Multi-Level Image Segmentation Based on Fuzzy Tsallis Entropy and Differential Evolution S. Sarkar ECE Department RCCIIT Kolkata - 700015, India Email:
[email protected]
S. Das ECS Unit Indian Statistical Institute Kolkata - 700108, India Email:
[email protected]
Abstract— This paper presents a fuzzy partition and Tsallis entropy based thresholding approach for multi-level image segmentation. Image segmentation is considered as one of the most critical tasks in image processing and pattern recognition area. However, discriminating many objects present in an image automatically is the most challenging one. As a result, multilevel thresholding based methods gain importance in recent times, because of its ability to split the image into more than one segments. Efficiency of these algorithms still remains a matter of concern. Over the years, fuzzy partition of 1-D histogram has been employed successfully in bi-level image segmentation to improve the separation between object and the background. Here a fuzzy based technique is adopted in multi-level image segmentation scenario using Tsallis entropy based thresholding. Differential Evolution, a widely used meta-heuristic in recent times, is used for lesser computation time of the proposed algorithm. Both visual and statistical comparison of outcomes between Tsallis and Fuzzy - Tsallis entropy based methods are given in this paper to establish the superiority of the technique. Keywords—Multi-level Image Segmentation, Multi-Level Thresholding, Tsallis Entropy, Fuzzy Entropy, Differential Evolution, MSSIM
I. INTRODUCTION Image segmentation, the process of discriminating objects from its background in pixel level, has become the utmost component of image analysis. Over the years, segmentation is being applied as a basic step for several computer vision applications like feature extraction, identification, image registration etc. Image segmentation done via bi-level thresholding, that subdivides the image into two homogenous regions, based on texture, histogram, edge etc., uses only one threshold value. In the year 2004, bi-level maximum Tsallis entropy (MTE) based image segmentation was introduced by Portes de Albuquerque [1], which was based on extended version of Havrda and Charvat’s 1967 paper [2], proposed by Tsallis in 2001 [3]. Later multi-level image segmentation gained popularity for its ability of sub-dividing the image into more than one segment. Image segmentation, done via multilevel
S. Paul, S. Polley, R. Burman, S. S. Chaudhuri ETCE Department Jadavpur University Kolkata - 700032, India
thresholding, splits the image into different classes by selecting multiple threshold points. Otsu (1979) [4] developed a non-parametric multi-level image segmentation algorithm, which was later modified by Kapur et. al. [5]. It makes the image more useful for later analysis and study. However, the computation complexity of these methods had increased to a significant amount. Recently, to reduce the computational time, several multilevel Tsallis entropy based image segmentation techniques were being proposed by using some state-of-art metaheuristics of recent time, like Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC) and Bacterial Foraging Optimization Algorithm (BFOA) [6, 7]. In this paper, Differential Evolution (DE) has been used to find the maximum Tsallis entropy for accurate and faster computation. DE is no doubt, a powerful and real parameter optimizer of current times [8, 9]. It has been shown that DE can outperform GA and PSO when it is used for multi-level thresholding based image segmentation problems [10, 11]. The fidelity of these methods for automatic separation between multiple classes persists as a vital concern to the researchers. In 1972, Luca and Termini [12] tried a modification to solve this problem and introduced fuzzy partition technique for image segmentation. A thorough literature survey reveals that fuzzy based approach got widely popular among the researchers for bi-level thresholding, but its utilization in multi-level thresholding problems remained unattended. Notably, in 2001 Zhao et al. [13] first applied a multi-level approach; outcome of his work was not much encouraging. He defined three membership functions for 3level thresholding i.e. dark, medium and bright. Then he derived a necessary condition for maximizing the objective function, which was as following: pbright = pmedium = pdark = 1/3. Based on this paper, in 2003 Tao et al. [14] proposed a 3-level fuzzy entropy based image segmentation technique, where he partitioned the image into dark, grey and white regions by using 3 different membership functions, Z-function, function and S-function, respectively. The threshold values were obtained by maximizing the total entropy, which made it
a real parameter optimization problem, and use of global optimization algorithms are inevitable in this scenario. A generalized fuzzy partition based Tsallis entropy thresholding technique for multilevel image segmentation has been presented here using the same membership functions as Zhao et. al. Here, DE is used to obtain the highest objective value and to boost the computational speed of the proposed scheme. Outcomes are compared against Tsallis entropy on 300 images under the Berkeley Segmentation Data Set (BSD 300) by using performance metrics like Probabilistic Rand Index (PRI), Variation of Information (VoI), Global Consistency Error (GCE), and Boundary Displacement Error (BDE) in context to multilevel thresholding. The basic concept of Tsallis entropy has been described in Section 2, which also describes the concept of multilevel image thresholding based on Tsallis entropy and fuzzy partition. A brief introduction of Differential Evolution (DE) is given in Section 3. The experimental results and comparative performances are presented in Section 4. Lastly, the paper is concluded in Section 5.
1 1
1
3
1
4
and 1 1 where and
The Tsallis global thresholding method can be further extended by using more than two classes e.g. class C1, class class Cn [2]. Let us define a vector C2 , …, , , , … , . The entropy of the system for (n-1) t threshold values can be defined by the following pseudo additive entropic rule:
…
1 1
Let ∆n =
,
,
, . . . .. ,
p1 ,p2, ,. . . , pn
pi ≥ 0, i=1,2,…, n, n ≥ 2,
=1}
is a set of discrete finite n-ary probability distributions. Havrda and Charvat defined entropy of degree α as [2]:
1
1 2
1
1
where ∑ni=1 pi
.
1
1 1
∆ ,
1
In 1988, independently Tsallis [3] proposed a one parameter generalization of the Shannon entropy as:
,
5
where
II. MULTI-LEVEL FUZZY-TSALLIS ENTROPY A. Multi-Level Tsallis Entropy
.
1 1
,
1
1
,
.
where number of segmentation levels would be n. For the 0 and ease of computation, two dummy thresholds 1 are introduced with . A global optimization technique is highly preferable here to satisfy equation 4 and hence get the optimum thresholds. B. Multi-Level Fuzzy-Tsallis Entropy
1 1
1
,
2
where α is a real positive parameter not equal to one For an image the entire distribution is divided into two classes, one for object (class C1) and another for background (class C2)). Then the priori Tsallis for each distribution can be defined as:
A simple trapezoidal membership function is used in this paper to estimate the membership of n segmented regions, , ,… , by using (n - 1) threshold values. The membership function will have (2 × (n – 1)) unknown parameters, namely a1, c1… an-1, cn-1 where 0 ≤ a1 ≤ c1 ≤ … ≤ an-1 ≤ cn-1 ≤ L - 1(Fig. 1.). Then the following membership function can be derived for (n - 1) level thresholding or n level segmentation.
1
∑
…, histogram.
6 0
. and P is the normalized
The optimum value of parameters can be obtained by maximizing the total entropy
0
, 1
,…,
, 1
6
0 0 6
…
8
Global optimization technique is needed to optimize equation no. (8) efficiently and also to reduce the time complexity of the proposed method. The (n - 1) no of threshold values can obtained using the fuzzy parameters in the following way: ,
1
.
,…,
.
9
III. DIFFERENTIAL EVALUATION (DE) The Differential Evolution (DE) [8, 9] algorithm recently emerged as a simple, yet very competitive evolutionary optimizer. Since the late 1990s, the DE family of algorithms has been frequently adopted to tackle multi-objective, constrained, dynamic, large-scale, and multimodal optimization problems. Fig. 1. Fuzzy membership function for n - level segmentation
The maximum fuzzy entropy for each segment of n –level segments can be defined by
DE is a population-based optimizer like other evolutionary individual (parameter vector) of the algorithms. In DE the population at generation (time-step) is a -dimensional vector containing a set of optimization parameters: ,
1 1 1 1 1 1
1
,
1
,
1
.
7
where ,
,
,
,
,…,
,
(10)
The population members are initialized randomly to cover the feasible search volume as much as possible. In each (say), first a generation, to change a population member donor vector is created through mutation. It is the method of creating this donor vector that distinguishes among various DE schemes. One of the earliest and simplest variants of DE, now called the DE/rand/1 scheme, is used to implement the algorithm for our purpose here. Under this for each member, three other scheme, to create parameter vectors (say the 1, , and -th vectors such that , , 1, and are chosen at random from the current population. The donor vector is then obtained by multiplying a scalar number F with the difference component of of any two of the three. The process for the vector may be expressed as: the
,
,
.
,
,
11
Next, a binomial crossover operation takes place to increase the potential diversity of the population. The binomial crossover is performed on each of the variables whenever a uniformly generated random number, between 0 and 1, is less than or equal to the crossover rate , that appears as a control parameter for DE. In this case, the number of parameters, inherited from the donor or mutant, has a (nearly) binomial , a trial vector distribution. Thus, for each target vector is created in the following fashion: ,
,
0,1
,
0,1
12
0,1 0,1 is the jth For j = 1, 2, … , D and evaluation of a uniform random number generator. 1, 2, … … , is a randomly chosen index to ensure that gets at least one component from . Finally selection is performed in order to determine which one between the target vector and trial vector will survive to the next generation i.e. at time 1. If the trial vector yields a better value of the fitness function, it replaces its target vector in the next generation; otherwise the parent is retained in the population. The selection process can be schematically outlined as: 1
13
where . is the function to be maximized. The new population members at t = t+1, are iterated through the steps of mutation, crossover, and selection. The iterations are repeated till some stopping criterion is satisfied. The DE scheme described through equations (10) – (13) is collectively known as DE/rand/1/bin in literature [8]. The proof of convergence of this algorithm has been recently provided in [15] under mild regularity assumptions. IV. RESULTS & DISCUSSION A. Experimental Set Up The simulations are performed with MATLAB R2011b in a workstation with Intel® Core™ i5 2.9 GHz processor. The DE/rand/1/bin scheme is used to compute the threshold levels efficiently. The parametric set up of DE is shown in Table I. Parametric settings for GA and PSO are listed in Tables II & III respectively. Best parametric set up is maintained for each
of the metaheuristics. Results of the meta-heuristic algorithms have been provided as the mean of 50 independent runs where each run was continued till the exhaustion of D×1000 number of Fitness Evaluations (FEs), D denoting the search space dimensionality. Note that for an n levels segmentation or (n 1) level thresholding problem, the dimensionality of the search space is D =2* (n – 1). For simplicity, we denoted the segmentation level as Lv. TABLE I: EXPERIMENTAL SETUP FOR THE DE ALGORITHM PARAMETER
VALUE
NP F Cr
10*D 0.5 0.9
TABLE II: EXPERIMENTAL SET UP FOR THE GA PARAMETER VALUE Population size 100 Crossover Probability 0.9 Mutation Probability 0.001 Encoding binary TABLE III: EXPERIMENTAL SET UP FOR THE PSO ALGORITHM PARAMETER VALUE Wmax ,Wmin 0.4, 0.1 C1, C2 2 Swarm size 20
B. Performance Evaluation One of the key aspects of effective performance of Tsallis entropy is the value of α. Here in this paper, we maintained α = 0.8 as proposed in [5]. For testing and analysis, 300 images were used from the Berkeley Segmentation Data Set and Benchmark (BSDS 300). For statistical comparison, performance evaluation matrices, from Berkeley image segmentation database, are used. The matrices are: Probabilistic Rand Index (PRI), Variation of Information (VoI), Global Consistency Error (GCE) and Boundary Displacement Error (BDE). Higher value of PRI indicates better segmentation, whereas for rest, lower values point out the same [16]. The comparison of computational speed, accuracy, and robustness between the three meta-heuristic algorithms considered here, have been indicated in Table IV, in terms of average computational time (T) in seconds, mean best-of-therun objective function values (fmean), and the related standard
deviations (STD) corresponding to segmentationn levels (Lv) 3 and 4 as computed over some example imaages from the Berkeley image database. Best results for eeach level are marked in boldface. Entries of Table IV inddicate that DE outperforms its competitive algorithms for tthe considered numbers of segmentation levels in a statisticallly meaningful way. Also DE yields results with minimum stanndard deviation over repeated runs and this indicates greater robbustness of the algorithm, despite the use of stochastic compponents in the algorithmic framework. The Convergence plot oof the different optimization techniques for an example image,, are displayed in Fig. 3 and 4, which support the results show wn in Table IV and hence clearly establish superiority of DE. Due to space limitation we only provided results on some imaages from BSD 300 datasets. TABLE IV: PERFORMACE OF DE, PSO, GA ( COMPUT TED ON SOME EXAMPLE IMAGE FROM BERKELEY IMAGE SEGME ENTATION DATABASE) IMAGE
Lv 3
23084 4
ALGOR -ITHM DE
T(SEC)
fmean
5.220
6.149×104
0
PSO
5.309
6.103×104
2.180×102
GA
19.910
6.126×104
2.320×102
DE
9.095
1.443×106
4.777×10-10
9.216
1.386×10
6
1.301×104
1.423×10
6
1.725×104
4
PSO GA
3 103070 4
DE
5.110
4.612×10
PSO
5.587
4.577×104
3.308×102
GA
20.225
4.608×104
1.187×102
DE
8.684
9.865×10
9.298
9.659×105
1.173×104
GA
34.431
9.804×105
4.721×103
PSO GA
108005
DE 4
PSO GA
3 210088
5.149 5.507 20.313 8.962 9.297 37.219
4.954×10
4
2.239×10-11
4.899×10
4
2.572×102
4.947×10
4
1.0374×102
1.131×10
6
2.388×10-10
1.097×10
6
9.071×103
1.121×10
6
3.834×103
4
DE
5.082
4.085×10
5.661
4.064×104
1.335×102
GA
20.892
4.081×104
0.884×102
PSO GA
8.599 9.311 33.574
(b)
(c)
(d)
Fig. 2. Example Images from BSD D 300, Image No. (a) 23084 (b) 103070 (c) 108005 5 (d) 210088.
The obtained fuzzy membersh hip functions of the example images (as shown in Fig. 2.), are displayed in Fig. 5. For both Lv = 3 and 4, the resulting thresh hold values, achieved for both Tsallis entropy and Fuzzy-Tsalliss entropy (using equation (9)) are presented in Table V. The ressulting segmented images are formed as index image and shown n in Fig. 6 and Fig. 7 for Lv = 3 and 4 respectively. Column (a) denotes results of Tsallis entropy and column (b) denottes results of Fuzzy-Tsallis entropy. It is quite clear from th hese two figures that FuzzyTsallis offered better results than Tsallis entropy.
0
PSO DE
4
5
0
PSO DE
3
33.995
STD
(a)
0
8.358×10
5
3.5832×10-10
8.087×10
5
1.479×104
8.253×10
5
4
1.043×10
Fig. 3. Convergence Plot of DE, PSO and GA for Lv = 3 on Image No. “210088.jjpg”.
To investigate further, we co ompared the performance of these two methods on 300 Imaages of BSD 300 using the famous performance measuremeent matrices of the Berkeley Image Segmentation Dataset. For the sake of better comparison, we took a new appro oach by counting the number of images for which each of the entropy based method gives better result than the other an nd finally we calculate the percentage on 300 images. The results r are displayed in Table VI and VII for Lv = 3 and 4 respectively. Except for PRI, Fuzzy-Tsallis entropy depicts better outcomes than the conventional Tsallis entropy. Fig. 4. Convergence Plot of DE, PSO and GA for Lv = 4 on Image No. “210088.jpg”.
(a) (a)
(b))
Fig. 5. Fuzzy membership function for example Images (ass in same order as Fig. 2.) from BSD 300 Images (a) 3 - level and (b) 4 – levvel Segmentation
(b)
Fig. 6. Example of 3 - level for im mages from BSD 300 dataset (a) Tsallis entropy (b) Fuzzy Tsallis Entro opy
TABLE V: THRESHOLD VALUES OBTAINED FOR 4 LEVEL SEGMENTATION Image
Lv
23084 103070 108005 210088 23084 103070 108005 210088
3
4
Tsallis Entropy 88 170 110 185 99 174 116 188 66 127 189 85 142 199 82 142 198 104 155 206
Fuzzy Tsallis Entropy 65 192 72 198 66 193 84 200 65 129 192 67 142 203 67 139 200 70 140 198
TABLE VI : PERCENTAGE OF METRICS VALUES FOR 3-LEVEL SEGMENTATION ( COMPUTED OVER 300 IMAGES FROM BARKELEY IMAGE SEGMENTATION DATABASE) METHOD
BDE
PRI
GCE
VoI
Tsallis
48.66
52.33
42.66
43.33
Fuzzy - Tsallis
51.33
47.66
57.33
56.66
TABLE VII : PERCENTAGE OF METRICS VALUES FOR 4-LEVEL SEGMENTATION ( COMPUTED OVER 300 IMAGES FROM BARKELEY IMAGE SEGMENTATION DATABASE) METHOD
BDE
PRI
GCE
VoI
Tsallis
47
53.66
39.66
46.33
Fuzzy - Tsallis
53
46.33
60.33
53.66
V. CONCLUSION It can be concluded from the above discussion that Tsallis entropy based thresholding methods, based on fuzzy cpartitions for multi-level segmentation, performs significantly better than Tsallis entropy based approaches. Fuzzy entropy based thresholding techniques delivers satisfactory results in case of both visual comparison and statistical comparison. Undisputedly, DE adds speed and accuracy to this algorithm. However, a better constrained DE or other upgraded variants of DE could be used to attain better performance. Also several other membership functions could be tested for better separation of the segmented regions. More image performance metrics could be used in future to prove the competence of segmentation algorithms. REFERENCES
(a)
(b)
Fig. 7. Example of 4 - level for images from BSD 300 dataset (a) Tsallis entropy (b) Fuzzy Tsallis Entropy
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