A Fuzzy Entropy Based Multi-Level Image Thresholding Using Differential Evolution S. Sarkar1(&), S. Paul3, R. Burman3, S. Das2, and S.S. Chaudhuri3 1
Department of Electronics and Communication Engineering, RCC Institute of Information Technology, Kolkata 700015, India
[email protected] 2 Electronics and Communication Sciences Unit, Indian Statistical Institute, Kolkata 700108, India
[email protected] 3 Department of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata 700032, India
[email protected]
Abstract. This paper presents a multi-level image thresholding approach based on fuzzy partition of the image histogram and entropy theory. Here a fuzzy entropy based approach is adopted in context to the multi-level image segmentation scenario. This entropy measure is then optimized to obtain the thresholds of the image. In order to solve the optimization problem, a meta-heuristic, Differential Evolution (DE) is used, which leads to a faster and accurate convergence towards the optima. The performance of DE is also measured with respect to some popular global optimization techniques like Particle Swarm Optimization (PSO) and Genetic Algorithms (GAs).The outcomes are compared with Shannon entropy, both visually and statistically in order to establish the perceptible difference in image. Keywords: Multilevel image segmentation Fuzzy entropy evolution CWSSIM MSSIM FSIM GSM
Differential
1 Introduction Image thresholding, the technique to discriminate objects from its background at pixel level is one of the most important tasks of image analysis. Automatic separation between objects and background remains the most difficult and intriguing domain in the field of image processing and pattern recognition. Scientific literature presents several image segmentation processes, such as gray level thresholding, interactive pixel classification, neural network based approaches, edge detection, and fuzzy based segmentation etc. Comprehensive surveys on such techniques can be found in [1–7]. Gray level global thresholding has been a popular segmentation method. There are many techniques available for this purpose e.g. entropy based global thresholding Kapur et al. [8]; Sahoo et al. [9]; Pal [10]; Li [11]; Rosin [12]. Further, techniques like Shannon entropy, Renyi’s entropy and Tsallis entropy provide the scope for investigating the process of efficient separation of the image between objects and background. © Springer International Publishing Switzerland 2015 B.K. Panigrahi et al. (Eds.): SEMCCO 2014, LNCS 8947, pp. 386–395, 2015. DOI: 10.1007/978-3-319-20294-5_34
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Image segmentation, done via multilevel thresholding, splits the image into different classes by selecting multiple threshold points. Otsu [13] developed a non-parametric multi-level image segmentation algorithm which was later modified by Kapur et al. [8].A majority of the proposed methods for thresholding are histogram dependent and as there are no clear boundary between regions, these histogram based segmented leads to some ambiguity. Luca and Termini [14] tried a modification to solve this problem and introduced a fuzzy partition technique for image segmentation. Bloch examined the applications of fuzzy spatial relationship in image processing and image interpretation area [15]. Although application of fuzzy partition technique in multi-level thresholding scenario remained unattended. Notably, in 2001 Zhao et al. [16] first applied a multi-level approach by defining three membership functions for 3-level thresholding i.e. dark, medium and bright. Based on this paper, in 2003 Tao et al. [17] proposed a 3-level fuzzy entropy based image segmentation technique, where 3 different membership functions, Z-function, F-function and S-function, were used. The threshold values were obtained by maximizing the total entropy by applying a popular global optimization technique, GA [18]. Both Zhao and Tao show the betterment of results by means of visual comparison, but lacks statistical evaluations. Another important reason of using metaheuristics is to minimize the computational time and complexity of the algorithm as much as possible [16]. To pursue with the present research interest a fuzzy-entropy based multi-level image segmentation process, boosted by Differential Evolution (DE) is proposed in this paper. DE is arguably one of the most powerful real parameter optimizers of current interest [19, 20]. It has been shown that DE can outperform GA and PSO when it is used for multi-level thresholding based image segmentation problems [21, 22]. Extensive simulations have been undertaken to demonstrate the efficiency and robustness of this DE based scheme in comparison with other popular global optimization techniques like GA and PSO in terms of computational time, mean objective value and standard deviation. Results are tested against Shannon’s entropy. Both visual and statistical comparison of the segmented images, are provided. Statistical comparison is done via state of art Image Quality Assessment (IQA) metrics. This paper has been organized as the following: - In Sect. 2 a brief introduction to fuzzy entropy and multi-level fuzzy entropy based on probability partition and its mathematical formulations are presented. Following this Differential Evolution is being discussed in Sect. 3. In Sect. 4, the test findings of our proposed method are presented along with their statistical analysis. Finally, Sect. 5 concludes the paper unearthing future avenues of research.
2 Concept of Multi-Level Fuzzy Entropy 2.1
Multi- Level Shannon Entropy Let P ¼ p1 ; p2; p3 ; :::::; pn 2 Dn ,where Dn ¼ f p1 ; p2; ; . . .; pn jpi 0; i ¼ 1; 2; . . .; P n; n 2; ni¼1 pi ¼ 1g is a set of discrete finite n-ary probability distributions. Then entropy of the total image can be defined as:
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H ðP Þ ¼
n X
pi log2 pi ;
ð1Þ
i¼1
I denote a 8 bit gray level digital image of dimension M × N. P is the normalized histogram for image with L = 255 gray levels. Now, if there are n 1 thresholds (t), partitioning the normalized histogram into n classes, then the entropy for each class may be computed as, H1 ðtÞ ¼
t1 X pi i¼0
H 2 ðt Þ ¼
P1
ln
pi ; P1
t2 X pi pi ln ; P P2 i¼t þ1 2 1
H n ðt Þ ¼
L1 X pi pi ln : P Pn i¼t þ1 n n1
where, P 1 ðt Þ ¼
t1 X
pi ; P2 ðtÞ ¼
t2 X
pi ; . . .; Pn ðtÞ ¼
i¼t1 þ1
i¼0
L1 X
pi :
i¼tn1 þ1
where For ease of computation two dummy thresholds t0 ¼ 0, tn ¼ L 1 are introduced with t0 \t1 \. . .\tn1 \tn . Then the optimum threshold value can be found by uðt1 ; t2 ; . . .; tn Þ ¼ Arg maxð½H1 ðtÞ þ H2 ðtÞ þ . . . þ Hn ðtÞÞ:
2.2
ð2Þ
Multi-Level Fuzzy Entropy
A classical set A can be defined as a collection of element that can either belong to or not belongs to set A. Whereas as according to fuzzy set, which is a generalization of classical set, an element can partially belongs to a set A. A can be defined as A ¼ fðx; lA ð xÞÞjx 2 X g;
ð3Þ
where, 0 lA ð xÞ 1 and lA ð xÞ is called the membership function, which measures the closeness of x to A. For simplicity trapezoidal membership function is used in this paper to estimate the membership of n segmented regions, l1 ; l2 ; . . .; ln by using 2 (n-1) unknown fuzzy 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 level thresholding
A Fuzzy Entropy Based Multi-Level Image Thresholding
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Fig. 1. Fuzzy membership function for n – level thresholding
8 > > >
> > :
8 0 > > > kan2 > < cn2 an2 ln1 ðkÞ ¼ 1 > kcn1 > > an1 > : cn1 0 ln ðkÞ ¼
8
