New Quantum Inspired Tabu Search for Multi-level ... - IEEE Xplore

New Quantum Inspired Tabu Search for Multi-level Colour Image Thresholding Sandip Dey Department of Information Technology Camellia Institute of Technology Kolkata - 700 129, India Email: [email protected]

Siddhartha Bhattacharyya Department of Information Technology RCC Institute of Information Technology Beliaghata, Kolkata-700015, India Email:dr.siddhartha.bhattacharyy [email protected]

Abstract— In this article, a Quantum Inspired Tabu Search for Multi-level thresholding for Colour Image has been developed to boost the possible effectiveness than that of its classical counterpart. The proposed algorithm has been applied to two true colour images to determine optimal threshold values at different levels using Otsu’s method as an objective function. The features of quantum mechanics are coupled with the basic constitution of a popular meta-heuristic algorithm, called tabu search to form the quantum inspired meta-heuristic algorithm. Between the participating algorithms, the proposed algorithm takes least time for execution. The usefulness of the proposed method is established in context of exactitude, resilience and computational time over its respective conventional method. In addition, a popular test, called one-tailed t-test, used for statistical measurement, demonstrates the efficiency of the proposed algorithm. Keywords— colour image segmentation, multi-level image thresholding, tabu search, quantum computing, Otsu’s method

I.

INTRODUCTION

To date, image segmentation has been extensively used as an effective and popular technique in image analysis and image processing. Of late, colour image segmentation became well reputable in many application areas [1]. Some typical examples of colour image segmentation may include object recognition, robotics and data compression to name a few [2]. Since each pixel of the colour image is recognized in multidimensional colour space, this leads to increase the processing complexity in real life applications. Unlike image segmentation in monochrome image where no parameter or a few parameters are adjusted, more number of parameters are needed to be tuned for optimality in colour image segmentation [2]. Among the several tools used in image segmentation, thresholding has the highest acceptability to incorporate in establishment. Thresholding is morally effortless to put into practice for bilevel and multi-level image segmentation and provides robust results in all circumstance [3].

c 978-93-80544-12-0/14/$31.00 2014 IEEE

Ujjwal Maulik Department of Computer Science & Engineering Jadavpur University Kolkata-700032, India Email: [email protected]

Thus far, numerous classical techniques have been developed for gray level image thresholding [1]. In [2][4], the author has presented a detailed survey over different classical/non classical method for image segmentation. Lately, an assortment of classical method and quantum theory is put in algorithmic formation to produce different quantum inspired classical algorithms. This idea leads to make faster for the execution of new quantum version of classical algorithm. A few distinctive examples in this regards may include, finding the factor of large number [5] and finding data from database [6]. Meta-heuristic algorithms are stochastic and nondeterministic search procedures that usually guide the process to find their optimal solutions. It eventually produces fairly accurate results by exploring its search space and sometime applies apposite mechanism to avoid local trap. Meta-heuristic algorithms can be effectively used for optimality in accordance with multi-level image thresholding. Some examples of this approach are available in the literature [7][8]. In this paper, the concepts of quantum mechanics have been fused with a popular meta-heuristic method to develop a new quantum inspired meta-heuristic algorithm, namely, Quantum Inspired Tabu Search for Multi-level Image Thresholding for Colour Image (QITSMLTCI). The proposed method uses the Otsu’s function as an objective function to determine the optimum threshold value of two true colour images. Later, this method is compared with its classical counterpart to find the precision and rigidity of the proposed algorithm. Finally, one-tailed ttest establishes effectiveness of it [9]. II.

OVERVIEW OF META-HEURISTIC ALGORITHMS

Meta-heuristic algorithms are eminent stochastically search procedure which are used for optimization purpose specially for optimizing different combinatorial problems which are deterministic in nature. These algorithms proficiently explore their search spaces to discover new paths which have not visited before. They are known to be equally effective to uncover the near optimal solutions for simple to any

311

complicated learning search procedure. The paradigm of most commonly used meta-heuristic algorithms in vogue are Simulated Annealing [10], Tabu Search [11], Variable Neighborhood Search [12] to name a few. Meta-heuristic algorithms are characterized firstly by encoding the input solution followed by the objective function used for evaluating the fitness and finally by tuning some essential parameters for execution [8]. A. Principles of a Tabu Search In 1986, Glover and Laguna first developed a renowned meta-heuristic algorithm called Tabu Search (TS) [11]. TS undergoes for iterative searching to explore its solution space to find the near optimal solution. TS is simple to implement, supple and powerful in nature. A proper mechanism must be employed to free TS from sticking over local optima. The only problem of TS is that one can face parameter adjustment at beforehand. Since parameters used in TS, have direct influence for the solution, a proper knowledge of parameter tuning is an essential and desirable condition to possess a good result. Some fundaments ingredients of TS can be described as follows. i) neighborhood: It consists of collection of adjacent solutions which are found from the existing solution ii) tabu: To utilize memory in TS that is to classify in neighborhood as entry in tabu list iii) attributes: It is referred to as classification that depend on the recent history iv) tabu list: Record the tabu moves and v) aspiration criterion: When the tabu classification need to be overridden using some relevant condition, it is called aspiration criterion for tabu search.

quantum communication between the participating particles. The effects of quantum entanglement affect the property of one particle when the characteristic of another linked particle is changed. IV.

OTSU’S METHOD AS OBJECTIVE FUNCTION FOR OPTIMIZATION

In the recent years, different methods have been used by plenty of researchers for image thresholding optimization. Sezgin and Sankar presented a detailed survey about various methods in their paper [4]. They have categorized all the existing methods into six groups. In the proposed method, a cluster-based method namely, Otsu’s method [16] has been employed as an objective function for optimization of multilevel thresholding for true colour images. Among all the on hand thresholding methods, Otsu’s method [16] is supposed to be the most popular and mostly used method for this purpose. The QITSMLTCI finds the predefined number of optimal threshold values by maximizing the between-class variances in the test images. Formally, the set of optimal threshold values are determined by using the formula given by D

θ op = γ {θ1 , θ 2 ,! ,θ D −1} = ¦ϖ i (ν i −ν )

( 3)

fk kp ,ν i = ¦ k M k∈Ci ϖ i

( 4)

i =1

where, ϖ i

=

¦ p ,p k

k

=

k∈Ci

and the pixel intensity values are grouped into D number of III.

QUANTUM COMPUTING RUDIMENTS

classes,

As the name suggest, Quantum computer (QC) exploits a bundle of quantum physical features for information processing [13,14]. The mechanism of quantum machine is overseen by Schrödinger's equation. Hilbert space has the impression to be the primary consideration for any quantum mechanical system. Suppose, a quantum system is accounted by using a wave function, states,

ψ

in Hilbert space, which consists of a set of basic

φi , i = 1, 2,! , k . Then the given wave function can

be defined by

ψ = d1 φ1 + d 2 φ2 + " + d k φk

(1)

where, each d i is a complex number satisfying the unitary condition given by

( 2)

d12 + d 2 2 + " + d i 2 = 1 In

quantum

system,

ψ

is

referred

to

as

linear

superposition of the given basic states, φi . In reference to wave function, the concept of coherence and decoherence can be recounted with linear superposition theory [15]. Until there is a constant phase relationship subsists between two waves, QC coheres. When the defined phase relationship is destroyed to rupture the wave superposed form, QC decoheres. Quantum Entanglement is an important property of QC. It is used for

312

C = {C1 , C2 ,! CD } .

pk and f k are the probability and frequency of kth pixel, respectively of an image. M is the total number of pixels in the entire image. For class Ci ,ϖ i represents the Here,

probability whereas, ν i signifies the mean of the respective class. ν represents the mean of all classes which are created after segmentation of the whole image. The optimal threshold values are determined by maximizing θ op . V.

PROPOSED METHODOLOGY

In this paper, a new quantum inspired tabu search algorithm for multi-level thresholding for true colour images (QITSMLTCI) is presented. The features of QC are used in collaboration with the conventional meta-heuristic algorithm, tabu search to form the aforesaid algorithm. The details of the proposed QITSMLTCI are outlined below: Step 1: Initialization: In the proposed QITSMLTCI, the pixel intensity values are selected from the true colour test images at random. These pixels are used to form the initial population, P, which contains one string for each individual colour component. The length of each component is taken as

L = Ll × Lw where, Ll and Lw are the length and height of the input image.

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Step 2: Pixel encoding: Using the theoretical concept of the basic unit of QC, a random positive floating number between (0,1) is assigned to each element in P. Let it be creates P’.

Step 16: The value of F is compared with the value of

Step 3: Quantum interference: Each element in P’ endures quantum interference to ensure some fundamental properties of QC. Let it creates P’’.

respectively.

Step 4: Find threshold values: Three different set of predefined number of threshold values are found out for three individual colour components in P’’. It should satisfying the relation P’’>random(0,1) to acquire the desired threshold values. Let it creates P*. Step 5: Fitness computation: Evaluate the fitness values of the strings in P* for each of different colour component using

F1 and record the respective threshold values in Trs , Tgs and Tbs ,

Step 17: Update the value of

c by 1.

Step 18: Go to step 10 until reaches the predefined number of generation, M xg . Step 19: Report the optimal threshold values,

θ ∈ Trs , Tgs and

Tbs .

Step 7: Record the sum of the best fitness value of each colour component in F .

A. Complexity analysis The worst case time Complexity analysis of QITSMLTCI is presented in this subsection. • A colour image comprises three basic components. Since, tabu search works on single string, the population, P, is initially populated with three different strings for three distinct colour components in the test image, each for individual colour component. Let, length of each string is assumed to be S = H × W , where, H and W are the height and width of the input image. Therefore, time complexity to create the population, P is turned out to

Step 8: Incorporate tabu memory: Initialize tabu memory, mem = ϕ .

• The time complexity to complete pixel encoding scheme as

Equation

( 3)

.

Step 6: Record best string with threshold value and fitness: Save the string as the best string for each individual colour component in Br , Bg and Bb , their respective threshold values in

Tr , Tg and Tb and the fitness values in Fr , Fg and

Fb , respectively.

described in step 2 is

Step 9: Initialize, c = 1 . Step 10: Set Vr = Br , Vg = Bg and

be O ( 3 × S ) = O ( S ) .

O(S ).

• To perform quantum interference as explained in step 3,

Vb = Bb .

the time complexity becomes O ( S ) .

Step 11: Find a set of neighbors: A set having three different subsets of string is formed by exploring the neighbors of the best strings for each individual colour component. Let it creates VN . Step 12: Update best string: For each string, v ∈ VN , if the fitness value is better than the corresponding fitness value in Vr , Vg or Vb and v ∉ mem , update Vr , Vg or Vb by v .

•

For executing step 4, time complexity to produce P*, turns

into O ( S ) .

• To compute fitness of string in QITSMLTCI, time complexity grows to be O ( S ) .

• After finding the best string at each generation, the neighbors of the string are explored. To find the set of neighbors, the time complexity becomes O (U ) , where, U

Step 13: Update tabu memory: The tabu memory is updated with the best individual string for each colour component as mem = mem ∪ Vr ∪ Vg ∪ Vb .

represents the number of neighbors. • For updating the best string as explained in step 12, the

Step 14: Repeat steps 3-5 to compute the fitness values of string in Vr , Vg and Vb . Record the fitness values in Fr1 , Fg1

• Hence, to run QITSMLTCI for a prefixed number of

and

Fb1 with the threshold values in, Tr1 , Tg1 and Tb1 ,

respectively.

F1 .

generations, the time complexity becomes O (U × S × G ) .

Here, G denotes the number of generations. Therefore, the overall worst case time complexity (recapping

Step 15: Repeat step 7 to record the sum of Fr1 , Fg1 and in

time complexity turns into O (U × S ) .

Fb1

all the steps stated above) happens to be O (U × S × G ) . VI.

EXPERIMENTAL RESULTS

To evaluate the performance of the proposed QITSMLTCI and the equivalent conventional meta-heuristic algorithm, the

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313

aforementioned two algorithms have been tested on two true colour images. For this reason, Otsu’s method [16] has been exploited as the objective function to find out the optimal threshold values at different levels for the test images. Otsu’s method [16] maximizes to determine optimal threshold values as given in Equation ( 3 ) . For experimental purpose, the proposed QITSMLTCI for multi-level thresholding and its conventional counterpart have been executed on Tulips and Barche, each of size 256X256, which are depicted in Figs. 1(a) and (b), respectively. The control parameters set for executing two algorithms, have been listed in Table I.

Table I. Values of input parameters for QITSMLTCI and TS QITSMLTCI

TS

1) Number of Generation: G = 100

1) Number of Generation: G = 150

2) No. of classes: D = 2, 4 and 6

2) No. of classes: D = 2, 4 and 6

Table II. Best result of QITSMLTCI and TS for multi-level thresholding for Tulips Tulips QITSMLTCI

D

(θ r )(θb ) (θ g )

(θ r ) , green (θ g ) and blue (θb ) components of the input image, best fitness values ( γ B )

2

(139) (125) (116)

10909.76

01.14

4

(56,124,195) (69,120,175) (61,116,181)

12792.40

13.22

among different runs and finally the execution time

6

13119.14

16.41

The best results among 20 different runs for each method have been reported in Tables II and III for D=2, 4 and 6. Each table is filled with the number of predefined classes (D), optimal threshold values for red

( t ) (measured

in seconds) for different values of D.

Furthermore, the average fitness value

( γ AV ) and

(38,63,105,154,210) (57,97,130,163,203) (47,85,120,154,199)

γB

t

TS

standard

(σ ) over all different runs are reported in Tables IV

D

(θ r )(θb ) (θ g )

γB

t

and V for D=2, 4 and 6. At last, one-tailed t-test, an eminent statistical significance test, has been performed between the two participating algorithms to establish the dominance of one of them over another. The test was performed at 5% confidence level, which means that the alternative hypothesis would be accepted as the test result if the desired p value becomes fewer than 0.05. The results of such test have been documented in Table VI.

2

(139) (125) (116)

10909.76

01.35

4

(57,122,199) (70,121,172) (62,113,176)

12791.73

22.12

13116.47

35.42

deviation

6

Table III. Best result of QITSMLTCI and TS for multi-level thresholding for Barche

Since, both algorithms present equal threshold values for all different colour components in all different runs for D=2, there is no change in the mean value ( γ AV

)

(48,68,108,173,211) (61,95,130,164,200) (49,79,119,159,204)

Barche QITSMLTCI

and standard

deviation (σ ) . Hence, for lowest level of thresholding, each

D

(θ r )(θb ) (θ g )

γB

t

algorithm sounds equally strong and accurate. The proposed algorithm finds better fitness value than that of its classical counterpart for D=4 and 6. On the other hand, it can be easily perceptible from the values in Tables IV and V that QITSMLTCI is more accurate and robust as well compare to its corresponding conventional meta-heuristic algorithm. Seeing as, the mean fitness values of the quantum inspired algorithm is more than other algorithm for higher level of thresholding for each image (except D=2), the alternative hypothesis is assumed to be better results for the quantum version of algorithm. The results of one-tailed t-test prove the alternative hypothesis that in turn establishes the effectiveness of the proposed algorithm over other algorithm. A computer with configuration, Intel(R) Core (TM) i3, 2.53GHz PC and 2GB RAM, has been incorporated for conducting the experiments. It can be easily substantiated from Tables II and III that the proposed quantum inspired algorithm is more time efficient compared to its classical version. The original test images are shown in Figs. 1(a) and (b), respectively. After exercising QITSMLTCI, the thresholded images at various levels are designed at Fig 2.

2

(106) (99) (92)

5949.75

01.03

4

(74,127,169) (54,105,150) (47,91,134)

7258.03

11.14

6

(64,107,142,164,187) (46,81,122,151,179) (46,78,104,133,152)

7472.60

15.19

314

TS D

(θ r )(θb ) (θ g )

γB

t

2

(106) (99) (92)

5949.75

01.15

4

(74,127,169) (55,110,150) (44,92,133)

7257.34

25.14

6

(61,106,141,172,193) (43,86,128,154,184) (42,77,101,131,153)

7470.81

37.20

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( γ AV ) and

Table IV. Average fitness

standard deviation

(σ )

of

QITSMLTCI and TS for multi-level thresholding for Tulips Tulips D

QITSMLTCI

TS

( γ AV )

(σ )

( γ AV )

(σ )

2

10909.76

0

10909.76

0

4

12791.98

0.27

12790.84

1.33

6

13117.03

1.81

13112.86

2.78

( γ AV ) and

Table V. Average fitness

standard deviation

(σ )

(c)

(d)

(e)

(f)

of

QITSMLTCI and TS for multi-level thresholding for Barche Barche D

QITSMLTCI

TS

( γ AV )

(σ )

( γ AV )

(σ )

2

5949.75

0

5949.75

0

4

7257.44

0.41

7255.72

1.01

6

7470.90

1.17

7468.61

1.85

Table VI. Test results of one-tailed t-test for Tulips and Barche at different levels of thresholding Tulips Barche D 2 4 6

QITSMLTCI & TS

QITSMLTCI & TS

Nan (Not-a-Number) 0.038 0.005

Nan (Not-a-Number) 0.016 0.023

Fig. 2. Test image after thresholding for D = 2, 4, 6 in (a), (c), (e) for Tulips and, (b), (d), (f) for Barche, respectively, after exercising QITSMLTCI for multi-level thresholding

VII. CONCLUSION AND FUTURE SCOPE The design of new Quantum Inspired Tabu Search for Multi-level thresholding for colour images is the dictum of this paper. The proposed algorithm uses Otsu’s method to determine optimal thresholds value for multi-level thresholding. QITSMLTCI demonstrates that it is faster with regards to its computation than the classical algorithm. Furthermore, the statistical significance test confirms that the proposed algorithm is more efficient as compared to other algorithm. The results of multi-level thresholding for QITSMLTCI appear promising and it persuades further researches to apply in multi-level and multi-objective optimization for colour images. REFERENCES [1] R. C. Gonzalez and R. E. Woods, Digital Image Processing, Englewood Cliffs, NJ: Prentice-Hal, 2002.

(a)

(b)

Fig. 1. Original test images (a) Tulips and (b) Barche

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(a)

(b)

[6] L. K. Grover, “Quantum computers can search rapidly by using almost any transformation,” Physical Review Letters, vol. 80, no. 19, pp. 4329–4332, 1998.

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[7] C. Blum and A. Roli, Metaheuristic in combinatorial optimization: overview and conceptual comparison, IRIDIA: Technical Report, 2001-13. [8] F. Glover and G. A. Kochenberger, Handbook on Metaheuristics, Kluwer Academic Publishers, 2003. [9] G. A. Ferguson and Y. Takane, Statistical Analysis in Psychology and Education, McGraw-Hill Ryerson Limited, 2005. [10] S. Kirkpatrik, C. D. Gelatt and M. P. Vecchi, “Optimization by simulated annealing”, Science, vol. 220, pp. 671–680, 1983. [11] F. Glover, “Tabu search, part I”, ORSA Journal on Computing, vol. 1, pp. 190–206, 1989. [12] P. Hansen and N. Mladenovic. “An introduction to variable neighborhood search”, In S. Voss et al. (eds), in Proc. of MIC 97 Conference, pp. 433-458, 1997. [13] D. Mcmohan, Quantum computing explained, Hoboken, New Jersey: John Wiley & Sons, Inc., 2008. [14] S. Bhattacharyya and S. Dey, “An efficient quantum inspired genetic algorithm with chaotic map model based interference and fuzzy objective function for gray level image thresholding,” in Proc. Int. Conf. on Computational Intelligence and Communication Systems, pp. 121–125, 2011. [15] D. Ventura and T. Martinez, “Quantum Associative Memory with Exponential Capacity,” in Proc. Int’l Jt. Conf. Neural Networks, pp. 509-513, May 1998. [16] N. Otsu, “A threshold selection method from gray level histograms”, IEEE Transactions on Systems, Man, and Cybernetics, vol. 9, pp. 62–66, 1979.

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