Performance Study of Harmony Search Algorithm for Multilevel Thresholding

1 Introduction

Image segmentation is a low-level image processing task that aims at partitioning an image into well-separated regions such that each one contains pixels with similar proprieties like gray level, color, texture, etc. Many image segmentation techniques have been reported in the literature [26]. Among them, thresholding is one of the simplest techniques for performing image segmentation. According to the number of thresholds, it can be as bi-level thresholding and multilevel thresholding. Bi-level thresholding divides the image into two regions, while multilevel thresholding divides the pixels into several regions.

Thresholding methods can be classified into parametric and nonparametric methods [28]. In the parametric approaches, the statistical parameters of the classes in the image are estimated. They are computationally expensive, and their performance may vary depending on the initial conditions. In the nonparametric approaches, the thresholds are determined by maximizing some criteria, such as between-class variance or entropy measures.

The application of entropic measures to image thresholding was first proposed by [27] and improved by Kapur [17]. Recent developments of statistical mechanics based on a concept of non-extensive entropy, also called Tsallis entropy, have intensified the interest of investigating a possible extension of Shannon’s entropy to information theory [32]. Tsallis entropy generalizes the Boltzmann/Gibbs’s traditional entropy to non-extensive physical systems by introducing a new parameter to measure the degree of nonextensivity [39]. In this paper, we use three objective functions: between-class variance, Kapur and Tsallis entropy as a criterion to select the optimal thresholds.

Most of the existing bi-level thresholding methods are easily extendable to multilevel thresholding as well. However, when the number of thresholds increases, complexity of the thresholding problem also will increase, and the traditional method requires more computational time. However, the computational complexity increases exponentially as the number of thresholds increases. For K thresholds, and L gray levels, the time complexity grows up to (L + K)!/L!K! [5]. It might be very difficult to derive a systematic and analytic solution when the number of thresholds increases.

Hence, in recent years, researchers have been attracted to bio-inspired metaheuristics for solving the multilevel thresholding problem. In this context, various thresholding algorithms are proposed, which use different metaheuristics. In this field, we can cite the genetic algorithms (GA) [10, 14, 37], particle swarm optimization (PSO) [11, 21, 38], differential evolution (DE) [9, 25, 30], artificial bee colony (ABC) [15, 16, 39], firefly algorithm (FA) [15]; bacterial foraging (BF) algorithm [31]; cuckoo search (CS) algorithm [4, 29]; bat algorithm [3, 23]. These methods give satisfactory performance when used to solve multilevel thresholding problem and use different objective functions.

The harmony search (HS) algorithm is an evolutionary algorithm proposed by Geem et al. [13].

Unlike current metaheuristic algorithms that imitate natural phenomena, i.e. physical annealing in simulated annealing, human memory in Tabu search, and evolution in evolutionary algorithms, HS algorithm is inspired from the improvisation process by which the musicians adjust their individual tones of their instruments through variation resulting in a better harmony. Musical performances seek to find pleasing harmony (a perfect state) as determined by an aesthetic standard, just as the optimization process seeks to find a global solution (a perfect state) as determined by an objective function. The pitch of each musical instrument determines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each decision variable [36].

Since its introduction in year 2000, the HS algorithm has been successfully applied to solve many optimization problems [1, 2, 6–8, 18, 19, 23, 33].

The purpose of this paper is to study the performance of HS algorithm to select the optimal thresholds in the image by maximizing between-class variance, Kapur and Tsallis entropy. Initially, each harmony (candidate solution) is built by taking random thresholds from a feasible search space inside the image histogram. The quality of the solutions has been evaluated using the objective function that is employed by the between-class variance, the Kapur or Tsallis method. Guided by these objective values, the set of candidate solutions are evolved using the harmony search operators until the optimal solution is found. Experimental results over a number of real images have validated the efficiency of the harmony search algorithm and its robustness against GA, PSO, and BFO algorithms. Comparison with the well-known metaheuristic ABC algorithm indicates the equal performance for all images when the number of thresholds M is equal to two, three, four, and five. When the number of thresholds is too high (M = 10, 15, 20), the ABC algorithm yields better results and better stability.

The rest of the paper is as follows: in Section 2, the multilevel thresholding problem is formulated using between-class variance, Kapur and Tsallis entropy. In Section 3, we describe the proposed HS algorithm for multilevel image thresholding. Experimental results are conducted in Section 4. A conclusion is given in Section 5.

2 Multilevel Thresholding Problem Formulation

Given a gray level image I to be segmented into meaningful regions, let there be L gray level values lying in the range {0, 1, 2, …, (L − 1)}. For each gray level i, we associate h(i), which represents the number of pixels having the i^th gray level as a value. Therefore, the probability P_i of the i^th gray level is defined as P_i = h(i)/N, where N denotes the total number of pixels in the image.

Suppose image I is composed of (M + 1) classes. Hence, M thresholds, {t₁, t₂, …, t_M} are required to achieve the subdivision of the image into classes: C₀ for [0, …, t₁ − 1], C₁ for [t₁, …, t₂ − 1], …, C_M for [t_M, …, L − 1], such that t₁ < t₂ … < t_{M − 1} < t_M. The thresholding problem consists in choosing the set of optimal thresholds (t1∗, t2∗, ...., tM∗) that maximizes the objective function f(t).

Optimal thresholds can be obtained by maximizing some desired criterion such that between-class variance Kapur and Tsallis entropy.

3 Adapted HS Algorithm for Multilevel Thresholding

The HS algorithm is an evolutionary algorithm inspired from the improvisation process by the musicians of the orchestra of jazz music [13].

In music improvisation, each player sounds any pitch within the possible range, together making one harmony vector. If all the pitches make a good harmony, that experience is stored in each player’s memory, and the possibility to make a good harmony is increased next time. Similarly, in engineering optimization, each decision variable initially chooses any value within the possible range, together making one solution vector. If all the values of decision variables make a good solution, that experience is stored in each variable’s memory, and the possibility to make a good solution is also increased next time. In real optimization, each musician can be replaced with each decision variable, and its preferred sound pitches can be replaced with the preferred value of each variable [20].

In the HS algorithm, each individual solution is a D dimensional vector and represents a harmony. First, an initial population of harmony memory solution (HMS) is generated randomly and is stored within a harmony memory (HM). A new candidate harmony is thus generated from the elements in the HM by using a memory consideration operation or by a random selection. In addition, a pitch adjustment operator is performed if the new solution is generated by the memory consideration operator. The new solution is then compared with the worst harmony solution stored in the HM. The worst harmony solution is replaced by the new generated solution if the latter provides a better fitness value. The above steps are repeated until a termination criterion is met.

The HS algorithm incorporates the structure of current metaheuristic optimization algorithms. It preserves the history of past vectors (HM) similar to the Tabu search and PSO. It considers several vectors simultaneously in a manner similar to the GA. However, the major difference between the GA and the HS algorithm is that the latter generates a new vector from all the existing vectors (all harmonies in the HM), while the GA generates a new vector from only two of the existing vectors (parents) [20].

For the multilevel thresholding problem, a population (x₁, x₂, …, x_hms) of solutions is created. Each individual solution x_i in generation G is formulated using equation (4).

The details of the adapted HS algorithm for the multilevel thresholding problem are given bellow.

Step 1: The position of each individual solution or harmony of the population is initialized using equation (5), then store them in the harmony memory (HM) such that HM = (x₁, x₂, …, x_hms)^T

where M parameters in the vector representation correspond to the M multiple thresholds, g_min and g_max are the minimum and the maximum gray levels in the image, respectively, and hms is the size of the population.

Step 2: for each solution x_ii = 1 … hms, we generate a new solution xinew using one of the two processes: memory consideration and a possible additional modification, or a random selection. In the memory consideration process, each threshold value ti,jnew  j = 1 … M of the new solution is set to the corresponding threshold value of one of the harmonies stored in the HM with the probability of harmony memory considering rate (HMCR). Otherwise, with a probability 1-HMCR, each threshold value of the new solution is generated randomly between the search bound [g_min, g_max].

In addition, if the new solution comes from the memorization process, an additional modification of the threshold value of the new solution is performed with pitch adjust rate (PAR) probability.

where BW is the bandwidth factor to control the local search around the selected elements of the HM.

Step 3: evaluate the new solution using equation (2) or (3). If the new solution xinew is better than the worst solution x^worst in HM, replace x^worst withxinew.

Step 4: repeat step 2 and step 3 until the maximum number of generation Max_gen is attained.

The outline of the HS algorithm for multilevel image thresholding is given in Algorithm 1.

4 Results and Discussions

To test the performance of the proposed HS algorithm for multilevel image thresholding, 12 real images taken from the Berkeley Segmentation Dataset [22] have been used as illustrated in Figure 1.

Figure 1:

Real Images Used in the Paper.

The results obtained by the HS algorithms have been compared with those generated by the state-of-the-art thresholding methods like (GA), bacterial foraging optimization (BFO), PSO algorithm, and artificial bee colony algorithm (ABC). All the algorithms have been used in their standard versions and have been employed separately. CPU time is also considered to examine the efficiency of the algorithms. Three different fitness criteria have been used such as Kapur’s entropy, between-class variance, and Tsallis entropy.

All metaheuristics used in this study have two common control parameters: the size of the population and the maximum number of generation Max_gen. The number of generations cannot be accepted as a time measure since the algorithms perform a different amount of work in their inner loops. For this reason, we use in this paper, the function evaluation number (Max_FEN) instead of the maximum number of generations. As the values of these two control parameters have a great impact on the convergence and on the computing time, they were chosen to be the same for all experiments in this paper. The size of the population was fixed to 50 and Max_FEN was fixed to 20,000.

The other parameters of the HS algorithm are:

hmcr = the rate of choosing a value from the harmony memory. It generally varies from 0.7 to 0.99. In this paper, hmcr was fixed to 0.8 for all experiments

PAR = the rate of choosing a neighboring value. It generally varies from 0.1 to 0.5. In this paper, the value of PAR was fixed to 0.4.

The parameter values of the GA, ABC, PSO, and BFO algorithms, which give the best results in terms of fitness values, are gathered in Table 1.

In order to have a fair comparison, an exhaustive search method has also been conducted to derive the optimal solution for each number M of thresholds using Kapur’s entropy, between-class variance, and Tsallis entropy. All the methods are validated through numerical simulations in Matlab on a computer having an Intel Core Duo processor (1.66 GHz) and 2 GB memory.

Quantitatively speaking, the quality of the segmented images has been evaluated using the two popular performance measures: SSIM and FSIM indices.

SSIM given by equation (8) evaluates the visual similarity between the original image and the reconstructed image [34]:

where μ_I is the mean intensity of the image I, μ_Seg is the mean of the image Seg, σI2 represents the variance of I, σSeg2 represents the variance of Seg, σ_{I, Seg} is the covariance of I and Seg, c₁ and c₂ are two constants, such that c₁ = (k₁L)² and c₂ = (k₂L)², k₁ = 0.01, k₂ = 0.03, and L is the dynamic range of the pixel values. A higher value of SSIM shows better performance.

The FSIM is used to calculate the similarity between two images [40]. It is calculated as:

and T₁ and T₂ are constants. Here, we choose T₁ = 0.85 and T₂ = 160. G represents the gradient magnitude (GM) of an image and is defined as:

G = Gx2 + Gy2PC  is  the phase congruence  and is defined as:PC(X) = E(X)(ε + ∑nAn(X))

A_n(X) is the local amplitude on scale n, and E(X) is the magnitude of the response vector at position X on scale n. ε is a small positive constant. A higher value of FSIM implies better performance.

The ground truth data obtained by the exhaustive search using Kapur’s entropy, between-class variance, and Tsallis entropy are summarized in Table 2 where the values of the thresholds along with the corresponding fitness function values and CPU time are given. It is apparent from Table 2 that the computation time of the exhaustive search method grows in many orders of magnitude with the number of required thresholds. When the number of thresholds is over 4, the consumption of CPU time of the exhaustive search becomes unbearable. For this reason, when the number of thresholds is >4, there are no correlative values for the exhaustive search listed in our experiments.

4.1 Multilevel Thresholding Results and Efficiency of Different Methods, With M = 2, 3, 4, 5

Results acquired for various test images using Kapur’s entropy, between-class variance, and Tsallis entropy are presented in Tables 3–5. Tables 3–5 tabulated the number of thresholds, objective values, and corresponding optimal threshold values obtained by GA, PSO, BFO, ABC, and HS methods for each objective function.

Table 3

The Best Thresholds and the Corresponding Best Fitness Obtained from the Kapur-Based Methods for Real Images.

Image	M	GA		PSO		BFO		HS		ABC
		Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values
101085	2	12.84678311	88 172	12.84708191	90 172	12.84708191	90 172	12.84708191	90 172	12.84708191	90 172
	3	15.96239615	74 126 180	15.96311893	76 128 180	15.95425336	78 134 180	15.96337103	74 127 180	15.96337103	74 127 180
	4	18.95330671	40 95 133 182	19.01019215	40 88 135 189	19.00570468	40 90 136 185	19.01366556	40 88 138 189	19.01366556	40 88 138 189
	5	21.69980662	40 83 115 162 199	21.72924709	40 88 128 164 200	21.70525954	37 76 119 161 202	21.74475542	37 83 122 161 196	21.74475542	37 83 122 161 196
Wherry	2	12.17769204	105 163	12.17769204	105 163	12.17769204	105 163	12.17769204	105 163	12.17769204	105 163
	3	15.12987155	105 148 191	15.14128282	99 148 190	15.14168887	100 148 190	15.14171160	100 148 191	15.14171160	100 148 191
	4	17.92991618	54 104 152 204	18.00112529	54 109 148 189	17.93735822	49 111 148 195	18.00465138	54 105 148 191	18.00465138	54 105 148 191
	5	20.66135353	56 110 159 204 238	20.67737969	48 84 118 148 192	20.61678261	73 113 156 201 238	20.69633836	48 85 120 152 193	20.69633836	48 85 120 152 193
Snake	2	12.38516237	87 171	12.38516237	87 171	12.38516237	87 171	12.38516237	87 171	12.38516237	87 171
	3	15.55893015	82 150 195	15.56911780	81 146 195	15.56147199	81 144 191	15.56911780	81 146 195	15.56911780	81 146 195
	4	18.47548738	70 121 160 199	18.50346120	72 117 163 204	18.50488722	72 119 163 204	18.50578163	75 121 164 204	18.50578163	75 121 164 204
	5	21.18481974	71 113 146 177 209	21.21500009	62 98 136 175 211	21.17820748	68 115 150 184 215	21.22318889	65 103 142 177 212	21.22318889	65 103 142 177 212
Zebra	2	12.25845978	93 169	12.25845978	93 169	12.25845978	93 169	12.25845978	93 169	12.25845978	93 169
	3	15.33370766	88 135 179	15.33831416	89 134 179	15.30710891	93 145 188	15.33831416	89 134 179	15.33831416	89 134 179
	4	18.16503843	49 93 133 177	18.16881438	49 92 135 181	18.17924768	49 93 136 179	18.17924768	49 93 136 179	18.17924768	49 93 136 179 0
	5	20.87464733	49 93 145 176 210	20.97089523	50 90 131 167 206	20.89884161	49 93 134 162 209	20.99674337	49 93 132 169 204	20.99674337	49 93 132 169 204
Butterfly	2	12.68395909	110 174	12.68395909	110 174	12.68395909	110 174	12.68395909	110 174	12.68395909	110 174
	3	15.79945716	76 122 176	15.80103221	75 123 176	15.80176741	74 123 176	15.80176741	74 123 176	15.80176741	74 123 176
	4	18.76442367	71 127 171 218	18.79815584	75 123 172 217	18.76147839	76 123 170 211	18.80421993	74 121 171 217	18.80421993	74 121 171 217
	5	21.38074243	67 99 133 179 224	21.46711960	57 97 134 173 217	21.36673134	56 84 128 169 208	21.48254318	60 96 133 174 217	21.48254318	60 96 133 174 217
Bird	2	12.08843694	91 168	12.08843694	91 168	12.08843694	91 168	12.08843694	91 168	12.08843694	91 168
	3	15.19285083	65 117 174	15.19896157	65 117 173	15.19896385	65 116 173	15.19896385	65 116 173	15.19896385	65 116 173
	4	17.98114471	55 97 141 181	17.98242669	56 98 144 183	17.96240587	52 92 133 173	17.98262358	55 98 144 183	17.98262358	55 98 144 183
	5	20.55778163	41 77 107 148 183	20.57898029	47 79 115 149 181	20.52164556	59 96 126 155 183	20.60405694	46 79 113 151 183	20.60405694	46 79 113 151 183
Landscape	2	11.55130425	96 148	11.55130425	96 148	11.55130425	96 148	11.55130425	96 148	11.55130425	96 148
	3	14.50602050	96 159 196	14.51149922	96 160 196	14.46815145	77 128 196	14.51573911	96 161 196	14.51573911	96 161 196
	4	17.28546277	81 120 162 196	17.31792194	75 120 162 196	17.21429823	91 124 161 195	17.32920654	75 123 162 196	17.32920654	75 123 162 196
	5	19.88854435	25 59 99 161 196	19.85706969	24 57 115 162 196	19.75888408	29 57 111 157 199	19.91178993	25 59 106 161 196	19.91178993	25 59 106 161 196
Ostrich	2	12.37287657	119 171	12.37287657	119 171	12.37287657	119 171	12.37287657	119 171	12.37287657	119 171
	3	15.46615176	73 122 171	15.46550180	73 119 171	15.45714947	73 122 177	15.46615176	73 122 171	15.46615176	73 122 171
	4	18.13001398	73 119 154 197	18.12887917	77 119 155 197	18.13095927	28 74 122 174	18.13902186	72 119 155 197	18.13902186	72 119 155 197
	5	20.72241619	28 64 113 148 190	20.76868817	28 69 120 165 204	20.69857390	57 90 119 152 192	20.80518220	28 74 119 155 197	20.80518220	28 74 119 155 197
86016	2	11.51653374	100 145	11.51653374	100 145	11.51653374	100 145	11.51653374	100 145	11.51653374	100 145
	3	14.21639503	98 139 173	14.21639503	98 139 173	14.21639503	98 139 173	14.21639503	98 139 173	14.21639503	98 139 173
	4	16.69930512	83 111 142 173	16.70035439	84 113 142 177	16.69342966	86 115 146 179	16.70412197	84 112 142 176	16.70412197	84 112 142 176
	5	19.03845470	63 98 128 153 179	19.01547067	63 88 113 139 174	19.01585443	81 107 133 154 184	19.04974825	63 90 118 146 177	19.04974825	63 90 118 146 177
Baboon	2	12.19613319	81 144	12.19613319	81 144	12.19613319	81 144	12.19613319	81 144	12.19613319	81 144
	3	15.22212866	56 106 156	15.22306822	51 103 153	15.22260623	56 105 155	15.22309422	51 103 154	15.22309422	51 103 154
	4	17.98272996	44 92 129 163	18.01389522	43 81 121 163	17.97736395	42 74 110 153	18.01445923	43 83 122 163	18.01445923	43 83 122 163
	5	20.61454980	33 74 105 136 171	20.63839360	37 70 104 142 175	20.62789349	38 72 103 133 166	20.65407061	38 72 106 139 172	20.65407061	38 72 106 139 172
Lake	2	12.45008781	94 163	12.45008781	94 163	12.45008781	94 163	12.45008781	94 163	12.45008781	94 163
	3	15.45884650	79 124 170	15.46359718	75 120 169	15.46366271	75 121 169	15.46366271	75 121 169	15.46366271	75 121 169
	4	18.23543886	74 115 159 193	18.24190940	71 111 155 192	18.18793509	75 125 165 194	18.24497025	72 113 156 194	18.24497025	72 113 156 194
	5	20.86829314	65 97 132 169 197	20.85095311	66 93 128 165 197	20.76696893	49 83 120 157 192	20.88114115	66 99 133 166 197	20.88114115	66 99 133 166 197
Woman	2	12.66160897	96 168	12.66160897	96 168	12.66160897	96 168	12.66160897	96 168	12.66160897	96 168
	3	15.73493103	72 124 177	15.73832596	75 126 177	15.73843511	76 126 177	15.73843511	76 126 177	15.73843511	76 126 177
	4	18.53027597	57 93 136 181	18.54483573	59 98 141 185	18.52703373	57 95 143 189	18.54603378	61 100 142 185	18.54603378	61 100 142 185
	5	21.15299222	61 100 138 179 207	21.20929358	61 96 133 174 211	21.17676917	66 95 132 171 210	21.22337209	58 95 133 172 210	21.22337209	58 95 133 172 210

Table 4

The Best Thresholds and the Corresponding Best Fitness Obtained from the Between-Class Variance Based Methods for Real Images.

Image	M	GA		PSO		BFO		HS		ABC
		Fitness value	Threshold values	Fitness value	Threshold values	Fitness Value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values
101085	2	2755.80780607	95 169	2755.80780607	95 169	2755.80780607	95 169	2755.80780607	95 169	2755.80780607	95 169
	3	2959.80935893	60 113 186	2960.73062947	61 114 183	2960.77194057	60 113 182	2960.77194057	60 113 182	2960.77194057	60 113 182
	4	3059.69110307	53 95 144 208	3063.47946029	52 92 140 197	3055.28231079	51 98 136 189	3063.49963857	52 92 139 197	3063.49963857	52 92 139 197
	5	3112.25453435	47 82 114 153 205	3115.04660156	43 70 106 147 201	3109.92412178	46 77 116 162 220	3115.75355168	44 73 108 150 202	3115.75355168	44 73 108 150 202
Wherry	2	3083.79418431	110 188	3083.79418431	110 188	3083.79418431	110 188	3083.79418431	110 188	3083.79418431	110 188
	3	3326.36233191	104 157 217	3326.63346304	102 158 216	3326.67830302	103 158 216	3326.67830302	103 158 216	3326.67830302	103 158 216
	4	3390.38017977	84 120 161 217 0	3390.24025965	85 120 162 218	3385.25282705	87 123 156 210	3390.47928530	85 121 161 217	3390.47928530	85 121 161 217
	5	3420.66002293	87 119 148 174 218	3422.16315882	82 116 149 179 225	3416.52007102	80 111 149 180 212	3422.87851658	83 118 149 179 222	3422.87851658	83 118 149 179 222
Snake	2	1110.02017316	87 134	1110.02017316	87 134	1110.02017316	87 134	1110.02017316	87 134	1110.02017316	87 134
	3	1219.19982173	78 114 151	1219.84832870	77 114 153	1219.84832870	77 114 153	1219.84832870	77 114 153	1219.84832870	77 114 153
	4	1274.09416423	70 102 129 167	1274.25201423	69 101 129 166	1274.29104142	70 101 129 166	1274.29104142	70 101 129 166	1274.29104142	70 101 129 166
	5	1296.75963283	61 88 114 132 160	1303.12748742	64 90 111 136 171	1300.52451087	57 84 108 136 175	1303.79390960	64 91 114 139 173	1303.79390960	64 91 114 139 173
Zebra	2	817.64291521	94 141	817.64291521	94 141	817.64291521	94 141	817.64291521	94 141	817.64291521	94 141
	3	915.10304297	90 125 173	915.10304297	90 125 173	915.07484319	90 125 172	915.10304297	90 125 173	915.10304297	90 125 173
	4	964.93187969	81 109 135 182	964.68072195	81 110 137 186	956.58667458	74 102 127 167	964.93591533	82 110 136 183	964.93591533	82 110 136 183
	5	984.80936682	78 107 123 152 195	989.47567975	75 100 121 146 194	987.01450004	79 105 127 152 189	990.04815474	75 99 119 142 187	990.04815474	75 99 119 142 187
Butterfly	2	3519.36925589	85 154	3519.36925589	85 154	3519.36925589	85 154	3519.36925589	85 154	3519.36925589	85 154
	3	3645.61374037	79 135 188	3646.01996751	80 137 189	3646.01996751	80 137 189	3646.01996751	80 137 189	3646.01996751	80 137 189
	4	3710.74969983	65 103 153 198	3711.09055933	69 107 152 195	3710.29091665	74 115 159 201	3712.05714618	67 103 150 196	3712.05714618	67 103 150 196
	5	3751.62772616	64 97 127 164 195	3759.00312301	57 89 126 168 201	3754.45907370	62 90 135 179 212	3760.33904545	59 90 128 169 205	3760.33904545	59 90 128 169 205
Bird	2	2535.45049890	90 160	2535.45049890	90 160	2535.45049890	90 160	2535.45049890	90 160	2535.45049890	90 160
	3	2611.29687484	77 136 178	2611.87729789	79 135 179	2611.87729789	79 135 179	2611.87729789	79 135 179	2611.87729789	79 135 179
	4	2649.61229402	63 104 145 182	2649.62355224	59 100 142 181	2649.43190559	62 102 143 182	2649.68644784	59 99 143 181	2649.68644784	59 99 143 181
	5	2667.49342213	53 92 131 167 190	2668.04242624	57 94 135 173 194	2667.54712153	55 95 138 168 189	2668.34104313	57 96 136 170 192	2668.34104313	57 96 136 170 192
Landscape	2	4419.79062113	56 146	4419.79062113	56 146	4419.79062113	56 146	4419.79062113	56 146	4419.79062113	56 146
	3	4720.65138746	49 111 173	4721.15000884	49 108 172	4721.18853555	49 109 173	4721.18853555	49 109 173	4721.18853555	49 109 173
	4	4782.31421537	47 96 136 188	4783.64328964	44 90 130 182	4783.10430766	47 95 133 184	4783.78331215	44 89 129 180	4783.78331215	44 89 129 180
	5	4823.29063693	42 80 106 139 187	4823.27727608	41 79 107 136 181	4823.78813836	42 80 107 139 182	4823.80426574	42 80 108 139 182	4823.80426574	42 80 108 139 182
Ostrish	2	1035.42809860	74 132	1035.42809860	74 132	1035.42809860	74 132	1035.42809860	74 132	1035.42809860	74 132
	3	1099.31606278	68 99 145	1099.37380408	67 98 145	1099.37380408	67 98 145	1099.38285377	67 98 144	1099.38285377	67 98 144
	4	1139.84113191	63 90 123 175	1139.67492367	64 91 125 174	1137.16820483	58 83 121 181	1139.97065299	62 89 124 176	1139.97065299	62 89 124 176
	5	1168.64814481	52 75 98 129 178	1167.59673276	54 76 96 127 178	1162.69144027	59 77 99 133 193	1168.71976730	51 74 98 129 179	1168.71976730	51 74 98 129 179
86016	2	1236.70799926	121 168	1236.70799926	121 168	1236.70799926	121 168	1236.70799926	121 168	1236.70799926	121 168
	3	1289.70039209	108 147 174	1289.94189074	105 145 174	1289.86634556	106 146 174	1289.94189074	105 145 174	1289.94189074	105 145 174
	4	1320.72542663	93 128 161 180	1321.59875801	91 125 157 178	1319.68599293	91 133 160 179	1321.74260380	92 127 157 178	1321.74260380	92 127 157 178
	5	1336.24411357	93 126 149 165 182	1337.12772982	91 121 149 168 183	1335.21762876	90 126 153 172 188	1337.34928666	90 121 149 167 183	1337.34928666	90 121 149 167 183
Baboon	2	1539.84605676	97 149	1539.84605676	97 149	1539.84605676	97 149	1539.84605676	97 149	1539.84605676	97 149
	3	1629.37795050	86 125 161	1629.30032139	86 126 162	1629.37795050	86 125 161	1629.37795050	86 125 161	1629.37795050	86 125 161
	4	1680.83536378	77 106 137 167	1682.12573591	72 105 135 168	1682.13859178	74 108 139 170	1682.67011109	72 106 137 168	1682.67011109	72 106 137 168
	5	1707.46411191	67 97 124 146 174	1707.46865933	63 95 122 146 173	1706.72229516	70 97 126 148 173	1708.21002664	68 99 125 149 174	1708.21002664	68 99 125 149 174
Lake	2	3741.34541715	88 155	3741.34541715	88 155	3741.34541715	88 155	3741.34541715	88 155	3741.34541715	88 155
	3	3875.90309922	83 142 193	3876.94522523	80 140 193	3876.91446608	79 139 192	3876.94624749	79 139 193	3876.94624749	79 139 193
	4	3942.36602132	69 112 157 194	3942.96557593	71 115 160 198	3942.95861601	70 113 157 196	3943.26884069	69 112 158 197	3943.26884069	69 112 158 197
	5	3973.37061149	53 77 116 161 198	3976.35082210	56 87 129 166 198	3969.76858013	68 106 137 165 197	3977.34404212	60 90 128 166 199	3977.34404212	60 90 128 166 199
Women	2	2604.63819548	82 147	2604.63819548	82 147	2604.63819548	82 147	2604.63819548	82 147	2604.63819548	82 147
	3	2780.76298088	75 128 176	2780.77911258	75 127 176	2780.77911258	75 127 176	2780.77911258	75 127 176	2780.77911258	75 127 176
	4	2849.06516595	69 113 146 185	2851.65969939	66 107 142 181	2846.33659459	58 99 136 176	2851.78602355	66 106 142 182	2851.78602355	66 106 142 182
	5	2882.94613070	60 86 113 147 187	2886.30746226	56 86 117 147 183	2885.00133497	52 84 118 149 186	2886.45195570	57 88 118 148 184	2886.45195570	57 88 118 148 184

Table 5

The Best Thresholds and the Corresponding Best Fitness Obtained from the Tsallis Based Methods for Real Images.

Image	M	GA		PSO		BFO		HS		ABC
		Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values	Fitness value	Threshold values
101085	2	0.88888454	88 142	0.88888454	88 142	0.88888454	88 142	0.88888454	88 142	0.88888454	88 142
	3	1.29627626	51 99 147	1.29627626	51 99 147	1.29627625	51 97 147	1.29627626	51 99 147	1.29627626	51 99 147
	4	1.65427349	53 95 131 165	1.65427681	51 87 124 161	1.65426897	51 94 130 171	1.65427731	51 88 125 161	1.65427731	51 88 125 161
	5	1.99578967	32 58 98 133 165	1.99579478	33 67 101 131 164	1.99578569	40 69 100 133 173	1.99579585	40 74 105 135 166	1.99579585	40 74 105 135 166
Wherry	2	0.88887305	85 122	0.88887305	85 122	0.88887305	85 122	0.88887305	85 122	0.88887305	85 122
	3	1.29623234	79 113 148	1.29623483	82 114 145	1.29622815	76 104 141	1.29623483	82 114 145	1.29623483	82 114 145
	4	1.65416609	86 120 150 175	1.65417660	76 100 122 148	1.65417307	72 96 122 157	1.65417874	73 98 122 149	1.65417874	73 98 122 149
	5	1.99563393	76 96 126 150 174	1.99564371	74 97 125 152 173	1.99560772	48 84 118 150 173	1.99565863	73 98 122 148 173	1.99565863	73 98 122 148 173
Snake	2	0.88888212	91 171	0.88888212	91 171	0.88888212	91 171	0.88888212	91 171	0.88888212	91 171
	3	1.29627860	77 124 179	1.29627867	75 122 177	1.29627850	75 122 180	1.29627868	77 122 177	1.29627868	77 122 177
	4	1.65427983	70 109 147 191	1.65428026	72 111 147 195	1.65428075	71 111 151 195	1.65428075	71 111 151 195	1.65428075	71 111 151 195
	5	1.99580310	59 89 118 154 202	1.99580847	63 101 132 166 204	1.99580916	62 95 128 164 198	1.99580987	64 98 130 165 202	1.99580987	64 98 130 165 202
Zebra	2	0.88888123	93 169	0.88888123	93 169	0.88888123	93 169	0.88888123	93 169	0.88888123	93 169
	3	1.29626858	89 132 176	1.29626862	91 132 176	1.29626705	88 132 170	1.29626862	91 132 176	1.29626862	91 132 176
	4	1.65424616	72 101 134 176	1.65424622	74 102 136 176	1.65424614	76 105 136 176	1.65424683	74 102 134 176	1.65424683	74 102 134 176
	5	1.99572696	69 93 122 155 196	1.99574296	74 100 132 167 198	1.99571792	71 104 127 169 193	1.99574709	74 102 132 162 194	1.99574709	74 102 132 162 194
Butterfly	2	0.88888526	113 174	0.88888526	113 174	0.88888526	113 174	0.88888526	113 174	0.88888526	113 174
	3	1.29628396	74 121 171	1.29628414	76 123 174	1.29628414	76 123 174	1.29628414	76 123 174	1.29628414	76 123 174
	4	1.65428403	74 117 163 201	1.65428450	79 121 160 201	1.65428338	79 125 169 204	1.65428512	74 118 159 200	1.65428512	74 118 159 200
	5	1.99581075	64 101 133 168 208	1.99581571	64 95 136 171 204	1.99580056	58 89 123 154 185	1.99581674	65 97 134 169 204	1.99581674	65 97 134 169 204
Bird	2	0.88887099	66 117	0.88887099	66 117	0.88887099	66 117	0.88887099	66 117	0.88887099	66 117
	3	1.29625264	62 109 162	1.29625321	58 100 156	1.29625318	59 100 154	1.29625326	59 102 156	1.29625326	59 102 156
	4	1.65424190	50 87 132 172	1.65424294	49 82 117 168	1.65424098	49 81 116 159	1.65424329	50 83 117 164	1.65424329	50 83 117 164
	5	1.99574841	46 75 102 147 181	1.99574922	45 75 100 140 172	1.99573963	48 72 97 143 173	1.99575182	46 75 102 141 173	1.99575182	46 75 102 141 173
Landscape	2	0.88885058	96 137	0.88885058	96 137	0.88885058	96 137	0.88885058	96 137	0.88885058	96 137
	3	1.29618571	96 126 155	1.29618564	95 127 155	1.29617471	88 124 155	1.29618603	95 126 155	1.29618603	95 126 155
	4	1.65412199	91 127 161 195	1.65412981	95 134 162 196	1.65410678	95 141 166 196	1.65413469	95 127 161 196	1.65413469	95 127 161 196
	5	1.99554512	83 110 140 162 192	1.99557455	77 107 134 162 196	1.99552383	74 100 127 148 196	1.99558005	75 104 134 162 196	1.99558005	75 104 134 162 196
Ostrich	2	0.88888197	122 171	0.88888197	122 171	0.88888197	122 171	0.88888197	122 171	0.88888197	122 171
	3	1.29627605	74 122 171	1.29627608	75 122 171	1.29627500	72 125 172	1.29627608	75 122 171	1.29627608	75 122 171
	4	1.65425592	61 88 129 171	1.65425554	60 87 131 171	1.65425666	60 88 123 171	1.65425696	60 86 123 171	1.65425696	60 86 123 171
	5	1.99575835	64 89 123 148 181	1.99576689	62 87 120 149 187	1.99575368	60 88 123 171 200	1.99577169	60 86 119 150 188	1.99577169	60 86 119 150 188
86016	2	0.88887766	99 142	0.88887766	99 142	0.88887766	99 142	0.88887766	99 142	0.88887766	99 142
	3	1.29624919	97 138 173	1.29624958	100 138 173	1.29624958	100 138 173	1.29624958	100 138 173	1.29624958	100 138 173
	4	1.65421083	88 113 142 176	1.65421244	87 111 139 172	1.65421119	88 113 142 172	1.65421297	87 112 139 173	1.65421297	87 112 139 173
	5	1.99564292	81 102 126 158 179	1.99565968	86 108 128 154 179	1.99564224	89 112 132 159 182	1.99566695	84 107 130 154 177	1.99566695	84 107 130 154 177
Baboon	2	0.88888388	92 147	0.88888388	92 147	0.88888388	92 147	0.88888388	92 147	0.88888388	92 147
	3	1.29627688	72 111 155	1.29627688	72 112 157	1.29627688	72 112 157	1.29627690	72 112 156	1.29627690	72 112 156
	4	1.65427196	66 98 136 169	1.65427224	71 103 136 167	1.65426610	57 99 140 171	1.65427304	66 101 135 167	1.65427304	66 101 135 167
	5	1.99578826	51 83 114 143 171	1.99578751	51 77 110 140 169	1.99577272	58 95 119 144 170	1.99579012	55 83 111 141 170	1.99579012	55 83 111 141 170
Lake	2	0.88888388	95 156	0.88888388	95 156	0.88888388	95 156	0.88888388	95 156	0.88888388	95 156
	3	1.29627859	79 123 168	1.29627861	77 121 168	1.29627855	79 124 168	1.29627861	77 122 168	1.29627861	77 122 168
	4	1.65427442	67 104 140 170	1.65427583	67 102 136 170	1.65427140	74 110 153 193	1.65427595	69 103 136 170	1.65427595	69 103 136 170
	5	1.99579112	67 100 130 155 194	1.99579248	66 99 138 168 197	1.99579702	69 102 132 163 196	1.99579745	66 97 128 160 195	1.99579745	66 97 128 160 195
Woman	2	0.88888604	97 168	0.88888604	97 168	0.88888604	97 168	0.88888604	97 168	0.88888604	97 168
	3	1.29628493	77 126 175	1.29628496	76 125 175	1.29628482	81 129 177	1.29628496	76 125 175	1.29628496	76 125 175
	4	1.65428859	59 102 146 184	1.65429028	61 98 137 178	1.65429032	59 97 138 184	1.65429055	60 97 137 181	1.65429055	60 97 137 181
	5	1.99581591	59 86 113 148 188	1.99582030	56 87 123 156 191	1.99581557	56 89 118 148 196	1.99582214	56 88 120 152 190	1.99582214	56 88 120 152 190

The results in these three tables indicate that all the methods performed equally when the number of thresholds M is equal to two. When the number M increases, numerical results indicate that HS and ABC metaheuristics have equal performance and outperforms the GA, BFO, and PSO algorithms for all real images and for the number of thresholds.

Apart from quality measurement, feature measurement assessment (FSIM and SSIM) obtained by each algorithm for the three objective functions are also checked and presented in Tables 6–8. It can be evidently realized from these tables that HS and ABC achieve the same quality of solutions in terms of FSIM and SSIM.

Table 6

SSIM, FSIM Metrics, and CPU Time of Real Images Segmented with the Thresholds Yielding the Best Kapur Entropy.

Image	M	SSIM					FSIM					CPU Time (s)
		GA	pso	BFO	HS	ABC	GA	PSO	BFO	HS	ABC	GA	PSO	BFO	HS	ABC
101085	2	0.5620	0.5620	0.5620	0.5620	0.5620	0.6606	0.6606	0.6606	0.6606	0.6606	1.925932	3.181786	7.500785	3.166484	3.014616
	3	0.6542	0.6753	0.6661	0.6761	0.6761	0.7363	0.7416	0.7392	0.7418	0.7418	1.951218	3.244262	7.819356	3.247470	3.025735
	4	0.7508	0.7765	0.7725	0.7772	0.7772	0.8126	0.8279	0.8238	0.8299	0.8299	2.022551	3.260813	8.070879	3.539006	3.026400
	5	0.7994	0.8058	0.8009	0.8221	0.8221	0.8594	0.8627	0.8618	0.8738	0.8738	2.161112	3.285346	8.460121	3.562860	3.183831
Wherry	2	0.6795	0.6795	0.6795	0.6795	0.6795	0.7490	0.7490	0.7490	0.7490	0.7490	1.796198	3.153836	7.107807	3.073505	2.942544
	3	0.7016	0.7039	0.7039	0.7135	0.7135	0.7837	0.7865	0.7868	0.7883	0.7883	1.814120	3.200489	7.821641	3.140663	2.948493
	4	0.7189	0.7193	0.7193	0.7267	0.7267	0.7970	0.8049	0.8013	0.8074	0.8074	1.821143	3.223766	7.840306	3.240036	2.968170
	5	0.7471	0.7863	0.7844	0.7938	0.7938	0.8328	0.8418	0.8415	0.8676	0.8676	1.850865	3.259420	7.859357	3.302673	2.972394
Snake	2	0.5827	0.5827	0.5827	0.5827	0.5827	0.6541	0.6541	0.6541	0.6541	0.6541	1.771291	3.223776	6.202439	3.106136	3.780838
	3	0.6754	0.7025	0.6918	0.7025	0.7025	0.7404	0.7616	0.7544	0.7616	0.7616	1.790890	3.277098	6.633986	3.185452	3.785939
	4	0.7931	0.7936	0.7940	0.7960	0.7960	0.8357	0.8357	0.8370	0.8386	0.8386	1.815854	3.246315	7.476230	3.303935	3.815253
	5	0.8236	0.8349	0.8346	0.8401	0.8401	0.8631	0.8794	0.8713	0.8831	0.8831	1.828897	3.249651	8.145582	3.327580	3.854868
Zebra	2	0.6064	0.6064	0.6064	0.6064	0.6064	0.6930	0.6930	0.6930	0.6930	0.6930	1.769225	3.139952	6.174534	3.049703	2.919237
	3	0.7284	0.7517	0.7467	0.7517	0.7517	0.7885	0.7995	0.7980	0.7995	0.7995	1.800487	3.171935	6.332212	3.143638	2.927716
	4	0.7598	0.7643	0.7643	0.7691	0.7691	0.8173	0.8195	0.8199	0.8199	0.8199	1.811150	3.192263	6.469981	3.219294	2.935833
	5	0.7374	0.7901	0.7899	0.7964	0.7964	0.8098	0.8399	0.8395	0.8425	0.8425	1.848724	3.247337	6.707575	3.320625	2.954885
Butterfly	2	0.7319	0.7319	0.7319	0.7319	0.7319	0.7084	0.7084	0.7084	0.7084	0.7084	1.807585	3.185566	7.552455	3.078996	2.952397
	3	0.8430	0.8430	0.8432	0.8432	0.8432	0.8061	0.8073	0.8102	0.8102	0.8102	1.820223	3.225876	8.645239	3.193072	2.954164
	4	0.8877	0.8953	0.8949	0.8960	0.8960	0.8716	0.8731	0.8719	0.8735	0.8735	1.857622	3.228833	9.751682	3.271846	2.968446
	5	0.9134	0.9203	0.9162	0.9209	0.9209	0.8909	0.9048	0.9039	0.9106	0.9106	1.890310	3.269106	10.392816	3.345437	3.001206
Bird	2	0.8708	0.8708	0.8708	0.8708	0.8708	0.8233	0.8233	0.8233	0.8233	0.8233	1.768740	3.117159	6.644387	3.039270	2.896517
	3	0.8914	0.8920	0.8924	0.8924	0.8924	0.8792	0.8798	0.8800	0.8800	0.8800	1.772488	3.167030	6.795271	3.133387	2.902262
	4	0.8890	0.8891	0.8890	0.9027	0.9027	0.8932	0.8943	0.8943	0.9013	0.9013	1.810974	3.191151	7.055027	3.248248	2.932165
	5	0.8935	0.8953	0.8949	0.9000	0.9000	0.8964	0.9071	0.9066	0.9106	0.9106	1.848200	3.239374	7.512748	3.250957	2.937233
Landscape	2	0.6314	0.6314	0.6314	0.6314	0.6314	0.7566	0.7566	0.7566	0.7566	0.7566	1.748772	3.123401	6.438242	3.046316	2.890496
	3	0.6802	0.6811	0.6802	0.6954	0.6954	0.7923	0.7998	0.7998	0.8002	0.8002	1.790520	3.179572	6.571338	3.112760	2.903656
	4	0.6940	0.7038	0.7022	0.7049	0.7049	0.8064	0.8109	0.8102	0.8221	0.8221	1.827448	3.189646	7.157381	3.197345	2.930407
	5	0.7851	0.7901	0.7883	0.7956	0.7956	0.8608	0.8717	0.8653	0.8729	0.8729	1.865216	3.229429	7.932049	3.279616	2.943174
Ostrish	2	0.6600	0.6600	0.6600	0.6600	0.6600	0.6891	0.6891	0.6891	0.6891	0.6891	1.786778	3.195110	7.294599	3.059399	2.912949
	3	0.7518	0.7522	0.7522	0.7541	0.7541	0.7541	0.7541	0.7541	0.7568	0.7568	1.815394	3.217608	7.761064	3.119983	2.934800
	4	0.7544	0.7687	0.7687	0.7755	0.7755	0.7632	0.7748	0.7748	0.7841	0.7841	1.805931	3.238389	9.096787	3.210060	2.937893
	5	0.7596	0.7721	0.7607	0.7845	0.7845	0.7706	0.7858	0.7758	0.8127	0.8127	1.863377	3.448028	10.045720	3.287202	2.965421
86016	2	0.6026	0.6026	0.6026	0.6026	0.6026	0.7161	0.7161	0.7161	0.7161	0.7161	1.635054	2.992476	7.208245	2.942271	2.762586
	3	0.7155	0.7155	0.7155	0.7155	0.7155	0.7926	0.7926	0.7926	0.7926	0.7926	1.644486	3.043141	7.954742	3.005113	2.766385
	4	0.7524	0.7597	0.7590	0.7714	0.7714	0.8405	0.8446	0.8446	0.8476	0.8476	1.669377	3.085785	8.356114	3.095727	2.799333
	5	0.7521	0.7762	0.7773	0.8037	0.8037	0.8441	0.8448	0.8503	0.8831	0.8831	1.686241	3.103798	8.800551	3.158408	2.810344
Baboon	2	0.7849	0.7849	0.7849	0.7849	0.7849	0.8526	0.8526	0.8526	0.8526	0.8526	1.717369	3.105095	9.129254	3.109727	3.271364
	3	0.8264	0.8273	0.8273	0.8327	0.8327	0.8831	0.8840	0.8840	0.8885	0.8885	1.751691	3.113413	9.152520	3.124247	3.251666
	4	0.8390	0.8617	0.8600	0.8701	0.8701	0.8927	0.9079	0.9065	0.9084	0.9084	1.766895	3.119534	9.898293	3.172944	3.316692
	5	0.8893	0.8990	0.8965	0.9012	0.9012	0.9196	0.9281	0.9249	0.9312	0.9312	1.791591	3.136692	10.139459	3.284885	3.330640
Lake	2	0.7865	0.7865	0.7865	0.7865	0.7865	0.8290	0.8290	0.8290	0.8290	0.8290	1.730446	3.085993	7.588667	3.131662	3.165144
	3	0.8263	0.8349	0.8353	0.8353	0.8353	0.8652	0.8730	0.8732	0.8732	0.8732	1.777940	3.133640	8.511788	3.179496	3.257353
	4	0.8677	0.8765	0.8727	0.8773	0.8773	0.8947	0.9009	0.8978	0.9015	0.9015	1.781042	3.154579	8.865637	3.233149	3.312249
	5	0.8924	0.9024	0.9007	0.9044	0.9044	0.9229	0.9244	0.9230	0.9267	0.9267	1.811895	3.203255	9.745394	3.457226	3.389604
Women	2	0.7214	0.7214	0.7214	0.7214	0.7214	0.7794	0.7794	0.7794	0.7794	0.7794	1.751702	3.091385	8.326582	3.080293	3.223500
	3	0.8089	0.8089	0.8104	0.8104	0.8104	0.8439	0.8439	0.8475	0.8475	0.8475	1.788292	3.151522	8.389617	3.249069	3.273957
	4	0.8549	0.8561	0.8561	0.8566	0.8566	0.8818	0.8822	0.8821	0.8824	0.8824	1.795823	3.174099	9.159723	3.429803	3.303330
	5	0.8789	0.8839	0.8793	0.8849	0.8849	0.8972	0.9047	0.9040	0.9060	0.9060	1.821274	3.175096	9.282727	3.567332	3.344907

Table 7

SSIM, FSIM Metrics, and CPU Time of Real Images Segmented with the Thresholds Yielding the Best Between-Class Variance.

Image	M	SSIM					FSIM					CPU time (s)
		GA	PSO	BFO	HS	ABC	GA	PSO	BFO	HS	ABC	GA	PSO	BFO	HS	ABC
101085	2	0.5440	0.5440	0.5440	0.5440	0.5440	0.6603	0.6603	0.6603	0.6603	0.6603	1.020995	2.314438	3.044573	2.395277	2.095347
	3	0.7116	0.7139	0.7157	0.7157	0.7157	0.7573	0.7628	0.7647	0.7647	0.7647	1.079955	2.326560	3.071098	2.579252	2.107340
	4	0.7783	0.7952	0.7835	0.7955	0.7955	0.8226	0.8344	0.8263	0.8368	0.8368	1.078366	2.368358	3.145167	2.615061	2.149461
	5	0.8328	0.8448	0.8376	0.8478	0.8478	0.8690	0.8717	0.8713	0.8745	0.8745	1.090071	2.405147	3.327075	2.751241	2.161365
Wherry	2	0.6498	0.6498	0.6498	0.6498	0.6498	0.7403	0.7403	0.7403	0.7403	0.7403	0.995861	2.306378	3.057472	2.390689	2.079305
	3	0.7071	0.7095	0.7139	0.7139	0.7139	0.7877	0.7880	0.7884	0.7884	0.7884	1.046402	2.316861	3.162198	2.543340	2.123097
	4	0.7831	0.7861	0.7855	0.7931	0.7931	0.8357	0.8384	0.8384	0.8393	0.8393	1.054891	2.368561	3.254486	2.562496	2.129340
	5	0.8058	0.8171	0.8152	0.8176	0.8176	0.8537	0.8621	0.8619	0.8624	0.8624	1.121071	2.402887	3.467269	2.757798	2.149760
Snake	2	0.5963	0.5963	0.5963	0.5963	0.5963	0.6623	0.6623	0.6623	0.6623	0.6623	0.979216	2.326212	3.012924	2.501655	2.464317
	3	0.7359	0.7423	0.7423	0.7423	0.7423	0.7639	0.7727	0.7727	0.7727	0.7727	1.012681	2.340947	3.015032	2.618946	2.535711
	4	0.8245	0.8245	0.8254	0.8254	0.8254	0.8403	0.8403	0.8425	0.8425	0.8425	1.032152	2.368083	3.068364	2.767460	2.642108
	5	0.8376	0.8642	0.8623	0.8682	0.8682	0.8520	0.8794	0.8748	0.8876	0.8876	1.065249	2.478885	3.353597	2.932905	2.889510
Zebra	2	0.6387	0.6387	0.6387	0.6387	0.6387	0.7065	0.7065	0.7065	0.7065	0.7065	0.966288	2.334942	3.025859	2.427918	2.068493
	3	0.7602	0.7624	0.7602	0.7624	0.7624	0.7941	0.7950	0.7941	0.7950	0.7950	1.002385	2.367029	3.146884	2.600479	2.133261
	4	0.8144	0.8230	0.8181	0.8238	0.8238	0.8481	0.8491	0.8491	0.8502	0.8502	1.047648	2.400261	3.139755	2.657824	2.138341
	5	0.8631	0.8657	0.8642	0.8780	0.8780	0.8760	0.8808	0.8799	0.8867	0.8867	1.057038	2.562087	3.233158	3.032822	2.140552
Butterfly	2	0.7867	0.7867	0.7867	0.7867	0.7867	0.7408	0.7408	0.7408	0.7408	0.7408	0.983765	2.335871	3.017444	2.406799	2.076239
	3	0.8455	0.8462	0.8462	0.8462	0.8462	0.8158	0.8177	0.8177	0.8177	0.8177	1.000196	2.351039	3.133638	2.589525	2.116139
	4	0.8750	0.8786	0.8786	0.8875	0.8875	0.8613	0.8674	0.8662	0.8735	0.8735	1.034771	2.362061	3.168692	2.590681	2.133415
	5	0.9002	0.9186	0.9143	0.9205	0.9205	0.8748	0.9071	0.9025	0.9113	0.9113	1.063389	2.455681	3.283641	2.739451	2.174935
Bird	2	0.8757	0.8757	0.8757	0.8757	0.8757	0.8294	0.8294	0.8294	0.8294	0.8294	0.961571	2.327173	3.027841	2.377722	2.084553
	3	0.8792	0.8813	0.8813	0.8813	0.8813	0.8623	0.8634	0.8634	0.8634	0.8634	1.022307	2.353864	3.058656	2.565041	2.128395
	4	0.8908	0.8933	0.8910	0.8934	0.8934	0.8900	0.8936	0.8906	0.8937	0.8937	1.028739	2.377752	3.125191	2.607020	2.134124
	5	0.8921	0.8960	0.8970	0.8976	0.8976	0.8947	0.8992	0.8952	0.8998	0.8998	1.105146	2.413394	3.129739	2.752888	2.137203
Landscape	2	0.6465	0.6465	0.6465	0.6465	0.6465	0.7712	0.7712	0.7712	0.7712	0.7712	0.964139	2.306994	3.033401	2.418163	2.085491
	3	0.7548	0.7555	0.7556	0.7556	0.7556	0.8381	0.8391	0.8395	0.8395	0.8395	1.001258	2.312851	3.066768	2.498554	2.129161
	4	0.7805	0.7820	0.7820	0.7965	0.7965	0.8653	0.8655	0.8655	0.8715	0.8715	1.039710	2.343849	3.121156	2.574302	2.131255
	5	0.8330	0.8345	0.8345	0.8372	0.8372	0.8921	0.8931	0.8931	0.8953	0.8953	1.067777	2.406257	3.149523	2.723763	2.137777
Ostrich	2	0.6848	0.6848	0.6848	0.6848	0.6848	0.6925	0.6925	0.6925	0.6925	0.6925	0.995993	2.323328	2.997650	2.363014	2.079943
	3	0.7093	0.7093	0.7099	0.7133	0.7133	0.7434	0.7434	0.7434	0.7461	0.7461	1.005875	2.361664	3.106313	2.574853	2.111293
	4	0.7730	0.7739	0.7730	0.7773	0.7773	0.7923	0.7935	0.7923	0.7960	0.7960	1.040034	2.390692	3.131443	2.631778	2.134214
	5	0.7946	0.7991	0.7957	0.8022	0.8022	0.8295	0.8327	0.8304	0.8329	0.8329	1.207087	2.410873	3.159066	2.765904	2.140974
86016	2	0.6192	0.6192	0.6192	0.6192	0.6192	0.7152	0.7152	0.7152	0.7152	0.7152	0.963235	2.298755	3.030301	2.420575	2.081054
	3	0.7072	0.7158	0.7137	0.7158	0.7158	0.7608	0.7758	0.7719	0.7758	0.7758	1.016719	2.301901	3.063830	2.561543	2.119711
	4	0.7985	0.7992	0.7985	0.8037	0.8037	0.8376	0.8385	0.8376	0.8400	0.8400	1.025760	2.359610	3.093613	2.654763	2.125234
	5	0.8383	0.8471	0.8455	0.8588	0.8588	0.8583	0.8697	0.8681	0.8895	0.8895	1.081344	2.390292	3.149113	2.747921	2.141562
Baboon	2	0.8010	0.8010	0.8010	0.8010	0.8010	0.8501	0.8501	0.8501	0.8501	0.8501	0.963566	2.251199	3.005878	2.563903	2.525971
	3	0.8508	0.8508	0.8536	0.8536	0.8536	0.8839	0.8839	0.8861	0.8861	0.8861	1.020810	2.308318	3.054847	2.778020	2.715676
	4	0.8831	0.8874	0.8874	0.8902	0.8902	0.9022	0.9099	0.9099	0.9113	0.9113	1.034441	2.337015	3.120403	2.895992	2.857749
	5	0.9057	0.9074	0.9067	0.9088	0.9088	0.9187	0.9234	0.9227	0.9242	0.9242	1.060675	2.372113	3.288941	3.02460	2.943297
Lake	2	0.7977	0.7977	0.7977	0.7977	0.7977	0.8353	0.8353	0.8353	0.8353	0.8353	1.000062	2.279080	3.015910	2.704825	2.643478
	3	0.8378	0.8418	0.8411	0.8424	0.8424	0.8674	0.8717	0.8712	0.8722	0.8722	1.020729	2.313239	3.081182	2.731592	2.693297
	4	0.8793	0.8830	0.8814	0.8840	0.8840	0.9061	0.9095	0.9076	0.9111	0.9111	1.030986	2.330575	3.130862	3.018311	2.918836
	5	0.8961	0.9092	0.9069	0.9109	0.9109	0.9199	0.9354	0.9354	0.9382	0.9382	1.063495	2.365575	3.168401	3.087520	3.010329
Women	2	0.7236	0.7236	0.7236	0.7236	0.7236	0.7643	0.7835	0.7835	0.7835	0.7835	0.984441	2.269656	3.009536	2.305761	2.280917
	3	0.8092	0.8093	0.8093	0.8093	0.8093	0.8451	0.8453	0.8453	0.8453	0.8453	1.019085	2.346640	3.092479	2.503620	2.464204
	4	0.8512	0.8539	0.8535	0.8542	0.8542	0.8780	0.8794	0.8794	0.8804	0.8804	1.030819	2.378506	3.106920	2.669935	2.641057
	5	0.8800	0.8809	0.8807	0.8813	0.8813	0.8982	0.9016	0.9014	0.9049	0.9049	1.057407	2.409270	3.139744	2.880946	2.716838

Table 8

SSIM, FSIM Metrics, and CPU Time of Real Images Segmented with the Thresholds Yielding the Best Tsallis Entropy.

Image	M	SSIM					FSIM					CPU time (s)
		GA	PSO	BFO	HS	ABC	GA	PSO	BFO	HS	ABC	GA	PSO	BFO	HS	ABC
101085	2	0.5485	0.5485	0.5485	0.5485	0.5485	0.6304	0.6304	0.6304	0.6304	0.6304	2.626421	2.795645	8.448097	3.748662	3.745046
	3	0.7249	0.7249	0.7187	0.7249	0.7249	0.7473	0.7473	0.7455	0.7473	0.7473	2.634630	3.222276	8.578100	3.797151	3.830164
	4	0.7724	0.7845	0.7813	0.7872	0.7872	0.8045	0.8056	0.8045	0.8208	0.8208	2.646673	3.969673	8.608713	3.903888	4.086058
	5	0.8083	0.8173	0.8154	0.8441	0.8441	0.8350	0.8364	0.8363	0.8570	0.8570	2.766825	4.179101	9.523008	3.990952	4.258310
Wherry	2	0.5581	0.5581	0.5581	0.5581	0.5581	0.6824	0.6824	0.6824	0.6824	0.6824	2.532295	3.830592	9.222887	3.754733	3.570691
	3	0.6184	0.6999	0.6707	0.6999	0.6999	0.7415	0.7777	0.7636	0.7777	0.7777	2.535511	3.346097	9.267731	3.630978	3.574615
	4	0.7257	0.7342	0.7342	0.7797	0.7797	0.7890	0.7952	0.7952	0.8107	0.8107	2.564897	3.928787	9.440848	3.666655	3.602458
	5	0.7818	0.8038	0.8014	0.8049	0.8049	0.8199	0.8250	0.8244	0.8268	0.8268	2.762089	2.985715	9.953193	3.800065	3.610382
Snake	2	0.5990	0.5990	0.5990	0.5990	0.5990	0.6606	0.6606	0.6606	0.6606	0.6606	2.434135	2.894742	7.338684	3.510395	3.458902
	3	0.7454	0.7521	0.7515	0.7536	0.7536	0.7924	0.7947	0.7947	0.7964	0.7964	2.488126	3.114613	7.339945	3.598088	3.487053
	4	0.8150	0.8150	0.8150	0.8198	0.8198	0.8514	0.8520	0.8520	0.8557	0.8557	2.495881	3.420465	8.414142	3.646438	3.585258
	5	0.8514	0.8555	0.8544	0.8577	0.8577	0.8903	0.8920	0.8919	0.8933	0.8933	2.515618	3.536209	8.543853	3.749874	3.946437
Zebra	2	0.6064	0.6064	0.6064	0.6064	0.6064	0.6930	0.6930	0.6930	0.6930	0.6930	2.439463	2.585987	8.616790	3.470826	3.530787
	3	0.7566	0.7650	0.7580	0.7650	0.7650	0.8029	0.8045	0.8033	0.8045	0.8045	2.467031	3.218596	8.725099	3.604165	3.550097
	4	0.8120	0.8164	0.8146	0.8213	0.8213	0.8543	0.8550	0.8550	0.8602	0.8602	2.473241	3.818603	8.753483	3.649342	3.554340
	5	0.8223	0.8408	0.8395	0.8533	0.8533	0.8563	0.8768	0.8751	0.8856	0.8856	2.507606	3.841220	8.800343	3.750795	3.623589
Butterfly	2	0.7235	0.7235	0.7235	0.7235	0.7235	0.6982	0.6982	0.6982	0.6982	0.6982	2.536436	2.927824	8.079085	3.551907	3.618250
	3	0.8389	0.8410	0.8410	0.8410	0.8410	0.7973	0.8010	0.8010	0.8010	0.8010	2.549586	3.134004	8.481825	3.757310	3.644737
	4	0.8849	0.8866	0.8866	0.8887	0.8887	0.8548	0.8648	0.8605	0.8676	0.8676	2.558520	3.908504	9.036458	3.966096	3.673664
	5	0.8824	0.9155	0.9145	0.9167	0.9167	0.8578	0.8956	0.8930	0.8973	0.8973	2.573502	3.924450	8.986290	4.043477	3.755942
Bird	2	0.8536	0.8536	0.8536	0.8536	0.8536	0.8290	0.8290	0.8290	0.8290	0.8290	2.430867	3.046916	7.144325	3.504329	3.512970
	3	0.8986	0.8992	0.8989	0.8993	0.8993	0.8846	0.8852	0.8846	0.8867	0.8867	2.432505	3.230546	7.705584	3.596368	3.530190
	4	0.9039	0.9058	0.9050	0.9059	0.9059	0.9017	0.9036	0.9029	0.9041	0.9041	2.453129	3.742993	8.283835	3.675192	3.682864
	5	0.8998	0.9078	0.9014	0.9090	0.9090	0.9079	0.9097	0.9093	0.9110	0.9110	2.473080	3.819853	8.810769	3.751724	3.734316
Landscape	2	0.6402	0.6402	0.6402	0.6402	0.6402	0.7722	0.7722	0.7722	0.7722	0.7722	2.414526	2.749872	7.540766	3.501467	3.504882
	3	0.6587	0.6806	0.6806	0.6836	0.6836	0.7729	0.7859	0.7854	0.7874	0.7874	2.416465	3.171597	8.479776	3.586881	3.505638
	4	0.7142	0.7296	0.7222	0.7306	0.7306	0.8251	0.8318	0.8251	0.8319	0.8319	2.452694	3.841072	8.752473	3.693602	3.541409
	5	0.7369	0.7625	0.7612	0.7655	0.7655	0.8285	0.8461	0.8447	0.8481	0.8481	2.478459	3.860845	8.789089	3.754390	3.554945
Ostrish	2	0.6538	0.6538	0.6538	0.6538	0.6538	0.6846	0.6846	0.6846	0.6846	0.6846	2.470205	3.259708	8.627786	3.543533	3.541746
	3	0.7477	0.7540	0.7535	0.7540	0.7540	0.7491	0.7579	0.7550	0.7579	0.7579	2.471557	3.823040	8.711803	3.622718	3.558267
	4	0.7570	0.7635	0.7734	0.7755	0.7755	0.7828	0.7889	0.7921	0.7926	0.7926	2.494043	3.861283	8.742884	3.725535	3.567410
	5	0.7799	0.7958	0.7933	0.8003	0.8003	0.8034	0.8182	0.8121	0.8189	0.8189	2.510326	2.916485	8.912017	3.762540	3.587248
86016	2	0.5967	0.5967	0.5967	0.5967	0.5967	0.7102	0.7102	0.7102	0.7102	0.7102	2.116886	3.498644	7.831090	3.273018	3.199819
	3	0.7108	0.7166	0.7166	0.7166	0.7166	0.7870	0.7968	0.7968	0.7968	0.7968	2.119636	3.499244	7.928025	3.360776	3.289376
	4	0.7408	0.7442	0.7442	0.7582	0.7582	0.8276	0.8307	0.8285	0.8360	0.8360	2.143438	2.815032	7.997756	3.442461	3.298251
	5	0.8068	0.8073	0.8069	0.8217	0.8217	0.8567	0.8605	0.8588	0.8727	0.8727	2.161149	3.089704	8.141272	3.486067	3.307082
Baboon	2	0.7968	0.7968	0.7968	0.7968	0.7968	0.8531	0.8531	0.8531	0.8531	0.8531	2.318719	2.754326	7.201327	4.776635	3.346769
	3	0.8359	0.8380	0.8380	0.8393	0.8393	0.8853	0.8867	0.8867	0.8883	0.8883	2.339003	3.677829	7.218220	4.812419	3.569012
	4	0.8856	0.8865	0.8865	0.8884	0.8884	0.9090	0.9129	0.9129	0.9177	0.9177	2.469003	3.708766	7.753493	4.839004	3.606355
	5	0.8967	0.8986	0.8975	0.8992	0.8992	0.9194	0.9239	0.9230	0.9246	0.9246	2.480883	3.183258	8.597607	4.949058	3.760195
Lake	2	0.7863	0.7863	0.7863	0.7863	0.7863	0.8247	0.8247	0.8247	0.8247	0.8247	2.394776	2.701248	7.079091	4.519010	3.509786
	3	0.8296	0.8339	0.8299	0.8339	0.8339	0.8653	0.8695	0.8654	0.8695	0.8695	2.431063	3.049446	8.478410	4.672146	3.643058
	4	0.8524	0.8527	0.8524	0.8729	0.8729	0.8866	0.8902	0.8866	0.8957	0.8957	2.543665	3.233192	8.602835	4.834773	3.646600
	5	0.8919	0.8954	0.8958	0.9020	0.9020	0.9127	0.9158	0.9158	0.9250	0.9250	2.893060	3.298347	8.844612	4.942581	3.798441
Women	2	0.7200	0.7200	0.7200	0.7200	0.7200	0.7779	0.7779	0.7779	0.7779	0.7779	2.453058	3.140329	7.159046	4.797852	3.479849
	3	0.8049	0.8070	0.8067	0.8070	0.8070	0.8386	0.8419	0.8410	0.8419	0.8419	2.460320	3.199159	8.373762	4.822158	3.630156
	4	0.8521	0.8544	0.8568	0.8576	0.8576	0.8795	0.8817	0.8817	0.8824	0.8824	2.574740	3.269772	8.699088	4.873256	3.659901
	5	0.8775	0.8800	0.8797	0.8807	0.8807	0.8998	0.9029	0.9025	0.9037	0.9037	2.634672	3.320769	8.764521	4.941415	3.819691

For visual evaluation, five segmented images by HS algorithm and their gray level histograms labeled with the best thresholds are given in Figures 2–6.

Figure 2:

Results of Snake Image Using HS Algorithm.

(A) Original snake image, (B) histogram of original image, (C–F) two-level to five-level thresholding-based segmented image with the best thresholds obtained from HS algorithm using Kapur’s entropy criterion, (G–J) two-level to five-level corresponding histogram labeled with the best threshold values obtained from HS algorithm based on the between-class variance criterion, (K–N) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Tsallis entropy criterion.

Figure 3:

Results of Butterfly Image Using the HS Algorithm.

(A) Original butterfly image, (B) histogram of original image, (C–F) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Kapur’s entropy criterion, (G–J) two-level to five-level corresponding histogram labeled with the best threshold values obtained from the HS algorithm based on the between-class variance criterion, (K–N) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Tsallis entropy criterion.

Figure 4:

Results of Bird Image Using the HS Algorithm.

(A) Original bird image, (B) histogram of original image, (C–F) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Kapur’s entropy criterion, (G–J) two-level to five-level corresponding histogram labeled with the best threshold values obtained from the HS algorithm based on the between-class variance criterion, (K–N) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Tsallis entropy criterion.

Figure 5:

Results of 86010 Image Using the HS Algorithm.

(A) Original 86010 image, (B) histogram of original image, (C–F) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Kapur’s entropy criterion, (G–J) two-level to five-level corresponding histogram labeled with the best threshold values obtained from the HS algorithm based on the between-class variance criterion, (K–N) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Tsallis entropy criterion.

Figure 6:

Results of Woman Image Using the HS Algorithm.

(A) Original woman image, (B) histogram of original image, (C–F) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Kapur’s entropy criterion, (G–J) two-level to five-level corresponding histogram labeled with the best threshold values obtained from the HS algorithm based on the between-class variance criterion, (K–N) two-level to five-level thresholding-based segmented image with the best thresholds obtained from the HS algorithm using Tsallis entropy criterion.

As the real-time applications need less running time in addition to high performance, the CPU time of each algorithm using Kapur’s entropy, between-class variance, and Tsallis entropy has been examined. The computation time for each evolutionary algorithm are listed in Tables 6–8, respectively. As indicated in these tables, computation time increases significantly as the threshold level increases. From these tables, it can be seen that algorithms based on between-class variance give the lowest CPU time. The results suggest that the BFO algorithm converges slowly, and the GA algorithm is the faster over all algorithms. We show also that the HS algorithm is scalable and that the running times of the algorithm seem to grow at a linear rate as the problem size increases.