MULTI-LEVEL THRESHOLDING BASED ON DIFFERENTIAL EVOLUTION-TSALLIS FUZZY ENTROPY

Transcription

1 MULTI-LEVEL THRESHOLDING BASED ON DIFFERENTIAL EVOLUTION-TSALLIS FUZZY ENTROPY ADITYA RAJ SITI NORUL HUDA SHEIKH ABDULLAH Fakulti Teknologi & Sains Maklumat, Universiti Kebangsaan Malaysia ABSTRACT This paper presents a Multi level image Thresholding technique, using Tsallis entropy and fuzzy partition with a novel threshold selection technique. A global search algorithm is required for the calculation of optimal threshold values; a popular Meta heuristic algorithm Differential evolution is used successfully for solving this problem. One of the most critical tasks in image processing is Image segmentation; the proposed threshold selection technique is compared with Shannon entropy or fuzzy entropy, and Tsallis-Fuzzy entropy with existing threshold selection techniques, using over 0 images from popular image database of Berkeley as benchmark images, these images have been selected with each having a unique gray level histogram with important image quality metrics such as SSIM, PSNR, and SNR, furthermore for statistical analysis Friedman test values and average ranks have been calculated which establishes the superiority of the proposed threshold selection technique over other existing threshold selection techniques for image segmentation. A second group of dataset has been taken from [24] and comparison between the DE based proposed thresholding technique with patch-levy based bees algorithm based kapur entropy criterion and Otsu between-class variance criterion, on the basis of SSIM values has been conducted for the assessment of the image quality of the thresholds, also the CPU time elapsed for converging to the optimal threshold values has been compared to determine a faster algorithm. Standard deviation has been compared to give a measure of stability and robustness. For statistical significance Wilcoxon and Friedman tests have been conducted and p-value evaluated. INTRODUCTION Image segmentation is a crucial step in image analysis. Image analysis essentially refers to extracting the desired features from the image. Image segmentation techniques result in a set of segments that collectively cover the entire image these segments are called binary images. Thresholding is a simplest method of image segmentation and hence is widely used. Image segmentation via thresholding can be a bi-level thresholding that produces only one threshold value, or a multi-level thresholding. In Bi-level thresholding the intensity of each pixel in an image is compared with a fixed constant and if the value is say greater than the set constant that pixel is replaced with a black pixel or otherwise a white pixel and multi-level thresholding techniques sub divide image into more than one segment and hence have an advantage of better manipulation of the binary images. Several approaches have been used to generate optimal values for thresholds with different Entropy, among these are Otsu [] in 979, developed a clustering based automatic image thresholding technique by maximizing the between class variance, although this method is time consuming, two dimensional entropy was used to calculate threshold by Abutaleb et al. (989)[2], which was later modified by Kapur et al.[3], the threshold is found by maximizing entropy of the histogram of gray levels of the resulting classes. Shannon [4] defined entropy which was a decreasing function of a scattering of random variable. However Shannon entropy cannot be used to distinguish weak features overlapping with much stronger features [5], this could be made possible by the use of tsallis entropy [6], here the value of q can be varied from a large positive to a large negative value, where positive value of q will be more sensitive to features that have a higher frequency of occurrence and a large negative value of q in tsallis entropy would be more sensitive towards features with less frequency of

2 occurrence. It makes the post processing of image more efficient in terms of extracting desired features. Various state of the art meta-heuristics have been proposed for multilevel tsallis entropy, it has been shown that DE can outperform GA and PSO when it is used for multi level thresholding based segmentation problems[ 7][8]. A fuzzy approach introduces uncertainty or a degree of fuzziness into the assessment of threshold, it corresponds to a stronger reliability and robustness.luca and termini in 972 [9], introduced fuzzy partition technique as a modification, for image segmentation for bi-level thresholding., later in 200 Zhao [0] first applied a multi-level approach using fuzzy c-partition, and Tao in 2003[] on the basis of Zhao s method designed a three level thresholding method for image segmentation. Sarkar et. al. (203) showed that tsallis entropy with fuzzy c-partition for image segmentation performs better than Tsallis entropy based image segmentation [2] 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. Here a fuzzy partition with a novel threshold selection method, based Tsallis entropy thresholding technique for multi level image segmentation has been proposed. To boost the computational speed and maintain high objective value with simple implementation, DE has been used. Fuzzy and Tsallisfuzzy entropy have also been compared against the proposed method with the first group of images which are over 0 images under the Berkeley segmentation Dataset, however for the scope of this paper 5 test images have been illustrated, each with unique gray-level histogram [3]. Image quality metrics such as SSIM, PSNR and SNR; have been estimated for number of Thresholds varying from three to twelve. Using Friedman test on the SSIM values average ranks have been calculated for 4 different algorithms used in this test and compared that show the superiority of the proposed method. SSIM as a quality metric has been used since; SSIM was designed to improve on traditional methods such as peak signal-to-noise ratio (PSNR) and mean squared error (MSE), which have proven to be inconsistent with human visual perception [20]. Alain hore et al [25] compared PSNR and SSIM with various kinds of image degradations such as Gaussian blur, additive Gaussian white noise, jpeg, jpeg2000 compression and stated as a final conclusion, that the values of the PSNR can be predicted from the SSIM and vice-versa. The PSNR and the SSIM mainly differ on their degree of sensitivity to image degradations. The second group of test images consist of 5 images that have been taken from Wassim et al[24]. In [24], threshold values of these images were calculated using Meta heuristic based on modified bees algorithm called the patch levy based bees algorithm (PLBA), Bacterial Foraging optimization (BFO) and modified Bacterial Foraging optimization (MBFO) using Kapur entropy and Otsu s methods. Here these thresholds have been used to calculate SSIM values for quality assessment and have been compared with the SSIM produced by the proposed thresholding technique, which has been later used to form the average rank table, to definitively justify the merit of the proposed algorithm. Objective of this paper is to investigate the performance of the proposed differential evolution based Tsallis-Fuzzy novel threshold selection technique in regard with deriving the optimal thresholds which improve the segmented image quality in multilevel image thresholding and at the same time to make the algorithm more efficient by reducing the time complexity and studying the stability and robustness of the algorithms using parameters such as standard deviation and objective function, against DE based Fuzzy and Tsallis-fuzzy entropy with existing membership functions, and other Metaheuristic based algorithms such as Patch-Levy Bees Algorithm (PLBA), Modified Bacterial

3 Foraging Optimization (MBFO) and Bacterial Foraging Optimization (BFO) based kapur s and otsu s entropy. For calculation of statistical significance Friedman and Wilcoxon tests have been conducted and comparisons have been made. The rest of the paper is organized as follows: the proposed fuzzy Tsallis entropy method is described in section II. The implementation of DE is briefly reviewed in section III. Experimental results and conclusion remarks have been given in section IV and V respectively. II MULTI-LEVEL FUZZY TSALLIS ENTROPY TSALLIS ENTROPY [4][0] Given a discrete set of probabilities p i, where p i is the probability of an event x i from the event X as {x x 2.. x n }, with the condition p i =, q is any real number such that the Tsallis entropy can be defined as :- H n q (P) = n q [ p q i= i ] () Where q is called the entropic index. Claude Shannon defined a measure of entropy as, n S = i= p i logp i (2) Tsallis entropy can be said a generalization of the Shannon entropy where the value of q. However where Shannon entropy is additive tsallis entropy is often termed as a pseudo-additive. Given two independent systems A and B, the joint probability density satisfies, P (A, B) = P (A)*P (B) (3) The tsallis entropy for the system with two independent systems is given by, H (A, B) = H (A) + H (B) + (-q)*h (A)*H (B). (4) The parameter (-q) is a measure of departure from additive nature. Let I be an image of dimensions MxN where at (x,y), a pixel location the grey level pixel is I(x,y), let D i = {(x,y): I(x,y) = i, (x,y) D}, where k is gray levels of the image that is i = 0,, h i = m i N M (5) Where m i is the number of elements ind i, 256 = {D 0, D, D 255 } is the probability partition of D with a probability distribution. p k = P(D k ) = h k, where k = 0,...255, if n- thresholds segment an image then it will have n segments from the above definitions we can have n = {D 0, D, D n } which is a probability partition of D with a probability distribution of, p d = P(D d ) = h d, where d = 0,...n-. Where n<255 is the number of gray levels in segmented image. MULTI-LEVEL TSALLIS ENTROPY For an image the tsallis entropy can be extended further for more than two systems, let there be n systems, A, A 2,.. A n, let us define a vector t = {t 0, t, t 2,.. t n }. The entropy of the system for (n-) threshold values can be defined by the following pseudo additive rule,

4 φ q (t) = Argmax([H q A (t) + H q A2 (t) H q An (t) + ( q)h q A (t)h q A2 (t).. H q An (t)]) (6) Where, Similarly H q A = H q A2 = q [ H q An = ( P(i) q t i=0 P (t)) (7) A t 2 ( P(i) q P (t)) A 2 [ q i= t + ] (8) L ( P(i) [ q i= t n + ] (9), where number P A n (t))q of segmentation levels would be n, and thresholds t 0 = 0, and t n = L such that t 0 < t <. < t n. Membership functions A membership function is a curve that defines how each point in the input space is mapped to a membership value. In this paper trapezoidal membership function and difference logistic function based membership function have been applied for calculating the fuzzy entropy and Tsallis-Fuzzy entropy. A} TRAPEZOIDAL MEMBERSHIP FUNCTION Here the trapezoidal membership function has been modelled to depend on n number of variables, a, a2, a3.., an where n represents the number of membership functions, these variable are calculated with the differential evolution algorithm, such that the fitness function i.e. the fuzzy entropy and the Tsallis fuzzy entropy is maximized. The set of all variables along with 0 and 255 are mentioned in P. 2n membership functions have to be defined for implementation of n threshold segmentation. Let µ represent the membership function. µ d = Z d (k, a, a 2, a 3, a 4 ) (0) P = {0, a, a 2, a n, 255} () 0 < a a 2.. a n < 255 (2) k = 0,,2,3,4. n (3) μ 0 (k) = { k a a 2 k a 2 a a k a 2 0 k > a 2 (4)

5 μ (k) = 0 k a k a a 2 a a k a 2 a 2 k a 3 a 4 k a 4 a 3 a 3 k a 4 { 0 k > a 4 (5) μ n (k) = { k a n 2 a n a n 2 0 k a n 2 a n 2 < k a n a n < k 255 (6) 0 a a 2 a 3 a Fig () Trapezoidal membership function The thresholds are given by, t i = P[2i]+ P[2i+] 2, where i = 0,,2,3... n- B} DIFFERENCE OF TWO LOGISTIC FUNCTIONS AS MEMBERSHIP FUNCTION Here we define a logistic function, as shown in figure(2), it is a more generalized form of a sigmoid function, and a sigmoid function has a wide application in machine learning, as it is easy to handle and mimics a realizable growth function as it is self limiting. f(x) = C + e k(x x 0) (7) Fig(2) logistic function Where C is the curve s maximum value, k is the steepness of the curve and atx = x 0, is the midpoint of the function, here we take C as C= so that the function ranges from 0 to, and x 0 controls where on the x-axis the growth should be, as for a given value of x 0, it defines the mid-point of the growth. hence when C =, k =, and x 0 = 0 a logistic function becomes a sigmoid function. Here we use

6 difference of two sigmoid functions, that in appearance looks similar to that of a trapezoidal membership function is used, as shown in figure(3), Let µ represent the membership function, similar to the trapezoidal membership function, µ d = Z d (k, a, a 2, a 3, a 4 ) (8) P = {0, a, a 2, a n, 255} (9) 0 < a a 2.. a n < 255 (20) k = 0,,2,3,4. n μ 0 = - +e (k) + e a (2) (k a2) (22) μ = + e a - (k a2) + e a 3(k a4) μ n 2 = + e a - n 2(k an ) + e a (23) n(k 255) μ n = + e a n (k an) +e 255(k 255) (24) 0 a a 2 a 3 a Fig (3) proposed membership function Here for the calculation of thresholds, t i = P[2i]+ P[2i+] 2 (25), where i= 0,,2,3... n-, The maximum Tsallis fuzzy entropy for each segment of n-level segments can be defined as, L ( P(i) μ 2(i) ) q P A 2 (t) i=0 ] (27) H q A (t) = L (P(i) μ (i) ) q [ q i=0 ] (26) P A (t) H q A2 (t) = q [

7 L ( P(i) μ n(i) ) q P A n (t) Similarly for A n, H q An (t) = q [ i=0 ] (28) the optimum value of parameters can be obtained by maximizing the total entropy, φ q (t) = Argmax([H q A (t) + H q A2 (t) H q An (t) + ( q)h q A (t)h q A2 (t).. H q An (t)]) (29) One of the popular Meta heuristics Differential Evolution has been used for optimization of the total entropy function φ q (t), as shown in equation (29), n- number of threshold values can be obtained from the fuzzy parameters calculated using DE, with equation (25). III DIFFERENTIAL EVOLUTION ALGORITHM Differential evolution is a stochastic, population based optimisation algorithm, which was introduced by storn and price in 996[5], developed to optimize real parameters and real valued functions. DE is a population based optimizer, the i th parameter vector of the population at generation G is a D- dimension vector containing a set of D optimization parameters. We also define the size of the population NP, and generation number G. The parameter vectors have the form as in (), for each parameter an initial value is uniformly selected randomly but within the defined upper and lower bounds. Each of the N parameters undergoes mutation, recombination and selection. x i,g = [x,i,g, x 2,i,G,. x D,i,G ] i =,2,3. N. (30) For each target vector or N parameter vector a mutant vector can be defined as shown in (2), mutation expands the search space, for a given parameter vector x i,g, three vectors are randomly selected such that their indices are distinct, for each i th member three parameter vectors say x r,g, x r2,g, x r3,g vectors such that r, r 2, r 3 [, NP], and r r 2 r 3 are chosen at random from the current population. x r3,g ) (3) v i,g+ = x r,g + F(x r2,g The weighted difference of two vectors is multiplied with a scalar number F and is added to the third this gives us the mutant vector. v i,g+ is also called a donor vector and the mutation factor F is a constant from [0, 2]. Here a simple variant of DE DE/rand/ scheme is used; depending on how the donor vector was created different variants of DE are distinguished. After mutation recombination takes place, it incorporates the successful solutions for the previous generation. A binomial crossover operation takes place to increase the potential diversity of the population, the binomial crossover is performed on each of the D variables, whenever a uniformly generated random number between 0 and is less than or equal to the crossover rate CR, that acts as a control parameter for DE. From each of the elements of target vector x i,g and the elements of the donor vectorv i,g+, the trial vector u i,g+ is developed and the elements of the donor vector enter the trial vector with probability CR. (32) u j,i,g+ = { v j,i,g+ if rand j,i CR or j = I rand x j,i,g if rand j,i > CR and j I rand

8 i =,2,3 N ; j =,2,3. D Where I rand is a random integer from [, 2, 3...D] to ensure that u i,g+ gets at least one component from x i and rand j [0, ] is the j th evaluation of a uniform random number generator. Selection determines whether the target or the trial vector survives to the next generation. The selection operation is described as (33) x i,g+ = { u i,g if f(u i,g ) f(x i,g ) x i,g otherwise Where f () is the objective function to be minimized. Therefore, if the objective of the new trial vector u i,g is equal to or less than the objective of the old trial vector, x i,g then x i,g+ is set to u i,g ; otherwise, the old value x i,g is retained. This process of mutation, recombination and selection will continue until some stopping criterion is reached, here that criteria is the number of iterations. The DE scheme described through equations is collectively known as DE/rand//bin. The proof of convergence of this algorithm has been recently provided in [6] under mild regularity assumptions. IV EXPERIMENTAL RESULTS EXPERIMENTAL SETUP: The multilevel thresholding problem deals with finding optimal thresholds within the range [0, L-] that maximizes the total entropy function φ q (t) (equation 29), the parameter q in total entropy function has been set to.2, T. Maszcyk et al.[5], compared the accuracy for Renyi entropy, Tsallis entropy and Shannon entropy per class on colon cancer data set and DLBCL dataset with varying q values, for both these datasets Tsallis entropy with q in the range from..3 gave the best results, improving both specificity and sensitivity. The DE/rand//bi scheme is used to compute the threshold levels efficiently. The simulations were performed with MATLAB R206a in a workstation with Intel(R) CORE(TM) i3,.70 GHz CPU; RAM 4GB with windows 0 64-bit operating system. 20 images were used here for the experiments, taken from the Berkeley segmentation dataset and benchmark [20], each of the image chosen has a unique gray level histogram, 5 of these images have been depicted in figure (4), the number of thresholds investigated are 2,3...2, with 00 number of iterations for each experiment for the images in first group, however for the images in the second group 50 number of iterations were used as used in [24]. The control parameters F and CR of the DE algorithms were set to 0.5 and 0.9 respectively [7,8]. DE algorithm the objective function evaluation is computed for NP x G, where NP is population size and G is the number of generations, the DE calls the Tsallis-fuzzy entropy function one time per generation. The number of generations for DE was set to 00. The value of NP was set to 0*D, where D is the search space dimensionality. For n levels segmentation the dimensionality of the search space D= 2*(n-). COMPARISON STRATEGIES AND PERFORMANCE EVALUATION: For testing and analysis two groups of images have been used, for the first set 0 images were used from the Berkeley segmentation data set and benchmark, with image size 48x32 out of which 5 images shown in table() have been shown for the scope of this paper. For statistical comparison, performance evaluation matrices were used. After segmenting the image with the chosen threshold

9 and the chosen procedure, the image quality measures are analysed using the famous image measures such as SSIM, PSNR and SNR as shown in table(2). The mathematical expressions of these parameters are as follows MSE = Mean Squared error, PSNR = peak signal-noise ratio, SSIM = structural similarity index (34) MSE (x,y) = MN H i= W j= [x(i, j) y(i, j)]2 255 PSNR (x,y) = 20log 0 ( ) db (35) MSE (x,y) SSIM (x,y) = (2μ x μ y + C )(2σ xy + C 2 ) (μ x 2 + μ y 2 C )(σ x 2 + σ y 2 + C 2 ) (36) 2 2 Where x and y are the original and segmented images;μ x and μ y are the average values, σ x and σ y are the variance values, σ xy is the covariance, C and C 2 are the regularization constants for the luminance and contrast terms, C = (k L) 2 and C 2 = (k 2 L) 2, with L = 256, k = 0.0 and k 2 = For the first set of images these before mentioned performance evaluators have been used to compare two algorithms Tsallis-fuzzy with trapezoidal membership function implemented with the meta heuristic DE, and Tsallis-fuzzy with the proposed membership function as shown in table(2) with the threshold values in table(3). The Friedman test is the non-parametric alternative to the one-way ANOVA with repeated measures. To statistically analyse the results obtained for the DE-Tsallis-fuzzy with proposed MF, the Friedman test was used. The Friedman test is the best known statistical method for testing the performance differences between more than two algorithms [22] [23]. Table (4) presents the value of ranks for the Friedman test with SSIM values, calculated for 4 algorithms fuzzy entropy with trapezoidal membership function, meta-heuristic DE, fuzzy entropy with the proposed membership function, Tsallis fuzzy entropy with the trapezoidal membership function and Tsallis fuzzy entropy with the proposed membership function, table (5) presents the average ranks for the SSIM values and the Friedman test value- p for images of the first set. The p-value calculated using the Friedman test showed very high significant differences among the other DE Based algorithms. It can be clearly seen that the proposed algorithm performs better than the other algorithms, and has the best rank. Table() contains the first set of test images, numbered from i-v, each has a unique gray level histogram as shown,

10 IMAGE(I) IMAGE(II) IMAGE(IV) IMAGE(III) IMAGE-(V) IMAGE(VI) IMAGE(VII) IMAGE(VIII) IMAGE(IX) IMAGE-(X) Table () First set of Test images with histograms Table (2): SSIM, PSNR and SNR values with DE-Tsallis Fuzzy entropy using trapmf and the proposed membership function with the 5 test images in the first set are shown:- TRAPMF PROPOSED MF Level SSIM PSNR SNR SSIM PSNR SNR

11 Table(3): the values of Multilevel thresholds varying from 2 to 2 with DE-Tsallis Fuzzy entropy using trapmf and the proposed membership function for the images in the first set are shown TRAPMF PROPOSED MF

12 66.5, , 44, , 26, 65, 26 45, 93, 35, 76, , 79.5, 08.75, 55.75, 83.5, , 7, 97.75, 30.75, 62.5, 89.5, , 66.5, 89, 8, 42, 73.5, 95.5, ,63,85.25,09.25,34.25,59,8.25,203.5, ,58.25,79,04.75,22.75,48.75,67.25, 88, , , 55, 73.5, 94.5, 2, 3.5, 54, 73.5, 92.5, 2, ,46.5,62.75,82.25,99.5,22.25,39.5,58.75,78.25,95.75,23.5, , ,33, ,23,62, ,3.5,44,75.25, ,84.5,8.25,5.5,76.75, ,8.25,06.75,39,63.25,93.75, ,78.75,0,25.5,48.25,73,93, ,73.5,97.5,23.25,45.75,67.75,88.75,20.5, , 69.75, 85, 06, 26.5, 47.25, 69.25, 90, 22.5, ,60.25,76.75,9.25,08.25,25.5,45.5,68.75,90.25,209.25, , 58.75, 77.25, 97.25, 2,27.5,46.75,64.75,83,204.5,22.5, , ,52, ,04,4.25, ,03,40.5,84.75, , 92, 3.5, 54, 85, ,92.5,3,33.25,57,86.25,223 35,75.5,92.25,4,34.5,59,86.5, ,59.5,76.25,93.25,2.75,33.5,58,86.25, ,62.25,78.75,95.25,4.5,33,54.25,83.75,96.75, , 59.5, 76, 93.25, 3.25, 34, 57, 83.25, 96.75, , 60, 74, 94.75, 3.25, 30, 44.5, 63.5, 79.5, 93.25, , , 5.75, , 02.75, 52, , 82.75, 9.25,54.5, , 70.75, 07.5, 43,75.75, , 65.5, 97, 27.75, 54.5, 85, , 60.75, 89.5, 09, 36, 67.75,90, ,5.75,78.5,97.25,24,48.75,74,94.25, ,54.25,73,95.75,5,4.5,62.25,84,99.25, ,47,68.5,87,05,26.75,46.25,65.75,85.75,205, ,45,65,8.25,96.75,3,32.75,5.75,69.25,9,20, , ,54.5, ,07,57.5,206 35,03,36.75,69, ,72.75,07,42.75,73, ,75.5,06,36,66.5,94, ,66,89,2.5,38,66,93.5, ,63.25,85.75,08.5,33.75,53.25,72.5,20.25, , 59.25, 80.75, 05.75,3,48.5,67.5,88.75,20.5, , 50, 70, 92.5,.5, 3.25,52,7,9.75,20.5, ,49.25,67,88,06.5,23.75,42.75,6.5,78.5,95.5,23.75, ,24 43,30.5, ,87.75,7,8 29.5,85.75,22.5,55.25, , 77, 2, 39.75,63.5,9 29.5, 8, 06, 28.5, 54, 78, ,62.25,83.5,06,28,53.75,78, ,58,84.25,06.25,29.25,50.25,78.75,95.25, ,54.75,7.25,94.25,5.5,40,56.5,78.75,95.75,2.5 5,4.75,60,77.5,08,26.75, 38, 53.75, 79, 95.5, ,39.5,53,76.25,90.5,0.25,29.75,42.5,57.25,7.5,97, , 32 57,4, ,2,40.5, ,96.5,25,54.5, ,00,26.5,55.75,93.25, ,97.5,23.75,53.25,80.25,2.75, , 83.25, 08, 32, 56.5, 82, 2.5, ,7.25,94.25,7.5,46,64.25,89.5,23.5, , 63, 87.5, 0.25, 28.75, 50, 68.75, 98.5,28.75, , 63.25, 84.75, 02, 6.25, 40.5, 63, 8.5,207.25,22, ,57.25,74.75,93.75,.25,29.25,47.25,66.5,84.25,206,220.25, , ,48.5, ,3.75,66.25, ,5.75,53.25,83.5, , 05, 30, 65, 9.75, , 86.5, 4, 40.75, 69.5, 95, ,76.75,97.75,28.25,52.5,82.5,204.25, , 74.5, 96.5, 2, 44, 65.5, 88.5, 2.25, ,72,90.25,3.25,30.75,52.5, 72.75, 96.5, 24.5, ,65.75,85.25,05.75,22,38.75,59,8,20.5,29.75, , 64, 79.75, 99.75, 3.75, 29.25, 48, 65, 80, 98.5, 26,239 55,6 4,5.75, ,07.5,43.5, ,95,22.5,57.75, ,80.25,3.75,43,77.75, ,78,08.25,42.25,74,98.75, ,70.25,92.5,6.5,4.75,73.5,204.75, ,66.25,92.75,6.5,4.5,63.25,85.5,208, ,55.5,74,99.5,6.25,4.75,65.25,84.5,206.75, ,48,64.25,89,05.5,27.75,49.75,64,85.5,208, ,48.25,74.25,85.75,07.75,24,44.5,62.25,80,97.25,27.25, , ,36.25, ,04.75,52.75, ,0.5,44.5,75, ,87.5,4.75,48.75,77, ,8.25,06.5,39.5,66,95.25, ,76.75,96.75,9.5,43,74.25,94.5, , 68.75, 9.25, 09.5,33.75,6.25,76.25,203.5, , 62.5, 82, 03.5, 25, 45.75, 62.75,79.75,208.5, ,33.25,5.5,76,95,2.75,44.25,68.25,86, , , 4,5,74.75, 93.25,09.75,33,53,68.25,200.25,28.5, Graph (), The following graph marks the ssim values of rest of the 5 images selected in the first group numbered from 5 to 0, values calculated with DE using Tsallis-Fuzzy proposed mf have been marked with blue, and are the values of SSIM calculated with DE using Tsallis-Fuzzy trapmf are marked with red as shown in

13 SSIM(ProposedMF) SSIM(TrapMF) SSIM(ProposedMF) SSIM(TrapMF) Graph. SSIM(ProposedMF) SSIM(TrapMF) SSIM(ProposedMF) SSIM(TrapMF) SSIM(ProposedMF) SSIM(TrapMF) In the statistical testing, a result has a statistical significance when it is very unlikely to have occurred given the null hypothesis, which is the default assumption that nothing changed. The significance level defined for a study α, is the probability of the study rejecting the null hypothesis, given that it were true, and the p-value of a result, p is the probability of obtaining a result at least as extreme, given that the null hypothesis were true. The result is statistically significant, by the standards of the study when p < α. The significance level for a study is chosen before data collection, and typically set to 5%. Here the significance level has been chosen as in order to avoid the type error. Under type error scenario we want to be very cautious about rejecting the null hypothesis, so we demand very strong evidence favouring the hypothesis before we reject the null hypothesis. Here for table 4. the Rank matrix of the test images in the first set for Freidman test on the quality metric SSIM for 4 DE based Algorithms, fuzzy entropy with trapezoidal MF and with the proposed MF, Tsallis-Fuzzy entropy with trapezoidal MF and with the proposed MF were calculated, where each value was ranked out of 4 with the highest value allotted rank and lowest rank 4.

14 Here the SSIM values of the proposed method have been tested statistically using the Friedman test, and p-value has been calculated, the observed p-values are less than pre-specified significance level. The p-values as shown in table (4) depict that the proposed scheme is statistically significant. From the table it is observed that group C has the least average rank that has been calculated considering SSIM values for thresholds from 2 to 2 levels, for the images in first group and hence the proposed method for multi-level thresholding method produces segments with considerably improved quality than other methods being compared. Table (4): Test Images A: DE-Fuzzy entropy(trapmf) B: DE-Tsallisfuzzy(trapmf) C: DE-Tsallisfuzzy(proposed-MF) D: DE-Fuzzy entropy(proposed-mf P-Value I <0.000 II <0.000 III <0.003 IV <0.000 V < The second group has 5 set of images taken from wassim et al[24], these 5 images have been used for comparison between the metaheuristic algorithm DE based proposed multi-level thresholding method with metaheuristic algorithms (PLBA, BFO, MBFO), based kapur entropy criterion and otsu s between-class variance criterion. The images from the first group used for the comparisons have been shown in the following table (5), each image selected has a unique gray-level histogram as shown. Table (5). Test images (52 52) employed in the second group and their corresponding histogram, a) Airplane (b) Hunter (c) Butterfly (d) living room (e) pepper. Following table (6), contains the threshold values calculated using meta heuristic (PLBA, MBFO, BFO), Based kapur entropy criterion from wassim et al[24], with the threshold values calculated using DE based proposed Tsallis-Fuzzy entropy criterion. (Levels: number of thresholds)

15 Image LEVELS PLBA MBFO BFO Proposed Airplane 2 76, 74 76, 74 76, 73 38, , 28, 82 66, 2, 82 66, 24, 86 35, 2, , 06, 44, 70, 07, 45, 84 7, 3, 49, 85 34, 04, 4, , 96, 27, 57, 87 66, 96, 27, 58, 88 68, 98, 3, 6, , 87.25, 7.25, 5, 82.5 Hunter 2 92, 79 92, 79 85, , , 7, 79 55, 9, 77 57, 04, , 92, , 90, 33, 79 48, 9, 33, 80 50, 98, 39, 80 22, 74.5, 20.75, , 9, 34, 79, 45, 9, 35, 79, 49, 93, 37, 79, 7.5, 5.5, 87.75, 40, Living Room 2 94, 75 94, 75 89, 70 46, , 03, 75 56, 27, 84 7, 24, 73 24, 9.5, , 98, 49, 97 47, 95, 38, 85 60, 04, 47, 89 24, 89, 3.75, , 85, 24, 62, 46, 89, 3, 65, 47, 94, 34, 69, 22, 73.25, 08.5, 53.75, Pepper 2 80, 50 80, 50 79, 49 39, , 3, 63 78, 9, 7 69, 00, 55 30, 98.75, , 0, 52, 97 52, 89, 30, 73 63, 09, 44, 78 28, 92.25, 36.5, , 85, 8, 57, 48, 86, 23, 60, 54, 89, 3, 64, 23.5, 72.25, 0.5, ,80.75 butterfly 2 96, 44 96, 44 97, , , 9, 52 87, 23, 64 75, 09, ,.5, , 05, 33, 64 77, 05, 33, 66 73, 97, 27, , 98.5, 23.5, , 96, 20, 44, 73, 96, 20, 44, 74, 97, 20, 44, 38.5, 94.25, 6, 37, Table 6. Following table (7), contains the threshold values calculated using meta heuristic (PLBA, MBFO, BFO), Based on Otsu s between-class variance criterion from wassim et al[24], with the threshold values calculated using DE based proposed Tsallis-Fuzzy entropy criterion. (Levels: number of thresholds) Image LEVELS PLBA MBFO BFO Proposed Airplane 2 6, 74 6, 74 7, 75 38, ,46, 9 95, 46, 9 9, 47, 90 35, 2, , 32, 74, , 32, 74, , 27, 69, , 04, 4, , 08, 43, 7, 08, 43, 79, 7, 0, 38, 75, 79, , 87.25, 7.25, 5, 82.5 Hunter 2 5, 6 5, 6 5, , , 86, 35 34, 86, 33 36, 86, , 92, , 72,, 46 30, 72,, 46 3, 80, 20, 52 22, 74.5, 20.75, , 53, 88, 22, 30, 72, 05, 34, 3, 73, 09, 4, 7.5, 5.5, 87.75, 40, Living Room 2 87, 45 87, 45 87, 46 46, , 23, 63 76, 23, 63 75, 24, 64 24, 9.5, , 97, 32, 68 56, 97, 32, 68 64, 02, 34, 72 24, 89, 3.75, , 88, 20, 46, 49, 88, 20, 46, 56, 94, 25, 48, 22, 73.25, 08.5, 53.75, Pepper 2 7, 38 7, 38 73, 37 39, , 22, 69 64, 22, 70 63, 25, 74 30, 98.75, , 88, 29, 72 50, 89, 29, 72 54, 89, 28, 7 28, 92.25, 36.5, , 85, 8, 50, 5, 86, 7, 50, 47, 86, 23, 58, 23.5, 72.25, 0.5, ,80.75 butterfly 2 98, 5 98, 52 99, , , 9, 60 8, 7, 59 78, 7, ,.5, 44.5

16 4 7, 98, 26, 62 7, 98, 26, 62 75, 05, 35, , 98.5, 23.5, , 99, 25, 53, 72, 99, 25, 53, 76, 04, 29, 54, 38.5, 94.25, 6, 37, Table 7. SSIM values for the given thresholds were calculated, and compared using the Friedman test, with the ssim values shown in table 8 and 9, the average rank values are shown in table 0, as the sample size is less p-values are not a good approximation, hence here we see the average ranks, with being the best choice and 4 the worst. Following table (8), contains the SSIM values using the thresholds calculated by meta heuristic (PLBA, MBFO, BFO), Based on kapur entropy criterion from wassim et al[24], with the SSIM values for the thresholds calculated using DE based proposed Tsallis-Fuzzy entropy criterion. (Levels: number of thresholds) Image LEVELS PLBA MBFO BFO Proposed Airplane Hunter Living Room Pepper butterfly Following table (9), contains the SSIM values using the thresholds calculated by meta heuristic (PLBA, MBFO, BFO), Based Otsu s between-class variance criterion from wassim et al[24], with the SSIM values for the thresholds calculated using DE based proposed Tsallis-Fuzzy entropy criterion. (Levels: number of thresholds) Image LEVELS PLBA MBFO BFO Proposed Airplane Hunter

17 Living Room Pepper Butterfly Table (0) contains the average ranks calculated for Friedman test, for the SSIM values in table (8) IMAGES PLBA MBFO BFO PROPOSED Airplane Hunter Living room Pepper Butterfly Table. Table () contains the average ranks for Friedman test, for the SSIM values in table (9), IMAGES PLBA MBFO BFO PROPOSED Airplane Hunter Living room Pepper Butterfly Table. Table(2) comparisons of CPU times(in seconds) elapsed for the PLBA based on kapur s entropy criteria, otsu s between class variance criteria [24] and the proposed DE based on Tsallis-Fuzzy with sigmoid based membership function, for the second group of test images.(levels: Number of thresholds) Image Levels PLBA(kapur) PLBA(otsu) DE:SystemA DE:SystemB Airplane Hunter Living room

18 Pepper Butterfly Table 2. For CPU times, the calculations for the proposed algorithm were made using a workstation with MATLAB 206a with System (A): intel(r) Core TM i CPU at 4.20GHz 4 core and 6G RAM on Microsoft windows 0 64-bit operating system, and with System (B): intel(r) Core TM i CPU@ 3.4GHz 4 Core 6G RAM the values for PLBA were taken from [24], were calculated with Intel(R) Core TM i GHz 4 core and 4G RAM with Microsoft Windows 7 64-bit operating system. For statistical significance Wilcoxon signed ranks test was conducted on the second group of test images between the proposed and PLBA based algorithms and the results have been presented in the tables 3 and 4. R + is the sum of ranks for which the DE outperformed PLBA and the CPU time was lesser and R is the sum of ranks for which PLBA outperformed DE and PLBA based CPU time was less. The significance value α was taken as 0.05 with the sample space of 20. The following table gives the result of Wilcoxon test with p-value for evaluation. Comparison R + R p-value SYSTEMA:DE-Proposed VS PLBA(Kapur) E-05 Statistically Significant SYSTEMB: DE- Proposed VS E-04 Statistically Significant PLBA(Kapur) Table 3. Comparison R + R p-value SystemA: DE-Proposed VS PLBA(Otsu) E-04 Statistically significant SystemB: DE-Proposed VS PLBA(Otsu) p-value > α( = 0.05) Table 4. From the Wilcoxon test it is clear that the proposed method out performed PLBA ( based kapur entropy criterion ) and PLBA(based Otsu method) when system A was employed, however For System B, we can observe from the table 2, that the proposed method performs considerably better than both PLBA based algorithms but Wilcoxon test suggests that for PLBA(Based otsu method), more data needs to be analyzed to be compared with for resolving its statistical significance. Standard deviation is an important tool to measure the robustness and stability of an algorithm, in the following tables 5 and 6 give comparisons between the standard deviation values of the images of the second group produced by the Metaheuristic algorithms ( BA based and BFO based algorithms) based on kapur s and otsu s criteria[24] respectively against the DE based proposed Tsallis-Fuzzy

19 algorithm. From the tables it can be clearly seen that the proposed algorithm is more stable as the values of standard deviation produced are consistent and less than the other Metaheuristics irrespective of the number of thresholds, that shows robustness of the proposed method. Table(5), comparison of standard deviations(std) of the results obtained by the optimization algorithms(plba,bfo,mbfo) based kapur s entropy criterion and DE, based proposed Tsallis- Fuzzy entropy using the second group of test images(levels : number of thresholds) Image LEVELS PLBA MBFO BFO Proposed Airplane e e e e e e e e-4 Hunter e e e e e e Living Room e e e e e e e e e-05 Pepper e e e e e e e e e e-4 Butterfly e e e e e e e e Table 5. Table(6), comparison of standard deviations(std) of the results obtained by the optimization algorithms(plba,bfo,mbfo) based otsu s between class variance criteria entropy criterion and DE, based proposed Tsallis-Fuzzy entropy using the second group of test images(levels : number of thresholds) Image LEVELS PLBA MBFO BFO Proposed Airplane 2.369e e e e e e e e-4 Hunter e e e e e Living Room e e e e e e e e-05 Pepper 2.890e e e e-5

20 4.3642e e e e-4 Butterfly e e e e e e e Table 6. From the table 5 and 6, the values that are better than PLBA has been written in bold, it can be observed that the proposed algorithm out performs MBFO and BFO and PLBA based Otsu s method but is slightly better than the PLBA based kapur s criterion. However in terms of CPU time and image quality metric, the proposed algorithm outperforms other algorithms, Hence it can be concluded it performs better than the compared methods and is slightly more stable and robust and this can be improved by applying the various different schemes in differential evolution with the proposed threshold selection technique. BA method has shown enough performance to handle different types of nonlinear problems in various fields but it has the downside of possibly being trapped in the local optimum, in each iteration newly generated particles explore new positions based on better opportunities. Differential evolution algorithm is diversified; all members have a possibility to move to global optimum, which grants this algorithm higher stability and better convergence time with at the same time producing optimal thresholds that indicate improved segmented image quality. From the following it can be clearly seen that the Proposed DE, Based Tsallis-Fuzzy entropy algorithm converges to the optimal thresholds much faster than compared to the PLBA based kapur Entropy and Otsu s thresholding algorithm. Also, it was observed that the proposed DE- Tsallis Fuzzy with sigmoid based membership function performed better than other algorithms such as fuzzy entropy with Trapezoidal membership function and the proposed membership function in regard with image quality metrics SSIM, PSNR and SNR, The algorithm proposed here also outperformed stateof-the-art metaheuristic based methods that employ Bacterial Foraging Optimization (BFO), Modified Bacterial Foraging Optimization (MBFO) and (PLBA) Patch-levy-Based Bees Algorithm adopted with Kapur s (ME) and Otsu s method in terms of image quality metric SSIM, however the PSNR values for the same were at best comparable. For measuring stability and robustness of the proposed algorithm standard deviation was calculated and compared with PLBA, MBFO and BFO based kapur entropy and Otsu s entropy criterion. The DE based proposed Tsallis-Fuzzy algorithm was found to be more stable and consistent than others. The proposed sigmoid based membership function is more controllable than trapezoidal membership function as its slope value can be easily manipulated and hence less rigid than the latter can intern produce better optimized values of thresholds. Also trapezoidal membership function can have points at which it might not be differentiable that can cause problems when using algorithms that utilize differential equations whereas sigmoid based membership function, bell shaped membership functions and Gaussian MFs are differentiable at all points. More study is required and more number of parameters can be tested to better evaluate and improve the performance of this method. V. CONCLUSION For multi level image thresholding, it is essential to not only produce better thresholds but also to do it in a reasonable computational complexity and with robustness, in this paper a novel sigmoid based

21 threshold selection technique with Tsallis Fuzzy, Differential Evolution based algorithm has been proposed. The proposed DE- Tsallis-fuzzy with sigmoid MF based entropy method has been compared with DEbased Tsallis Fuzzy, trapezoidal MF based entropy, where image quality metrics SSIM, PSNR and SNR were compared. Comparisons were made also between fuzzy entropy and tsallis entropy with the two MFs. Finally image quality metric SSIM, CPU time and standard deviation was compared with some state-of-the-art metaheuristic algorithms based on the bacterial foraging optimization (BFO) and those based on Bees algorithm (PLBA), Patch-levy nased bees algorithm. For statistical significance Friedman s test and Wilcoxon tests were performed. Consequently it can be concluded that the DEbased Tsallis-Fuzzy entropy method produced better results than other membership function(trapezoidal MF) and fuzzy entropy in terms of quality of the segmented image. Also the proposed algorithm was more stable than the algorithms PLBA, MBFO and BFO. DE-based proposed Tsallis Fuzzy entropy method performed better than PLBA, MBFO and BFO in terms of the image quality of the segmented image, and was able to converge to the optimal values of thresholds much faster. Proposed method was more robust and stable than MBFO, BFO and PLBA(based Otsu s criterion), as was observed from the standard deviation values, however it did slightly better than PLBA( based Kapur s method). The results obtained from the proposed algorithm is promising, different variants of Differntial evolution with novel sigmoid based membership functions could perform better in the future. The author would therefore like to apply them to complex and real time image processing and recognition tasks. ACKNOWLEDGMENT The author would like to thank the Ministry of Higher Education, Malaysia for providing facilities and financial support under Prototyping Research Grant Scheme No. PRGS//206/ICT02/UKM/02/ entitled Intelligent Vehicle Identity Recognition for Surveillance and UKM DIP entitled Object Descriptor via Optimized Unsupervised Learning Approaches. Not forgetting research team members at Digital Forensic Lab and Medical and Health Informatics Lab at Faculty of Information Science and Technology, Universiti Kebangsaan Malaysia. REFERENCES []. Otsu, A threshold selection method for grey level histograms, IEEE Transactions on System, Man and Cybernetics SMC, vol. 9, no., pp , 979. [2]. Abutaleb, A.S., 989. Automatic thresholding of gray level pictures using two dimensional entropy. Computer Vision Graphics Image Process. 47, [3]. Kapur, J.N., Sahoo, P.K., Wong, A.K., 985. A new Method for Gray-Level Picture Thresholding Using the Entropy of the Histogram. Computer Vision, Graphics, and Image Processing, 29, , 985. [4]. Shannon, C., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana, Ill. (964)

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