.025 0.026 0.075 0.053

12754.4 16652.8 17891.9 15674.1 18179.6 21039.2 17795.1 11325.9 11960.6 13051.7 0 7114.3 6889.9 8926.3 2533.8 8196.5 2548.4 7102.1 4922

166


Table 5 Transformation parameters provided by SPSO, and their corresponding SE, for the registration of each couple of images. Couple of images

s1

s2

tx

ty

SE × 105


1.021 1.002 0.993 0.972 1.011 0.999 1.02 1.011 0.961 0.99 0.998 1.016 1.018 0.974 1.006 1.034 1.004 1.013 1.009

0.999 0.977 0.987 1.008 0.995 0.995 0.992 0.995 1 0.997 1 0.994 0.993 1.007 0.999 0.986 0.999 0.998 0.997

−0.012 0.001 −0.003 0.018 0 −0.006 −0.016 −0.01 0.001 −0.007 0.001 −0.014 −0.016 0.015 −0.003 −0.027 −0.002 −0.008 −0.005

−0.005 −0.022 0.042 −0.021 −0.1 −0.18 −0.268 −0.15 −0.174 −0.131 −0.001 −0.015 −0.04 −0.018 0 0.007 0 −0.002 −0.003

0.149 0.334 0.213 0.125 0.12 0.035 0.006 0.002 0.001 −0.001 0 0.023 −0.003 0.078 0.024 0.025 0.026 0.075 0.053

12755.3 16652.9 17892 15674.8 18188.1 21039.3 17795.5 11326.3 11960.8 13051.7 2.5 7117 6889.9 8926.6 2533.9 8196.7 2548.7 7102.1 4922.4

Table 6 Transformation parameters provided by DE, and their corresponding SE, for the registration of each couple of images. Couple of images

s1

s2

tx

ty

SE × 105


1.02 1.002 0.994 0.972 1.008 0.998 1.02 1.01 0.96 0.99 1.001 1.015 1.018 0.975 1.005 1.034 1.004 1.013 1.009

1 0.977 0.987 1.008 0.996 0.995 0.992 0.995 1 0.997 1 0.994 0.993 1.007 0.999 0.986 0.999 0.998 0.997

−0.011 0.001 −0.003 0.017 0.002 −0.006 −0.016 −0.01 0.001 −0.007 0 −0.014 −0.016 0.015 −0.003 −0.027 −0.002 −0.007 −0.005

−0.004 −0.021 0.041 −0.022 −0.097 −0.181 −0.267 −0.15 −0.173 −0.131 −0.001 −0.012 −0.04 −0.019 −0.001 0.009 −0.001 −0.002 −0.002

0.148 0.334 0.212 0.122 0.116 0.035 0.006 0.002 0 0 0 0.023 −0.003 0.077 0.025 0.025 0.026 0.075 0.053

12754.5 16652.9 17893.4 15674.3 18179.7 21039.5 17795.2 11325.9 11961.1 13051.7 0.4 7114.3 6889.9 8926.4 2533.9 8196.5 2548.5 7102.2 4922

segmented. The method 2DLE failed totally to segment this sequence (Fig. 17). In Fig. 18, the segmentation using K-means allows to find the limit between the 2 regions (CSF and wall of the brain ventricle) in the case of 2 images of the sequence. The segmentation using the proposed method is presented in Fig. 19, where we can see that, in the whole sequence, the two regions are well separated.

5.2.3. Optimization algorithm (registration step) In this paragraph, we present a comparison of the obtained registration results using D-CMAES compared to those of two other provided metaheuristics. The fitting of these metaheuristics is given below. For D-CMAES, we used the recommended parameter fitting of CMAES, we set: the initial step size to 0.3, the population size to 10, and the number of selected individuals to 5. The archive size, that is a parameter of D-CMAES, was set to 5. SPSO-07 (Standard Particle Swarm Optimization in its 2007 version) [34], was used using the recommended parameter fitting, except for the number S of particles (S = 6) and for the parameter K used to generate the particles neighborhood (K = 5). DE (Differential Evolution) [35] was used with the “DE/targetto-best/1/bin” strategy, a number of parents equal to NP = 12, a weighting factor F = 0:8, and a crossover constant CR = 0.9.

In order to have a comparison on only the performance of the optimization algorithm, we assume that the segmentation results are perfect. Then, in Tables 4–6, the transformations obtained by all algorithms are not significantly different: the values found for the parameters s1 , s2 , , tx and ty do not differ more than 5.9E−3, 1.8E−3, 3.0E−3, 3.7E−3 and 4.5E−3, respectively. Afterwards, we compared the number of evaluations of the objective function (15). The obtained results are summarized in Table 7, where the number of evaluations of the objective function performed by the proposed D-CMAES is significantly lower than the ones of the static optimization algorithms. 5.2.4. Performance illustrations The framework was also tested on other parts of the third ventricle, as shown in Figs. 20 and 21. The obtained maximum amplitude Table 7 Average number of evaluations for the registration of all couples of contours (images). Algorithm

Number of evaluations of the objective function

D-CMAES CMAES SPSO DE

15650.31 17651.65 19611.70 21749.60

± ± ± ±

145.3 288.10 994.13 869.46


167

Table 8 Example 2 of obtained results. We only show the points that have maximum of displacement. Position of the point in the contour C1

Time




1 2 4 24 34 37 40 43 46 49 52 56 64

8 8 8 8 6 6 6 6 6 6 6 6 7

0 0 0 0 0 0 0 0 0 0 0 0 2

−2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 0

0 0 0 0 0 0 0 0 0 0 0 0 1.25

Vertical displacement (mm) −125 −125 −125 −125 −125 −125 −125 −125 −125 −125 −125 −125 0

Total displacement (mm) 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25

Table 9 Example 3 of obtained results. We only show the points that have maximum of displacement. Position of the point in the contour C1

Time




Vertical displacement (mm)

Total displacement (mm)

33 38

10 10

0 0

−5 −5

0 0

−3.125 −3.125

3.125 3.125

of displacement on the selected regions are given in Tables 4 and 8, respectively. Further tests were made on other image sequences of patients and sequences of witnesses with no abnormalities in the third ventricle. We found that our method depends on the spatial resolution of images used. When the resolution is low, the quantification result is better. After several tests, we noticed that, for a good quantification result of the movement of the walls of the third ventricle, it is necessary that the spatial resolution of a pixel is smaller than or equal to 0.6 mm (Table 9). 6. Conclusion In this paper, we are interested in quantifying the movements of the third cerebral ventricle in a cine-MRI sequence. To solve the problem of quantifying the movements of the third cerebral ventricle, we propose a method for quantification of movement based on fractional integration and dynamic evolutionary optimization. We proposed a new segmentation method based on the fractional integration to segment quickly all sequences. In this step, the thresholding method provided good results for detecting the contours of the images. For the registration step, we proposed a new Dynamic Covariance Matrix Adaptation Evolution Strategy, called D-CMAES, that allows, first, to build a distortion model representing the distortion of the ROI and, secondly, to quantify its movements, without restarting the optimization process from the beginning. The deformation model is obtained by connecting two successive contours across the sequence, it is considered as an affine transformation and its parameters are calculated using the DCMAES algorithm. The obtained results were considered good and represent the movement of the ROI rather well. In order to take into account the third dimension, the design of a new acquisition technique is under progress. References [1] Hodel J, Decq P, Rahmouni A, Bastuji-Garin S, Maraval A, Combes C, et al. Brain ventricular wall movement assessed by a gated cine MR true FISP sequence in patients treated with endoscopic third ventriculostomy. European Radiology 2009;19(12):2789–97. [2] Kurtcuoglu V, Soellinger M, Summers P, Boomsma K, Poulikakos D, Boesiger P, et al. Computational investigation of subject-specific cerebrospinal fluid flow in the third ventricle and aqueduct of Sylvius. Journal of Biomechanics 2007;40:1235–45.

[3] Kurtcuoglu V, Soellinger M, Summers P, Poulikakos D, Boesiger P. Mixing and modes of mass transfer in the third cerebral ventricle: a computational analysis. Journal of Biomechanical Engineering 2007;129:695–702. [4] Nakib A, Aiboud F, Hodel J, Siarry P, Decq P. Third brain ventricle deformation analysis using fractional differentiation and evolution strategy in brain cineMRI. In: SPIE medical imaging, vol. 7623. 2010. pp. 76232l–76232l-10. [5] Mayer A, Greenspan H. An adaptive mean-shift framework for MRI brain segmentation. IEEE Transactions on Medical Imaging 2009;28(8): 1238–50. [6] Paragios N, Osher S. Geometric level set methods in imaging vision and graphics. Springer; 2003. [7] He L, Peng Z, Everding B, Wang X, Han CY, Weiss KL, et al. A comparative study of deformable contour methods on medical image segmentation. Image and Vision Computing 2008;26(2):141–63. [8] Sezgin M, Sankur B. Survey over image thresholding techniques and quantitative performance evaluation. Journal of Electronic Imaging 2004;13(1):146–65. [9] Chenoune Y, Deléchelle E, Petit E, Goissen T, Garot J, Rahmouni A. Segmentation of cardiac cine-MR images and myocardial deformation assessment using level set methods. Computerized Medical Imaging and Graphics 2005;29(8):607–16. [10] Budoff MJ, Ahmadi N, Sarraf G, Gao Y, Chow D, Flores F, et al. Determination of left ventricular mass on cardiac computed tomographic angiography. Academic Radiology 2009;16(6):726–32. [11] Sundar H, Litt H, Shen D. Estimating myocardial motion by 4D image warping. Pattern Recognition 2009;42(11):2514–26. [12] Nakib A, Oulhadj H, Siarry P. Fractional differentiation and non-Pareto multiobjecitve optimization for image thresholding. Engineering Applications of Artificial Intelligence 2009;22(2):236–49. [13] Nakib A, Oulhadj H, Siarry P. A thresholding method based on two-dimensional fractional differentiation. Image and Vision Computing 2009;27(9):1343–57. [14] Hansen N, Ostermeier A. Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation. In: Proceedings of IEEE conference on evolutionary computation (CEC’96). 1996. p. 312–7. [15] Benchellal A, Poinot T, Trigeassou J-C. Approximation and identification of diffusive interfaces by fractional models. Signal Processing 2006;86(10):2712–27. [16] Mansouri D, Bettayeb D. Optimal low order model identification of fractional dynamic systems. Applied Mathematics and Computation 2008;206:543–54. [17] Nakib A, Ferdi Y. Entropy reduction in fractionally differentiated electrocardiogram signals. In: European medical and biological engineering conference EMBEC’02, vol. 3. 2002. p. 562–3. [18] Mathieu B, Melchior P, Oustaloup A, Ceyral Ch. Fractional differentiation for edge detection. Signal Processing 2003;83(11):2421–32. [19] Cocosco CA, Evans AC. A fully automatic and robust brain MRI tissue classification. Medical Image Analysis 2003;7(4):513–27. [20] Zhang Z. Iterative point matching for registration of free-form curves and surfaces. International Journal of Computer Vision 1994;13(2):119–52. [21] Zitová B, Flusser J. Image registration methods: a survey. Image and Vision Computing 2003;21(11):977–1000. [22] Noblet V, Heinrich C, Heitz F, Armspach J-P. 3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization. IEEE Transactions on Image Processing 2006;14(5):553–66. [23] Besle PJ, McKay ND. A method for registration of 3D shape. IEEE Transactions on Pattern Analysis and Machine Intelligence 1992;14(2):239–56. [24] Malandin G, Bardinet E, Nelissen K, Vanduffel W. Fusion of autoradiographs with an MR volume using 2-D and 3-D linear transformations. NeuroImage 2004;23(1):111–27.

168


[25] Delhay B, Clarysse P, Bonnet S, Grangeat P, Magnin IE. Combined 3D object motion estimation in medical sequences. In: IEEE international conference on image processing. 2004. p. 1461–4. [26] Bâty X, Cavaro-Menard C, Le Jeune J-J. Automatic multimodal registration of gated cardiac PET, CT and MR sequences. In: 6th IFAC symposium on modelling and control in biomedical systems. 2006. p. 35–40. [27] De Jong K, Fogel DB, Schwefel H-P. A history of evolutionary computation. New York: Oxford University Press/Institute of Physics Publishing; 1997. [28] Ros R, Hansen N. A simple modification in CMA-ES achieving linear time and space complexity. Lecture Notes in Computer Science 2008;5199:296–305. [29] Althouse MLG, Chang C-I. Image segmentation by local entropy methods. In: ICIP’95 Proceedings of the 1995 international conference on image, vol. 3, no. 3. 1995. p. 61–4.

[30] Otsu N. A threshold selection method from gray-level histograms. IEEE Transactions on Systems Man and Cybernetics 1979;9(1):62–6. [31] Abutaleb AS. Automatic thresholding of gray-level pictures using twodimensional entropy. Computer Vision Graphics and Image Processing 1989;47:22–32. [32] Qiao Y, Hu Q, Qian G, Luo S, Nowinski WL. Thresholding based on variance and intensity contrast. Pattern Recognition 2007;40:596–608. [33] Moses K, Muthukannan MM. Color image segmentation using k-means clustering and optimal fuzzy C-means clustering. In: Conference on communication and computational intelligence (INCOCCI). 2010. p. 229–34. [34] Clerc M. The Particle Swarm website [Online]. http://www.particleswarm.info/. [35] Price K, Storn R, Lampinen J. Differential evolution – a practical approach to global optimization. Springer; 2005.

Appendix D

Image thresholding based on Pareto multiobjective optimization This article was published in Engineering Applications of Artificial Intelligence. @articleNak10, author = Amir Nakib and Hamouche Oulhadj and Patrick Siarry, title = Image thresholding based on Pareto multiobjective optimization, journal = Eng. Appl. of AI, volume = 23, number = 3, pages = 313–320, year = 2010,

161

ARTICLE IN PRESS Engineering Applications of Artificial Intelligence 23 (2010) 313–320

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Image thresholding based on Pareto multiobjective optimization A. Nakib, H. Oulhadj, P. Siarry n ´teil, France Laboratoire Images, Signaux et Syste mes Intelligents (LiSSi, E. A. 3956) Universite´ de Paris 12, 61 avenue du Geńe´ral de Gaulle, 94010 Cre

a r t i c l e in fo

abstract

Article history: Received 30 October 2008 Received in revised form 7 July 2009 Accepted 30 September 2009 Available online 24 October 2009

A new image thresholding method based on multiobjective optimization following the Pareto approach is presented. This method allows to optimize several segmentation criteria simultaneously, in order to improve the quality of the segmentation. To obtain the Pareto front and then the optimal Pareto solution, we adapted the evolutionary algorithm NSGA-II (Deb et al., 2002). The final solution or Pareto solution corresponds to that allowing a compromise between the different segmentation criteria, without favouring any one. The proposed method was evaluated on various types of images. The obtained results show the robustness of the method, and its non dependence towards the kind of the image to be segmented. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Image segmentation Image thresholding Multiobjective optimization Pareto approach Genetic algorithms Evolutionary algorithms

1. Introduction Image segmentation is the most important component of any image analysis system. Consequently, this problem received a great deal of attention; as a consequence, giving a complete survey of the literature in this paper would be too spaceconsuming. However, the most important thresholding methods can be found in this recently published review (Sezgin and Sankur, 2004). Image thresholding is definitely one of the most popular segmentation approaches to extract objects from image, e.g. (Gonzalez et al., 2004). It is based on the assumption that the objects can be distinguished by their gray levels. In the case of a simple thresholding (two classes), the optimal threshold is the one that permits to separate different objects from each other or different objects from the background (Gonzalez et al., 2004; Otsu, 1979). The automatic fitting of this threshold is one of the main challenges of image thresholding. The progress due to optimization techniques was very important during these last years. Many metaheuristic methods, such as genetic algorithms (Bhanu et al., 1995) and recently particle swarm optimization (Nakib et al., 2007; Zahara et al., 2004) have been applied to image segmentation problems. Zahara et al. proposed a hybrid Nelder-Mead particle swarm optimization method to handle the objective function associated to Gaussian curve fitting for multi-level thresholding. However, curve fitting is

n

Corresponding author. E-mail addresses: [email protected] (A. Nakib), [email protected] (H. Oulhadj), [email protected] (P. Siarry). 0952-1976/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2009.09.002

unusually time-consuming for multilevel thresholding and when the optimal threshold is not located at the intersection of the Gaussian curves it may never be found. In Bhanu et al. (1995) the authors attempt to use a genetic-based segmentation algorithm in a real-time application. However, the segmentation result depends on the segmentation algorithm which is employed and the executing times are far from real-time processing rates. In this work, we show that image segmentation based on the simultaneous optimization of some criteria gives satisfactory results and increases the ability to apply one same technique to a wide variety of images. We also show that segmenting with a threshold which optimizes more than one criterion helps to overcome the weaknesses of these criteria, when used separately. The multiobjective optimization (MO) (also known as Pareto optimization) is an extension of the optimization technique allowing several objectives to be optimized simultaneously. Several metaheuristics have been developed for MO problems, including particle swarm optimization (Coello-Coello and Salazar Lechuga, 2002), ant colony optimization (Doerner et al., 2004), genetic algorithms (Deb et al., 2002), simulated annealing (Engrand, 1997), and many others, a description of some of them can be found in Collette and Siarry (2002). In this paper, a new multi-level image thresholding technique, called Thresholding using Pareto MO (TPMO), is proposed, that combines the flexibility of multiobjective fitness functions with the power of an enhanced version of the most popular multiobjective genetic optimization algorithm NSGA-II (Deb et al., 2002) for searching vast combinatorial state spaces. This paper is organized as follows: in the next section, we introduce the problem of multi-level image thresholding as a multiobjective problem. In Section 3, the mathematical

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formulation of the different criteria is given in the first part of the section; in the second part, we present the proposed algorithm based on Pareto MO. In Section 4, we illustrate the obtained results through the proposed image thresholding algorithm. The paper ends with a brief concluding section.

2. Formulation of multi-level image thresholding as a MO problem In this section, we show that the segmentation problem can be formulated as a multiobjective optimization problem. In most cases this problem is NP-hard. The segmentation of an image I, using, for example, a homogeneity criterion A, is defined as a partition P= R1, y, Rn of I where: n is the number of regions. I ¼ [ Ri ;

iA ½1; n

Ri connected region; AðRi Þ ¼ True;

8 i A ½1; n

In this section, we describe the proposed segmentation method. In the first part, we define the different used criteria. In the second part, the optimization technique is presented, then a brief presentation of NSGA-II is given, followed by its enhancement. The proposed method is based on the assumption that the number of segmentation classes is known. Fig. 1 illustrates the principle of the proposed method. A detailed description of the different parts is presented thereafter. 3.1. Segmentation criteria 3.1.1. The biased intraclass variance criterion The biased intraclass variance results from the changes applied to the interclass variance used in the Otsu method (Otsu, 1979). Improvements include the addition of parameters which guarantee a segmentation of higher quality. The new expression of the segmentation criterion is: ! N X s2int ðjÞ ð1Þ þ gj MVarðTÞ ¼ a j¼1

8 i A ½1; n

AðRi [ Rj Þ ¼ False;

8 i A ½1; n; for all connected regionsðRi ; Rj Þ:

We note that the uniqueness of the segmentation result is not guaranteed by the conditions above. The results of segmentation do not only depend on the information contained in the image, but also on the used method to find the optimal solution. Generally, to force the uniqueness of the solution, regularization is obtained by constraining the problem. This constraint consists in optimizing a function F, characterizing the quality of a good segmentation. Then, the last condition is: FðP Þ ¼ Min FðPÞ P A PA ðIÞ

where F is a decreasing function and PA(I) is the set of all possible partitions of I. It is clear that this condition does not entirely solve the problem of uniqueness of the segmentation. Indeed, some segmentations may have the same optimal solution Pn. This explains the need for using metaheuristics to solve the optimization problem. In the literature several criteria to regularize the segmentation problem are presented (Sezgin and Sankur, 2004). In the following, we will deal with the non-parametric criteria based on the image histogram analysis. The advantage of using nonparametric criteria is that the method can be used independently of the shape of the image histogram: multimodal or unimodal. Then, the optimal vector of thresholds can be searched in the gray-level space. Looking at the literature, we can say that there is no single criterion able to regularize the segmentation problem for all kinds of images. Then, our approach consists in optimizing several functions simultaneously, in order to have a good segmentation on more kinds of images than when the criteria are used separately. Then, to optimize simultaneously some criteria, we used Pareto MO technique. Consequently, the segmentation problem can be formulated as an MO problem: Minimize=Maximize Subject to

3. Proposed method

fm ðt1 ; :::; tN1 Þ

m ¼ 1; 2; :::; M

bj

where T (t1, y, tN 1) is the vector of thresholds and N the number of classes to segment s2int is the intraclass variance P defined bys2int ðjÞ ¼ i A Cj pi ðxi mj Þ2, where mj is the barycentre of the class Cj and pi the pffiffiffiffiffiffiffiprobability of the gray level xi. a is a coefficient equal to NR=ð10 000 MÞ, where M is the total number of pixels in the image and NR the number of regions (connected components) within the segmented image. This coefficient is used to normalize and to penalize the segmentation which forms too many regions. It is chosen so that the term s2int ðjÞ=bj is low for the large classes. 1 The term bj ¼ 1=1þ logðNj Þ is high for non-homogeneous regions (typically, large ones), Nj being the number of pixels in the class j, whilegj ¼ ðRðNj Þ=Nj Þ2 is high only for regions of which area Nj is equal to the area of many other regions in the segmented image (typically, small regions). R(Nj) is the number of regions of which cardinal is equal to Nj (Borsotti et al., 1998). In all cases, the denominator Nj forces gj to be close to zero for large regions. Then, the problem of the segmentation of an image consists in finding the optimal thresholds minimizing expression (1): T ¼ MinfMVarðt1 ; :::; tN1 Þg

ð2Þ

where 1ot1 ot2 oyo225. 3.1.2. Shannon entropy criterion The entropy of an image is given by the relation: L X

H¼

pi log2 pi

ð3Þ

i¼0

where pi is the probability of the gray-level i in the image, represented by its normalized histogram, and L is the total number of gray-levels. In the case of two classes segmentation, we

f1 Original image

Segmented Image

f2 fp

Pareto front generation

0 o t1 o o tN1 o L

where M is the number of criteria used for the segmentation, ti the segmentation thresholds, and L the number of gray levels.

Fig. 1. The basic principle of multiobjective Pareto approach for image segmentation. f1, y, fp are the different criteria used for segmentation.

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denote the first class by A and the second by B and we assume that the first class corresponds to gray-levels greater than the threshold value (t) and the second one to gray-levels lower than t. Then, the method is based on the assumption that the background and the objects have two independent probability densities: Class A :

p1 p2 pt ; ; :::; PA PA PA

ð4Þ

Class B :

pt þ 1 pt þ 2 pL ; ; :::; PB PB PB

ð5Þ

where PA and PB denote the accumulated probabilities of the two classes A and B. We define the entropies Ht(A) and Ht(B) corresponding to the classes A and B, respectively, by Ht ðAÞ ¼

t X pi p log2 i P PA i¼1 A

ð6Þ

Ht ðBÞ ¼

L X pi p log2 i P P B B i ¼ tþ1

ð7Þ

315

3.1.3. The two-dimensional entropy criterion Abutaleb (1989) and Pal and Pal (1989) generalized this approach to the two dimensional case by adding more information about the spatial pixel distribution. It consisted in a two dimensional image histogram, by adding a matrix that characterizes the spatial distribution of the pixels (the co-occurrence matrix of pixels, filtered image, etc.). In our method, the co-occurrence matrix of the image is used. It allows highlighting the textures within the image. An element of the co-occurrence matrix CM(x, y) represents the number of times the gray-level x is adjacent to the gray-level y in the horizontal direction; a detailed analysis of the co-occurrence matrix is given in (Corneloup et al., 1996). The Fig. 2 illustrates the shape of the co-occurrence matrices of test images house and plane. These test images are from the Berkeley segmentation database (Berkekey, 2007). Fig. 3 displays a division with a threshold of the matrix. In the quadrant 1, the included coefficients belong to the image

Then, the total entropy Ht(I) of an image I is given by HT ðIÞ ¼ Ht ðAÞ þHt ðBÞ

ð8Þ

Consequently, in the case of the segmentation of an image I into N classes, the total entropy is defined by HT ðIÞ ¼

N X

Hti ðCi Þ

Min

1 o t1 o t2 o o L

2

1

4

t

where T= (t1,t2, y, tN 1) represents the thresholds vector and Ci is the class number i. The optimal segmentation threshold vector is that maximizing the total entropy defined in (9). Therefore, the segmentation problem can be formulated as an optimization problem: T ¼

3 ð9Þ

i¼1

L

T

fH ðIÞg

ð10Þ

0

t

L

Fig. 3. Two-dimensional histogram.

Fig. 2. Cooccurrence matrices of test images: (a) house and (b) plane.

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background, on the contrary in the quadrant 2 the coefficients correspond to the foreground of the image or vice versa. However, the quadrants 3 and 4, made identical by construction of a symmetric matrix, contain coefficients related to the transitions between the image background and foreground. These coefficients are mostly low and negligible (Corneloup et al., 1996). The two dimensional entropy of one class k is given by: tkX þ 1 1 tkX þ 1 1

Gtk ðCk Þ ¼

i ¼ tk

j ¼ tn

pij pij log Pk Pk

N X

Gti ðCi Þ

3.3. Parameter fitting

ð12Þ

i¼1

The optimal segmentation thresholds are those maximizing the total entropy 2D given in (12). Therefore, the formulation of the problem as an optimization problem is given by: T ¼

Min

1 o t1 o t2 o ::: o L

fGT ðIÞg

si ¼ si þ asi Nð0; 1Þ where a = 1.50 (set empirically). The choice of NSGA-II is motivated by its good performance on various multiobjective optimization problems, as it was shown in (Shukla and Deb, 2007).

ð11Þ

where pij are normalized coefficients of the co-occurrence matrix and Pk is the cumulative probability of the k class. Then the total entropy in the case of N classes is GT ðIÞ ¼

and the variance s2i is obtained by mutating the strategy parameters (step sizes) sI according to:

ð13Þ

where L is the total number of gray-levels. 3.2. TMPO algorithm The genetic algorithm we used is based on NSGA-II (elitist nondominated sorting genetic algorithm). The aim of the improved NSGA-II module is to figure out the exact Pareto points effectively and maintain diversity in the meantime. The original NSGA-II algorithm consists of five operators: initialization, constrained nondominated sorting, crossover, mutation and elitist crowdedcomparison operator. The details of the complete method can be found in Deb’s paper (Deb et al., 2002). The improved module does not maintain all the operators of the original NSGA-II, and the modification is operated on the mutation operator. The mutation used is inspired from the Schwefel mutation ¨ and Schwefel, 2000). Each variable is mutated according (Schutz to a normally distributed random numberdi Nð0; s2i Þ: ti ¼ ti þdi

Give the number of classes: k. Initialize randomly the initial population P0 of size N = 10 ⋅ k. Calculate the various fronts Fi of the population P0.

The stopping criterion we used is based on the number Ng of generations. This number is defined by: Ng = 5 Nc, where Nc is the number of classes, assumed known a priori. The used selection is that implemented in the original NSGA-II: tournament selection, and we fixed the number of tournaments at 5, empirically. The goal of this operator is to increase the diversification in the population. If we take more than 5 tournaments, the result will not be improved, since the best result is already reached and the computation cost will be increased. However, if we take less than 5 tournaments, the population will not be enough diversified and we will not be sure to obtain the best solution. The crossover parameter is fixed at 0.5. The Fig. 4 illustrates the different steps of the optimization algorithm. The population size is defined by N= 20 Nc. 3.4. The thresholding algorithm From previous subsections, the proposed algorithm consists in the optimization simultaneously of the functions MVar, H and G. Then, the optimal threshold vectors correspond to the Pareto solution.

4. Experimental results and discussion In this section, we present the results obtained via the application of our method to different kinds of images. This section is divided into two parts; in the first subsection, we evaluate the performance results in comparison with those of five other methods: EM algorithm based method (EM) (Bazi et al., 2007), one method based on valley-emphasis (VE) (Ng, 2006), Otsu method (1979), Kapur et al. method (1985) and Sahoo and Arora (2006) method based on two-dimensional Tsallis entropy (TE). The algorithms are coded in Matlab version 7 and are run on a 2.26 GHz Pentium4 personal computer, under Microsoft Windows XP pro Operating system.

Select in P0 and create Q0 by applying genetic operators. While the stopping criterion is unsatisfied do: Create Rt = Pt ∪ Qt Calculate the various fronts Fi of the population Rt Make Pt+1=0 et i=1 While Card(Pt+1)

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