Multilevel Image Thresholding Based on Tsallis Entropy and ...

Multilevel Image Thresholding Based on Tsallis Entropy and Differential Evolution Soham Sarkar1, Swagatam Das2, and Sheli Sinha Chaudhuri3 1

Electronics and Communication Engineering Department, RCC Institute of Information Technology, Kolkata – 700015, India 2 Electronics and Communication Sciences Unit, Indian Statistical Institute, Kolkata – 700108, India 3 Electronics and Telecommunication Engineering Department, Jadavpur University, Kolkata – 700032, India [email protected], [email protected], [email protected]

Abstract. Image segmentation is known as one of the most critical task in image processing and pattern recognition in contemporary time, for this purpose Multi Level Thresholding based approach has been an acclaimed way out. Endeavor of this paper is to focus on obtaining the optimal threshold points by using Tsallis Entropy. In this paper, we have incorporated a Differential Evolution (DE) based technique to acquire optimal threshold values. Furthermore, results are compared with two state-of-art algorithms- a. Particle Swarm Optimization (PSO), and b. Genetic Algorithm (GA). Several image quality assessment indices are applied for the performance analysis of the outcome derived by applying the proposed algorithm. Keywords: Multilevel Image Segmentation, Tsallis Entropy, Differential Evolution, MSSIM, WPSNR.

1

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 proposed by Portes de Albuquerque et al [1]. Proposed technique of image segmentation is based on work done by Tsallis et. al. [2], which is an extended version of Havrda and Charva’t paper in 1967 [3]. Later multi-level image segmentation gained popularity for its ability for sub-dividing the image into more than one segment. It makes the image more useful for the later analysis and studies. However, the computation complexity of these methods had increased to a significant B.K. Panigrahi et al. (Eds.): SEMCCO 2012, LNCS 7677, pp. 17–24, 2012. © Springer-Verlag Berlin Heidelberg 2012

18

S. Sarkar, S. Das, and S.S. Chaudhuri

amount [4]. Now, to reduce the computational time, several multi–level Tsallis entropy based image segmentation techniques are being proposed by using some state-of-art metaheuristics of recent time, like Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Bacterial Foraging Algorithm (BFA) [5]. 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 time [6, 7]. It has been shown that DE can outperform GA and PSO when it is used for multi-level thresholding based image segmentation problems [8]. The rest of the paper includes the basic concept of Tsallis entropy -Section 2. Section 3 describes multi-level image thresholding. A brief introduction of Differential Evolution (DE) is given in Section 4. The experimental results and comparative performance are presented in Section 5. Lastly the paper is concluded in Section 6.

2 Let

Tsallis Entropy ,

,

∆n =

, . . . .. ,

∆ , where

p1 ,p2, ,. . . , pn

n

pi ≥0, i =1,2,…, n, n ≥ 2,

pi = 1} i=1

is a set of discrete finite n-ary probability distributions. Havrda and Charva’t defined entropy of degree α as [3]: 1 2

1

1

.

1

In 1988, Independently Tsallis proposed a one parameter generalization of the Shannon entropy as [16] 1 1

1

,

2

where α is a real positive parameter not equal to one. Both (1) and (2) have similar expression except the normalization factor. The Tsallis entropy is non-extensive in such a way that for a statistical independent system, the entropy of the system is defined by the following pseudo additive entropic rule [1] 1

3

For an image the entire distribution is divided into classes, one for object (class A) and another for background (class B).Then the priori Tsallis for each distribution can be defined as 1 1

1

,

4

Multilevel Image Thresholding Based on Tsallis Entropy and Differential Evolution

19

And 1 1

1

,

5

where

The optimum threshold value can be determined by: 1

3

6

Multi-level Tsallis Entropy

The Tsallis global thresholding method can be further extended by using more than two classes e.g. class A, class B and class C [9, 10, 11]. The entropy of the system for threshold values t1 and t2 is defined by the following pseudo additive entropic rule 1

,

7

where 1

1

1 1 1 1 1

,

,

1

,

,

1

,

.

A more generalized equation can be reformed from (7) for n threshold values: , ,…, 1

.

.….

,

8

where no of segmentation level would be n+1. For the ease of computation two 0, 1 and , are being dummy threshold incorporated. DE is used to find the threshold values by maximizing equation (8).

4

Differential Evolution (DE)

DE, a population-based global optimization algorithm, was proposed by Storn in individual (parameter vector) of the population at generation (time) 1997. The is a -dimensional vector containing a set of optimization parameters:

20


,

,

,……,

,

(9)

,

In each generation to change the population members (say), a donor vector is created. It is the method of creating this donor vector that distinguishes the various DE schemes. In one of the earliest variants of DE, now called DE/rand/1 for each member, three other parameter vectors (say the scheme, to create , and -th vectors such that , , 1, and ) are chosen 1, at random from the current population. The donor vector is then obtained multiplying a scalar number F with the difference of any two of the three. The process component of the vector may be expressed as, for the ,

.

,

,

(10)

,

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 randomly picked number between 0 and 1 is within the value. In this case the number of parameters inherited from the mutant has a (nearly) binomial distribution. Thus for each target vector , a trial vector is created in the following fashion: ,

0,1 0,1

, ,

(11)

0,1 0,1 is the jth evaluation of a uniform random For j = 1, 2, ….., D and number generator. 1, 2, … … , is a randomly chosen index to ensures 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 in 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: 1 (12) where

5

. is the function to be minimized.

Experimental Results

The simulations are performed with MATLAB R2011b in a workstation with Intel® Core™ i3 3.2 GHz processor. For testing and analysis, 4 images are used from the Berkeley Segmentation Dataset and Benchmark (obtainable from http://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/). The original gray scale images, their 1-D histograms are shown in Figure 1. The segmented greyscale image is formed by using a generalized equation. , where k = 1, 2, ⁄ ,n+1.

1 1

,

,

,

,

13


21

The performance of DE based method is compared with other efficient global optimization techniques like GA and PSO. In case of DE, the following parametric setup is used for all the test images: 0.9, 0.5. Best possible parametric setup is also maintained for GA and PSO. The used parametric setup for PSO is: C1 = C2 = 2 and w = 0.9 and for GA: crossover probability = 0.85 and mutation probability = 0.1 Results are reported as the mean of the objective functions of 50 independent runs. Each run contains 200 generations. The value of α is set to 0.3 .Through detail analysis it is found that Tsallis entropy based multi-level segmentation performs efficiently for α’s value below 0.4.

(a)

(b)

(c)

(d) Fig. 1. Test Images and their histograms (a) 12074 (b) 101087 (c) 209070 (d) 300091

22


Table 1 shows the required computational time for each levels of image “12074”, using DE, GA and PSO. Results clearly show that DE requires lesser computation time than the other algorithms. In addition, mean objective function value ( ) and standard deviation ( ), shown in Table 2, establish the superiority of DE over modern day’s state-of-art global optimization techniques. (For convenience, best results are shown in bold letters.) Table 1. Average Computational time in Seconds (Image No. 12074) for 200 generation in a single run No. Of Levels 2 3 4

DE 1.1019 1.4901 1.5390

PSO 1.1595 1.9173 2.2348

Table 2. Mean objective function value ( DE

L 2 3 4

GA 1.6945 2.4987 3.4223

) and standard deviation (

) of “12074”

PSO

1.1889e+004 1.8243e+005 1.8436e+006

0.0 0.0 0.0

1.1886e+004 1.8165e+005 1.8040e+006

2.5793 6.3079e+002 1.7922e+004

GA 1.1880e+004 1.8236e+005 1.8311e+006

7.640 45.1361 5.4529e+003

Different threshold values of test images, obtained by using meta-heuristics, are given in Table 3. Fig. 2 shows the segmented gray level test images. Weighted Peak Signal to Noise Ratio (WPSNR) [14] and Mean Structural Similarity Index Measurement (MSSIM) [15] (between original grayscale image and segmented grayscale image) are indicated to establish the betterment of results. Table 3. Threshold values acquired by using DE, GA, PSO Image 12074

101087

209070

300091

No. of Levels 2 3 4 2 3 4 2 3 4 2 3 4

DE 95 175 78 137 68

115

196 162

209

159 130

174

201 168

211

181 153

203

68 140 51 42

105 85

109 185 90 79

146 123

99 173 53 53

115 102

Threshold values PSO 95 175 77 138 189 64 112 156 207 67 140 51 105 159 46 87 133 177 109 185 91 146 198 79 122 165 209 99 173 53 117 184 53 102 151 205

99 79 70 67 54 41 110 90 80 99 53 53

GA 177 136 196 118 163 209 140 109 162 82 128 173 187 150 202 126 169 211 173 117 184 103 154 203


a. 1.

a. 2.

a. 3.

b. 1.

b. 2.

b. 3.

c. 1.

c. 2.

c. 3.

d. 1.

d. 2.

d. 3.

23

Fig. 2. Segmented images obtained by MTE-DE method ((a. 1.), (b. 1.), (c. 1.), (d. 1.) 3-level thresholding, (a. 2.), (b. 2.), (c. 2.), (d. 2.) 4-level thresholding, (a. 3.), (b. 3.), (c. 3.), (d. 3.) 5level thresholding Table 4. WPSNR and MSSIM of test images using DE Name of Images 12074 101087 209070 300091

WPSNR(dB) 2 3 4 22.0073 23.1569 23.9040 21.7986 24.0456 26.1610 20.0278 21.8441 22.8655 18.6277 27.7601 30.0469

2 0.1770 0.6749 0.3302 0.2010

MSSIM 3 0.2283 0.7495 0.4363 0.6748

4 0.2694 0.7762 0.4999 0.7080

24

6


Conclusion

In this paper we have proposed a scheme based on differential evolution for MultiLevel Thresholding by using MTE. DE approach has definitely increased the speed and accuracy of our selected algorithm. This technique was applied to various real images; the results demonstrated the efficiency of working of the algorithm, and the feasibility of our proposal. This finding could encourage further researches, still 2-D histogram based approaches and fuzzy based approach could be implemented to achieve better performance.

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