# A multilevel image thresholding using the animal migration optimization algorithm

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## Abstract

Thresholding is an important and well-known technique that plays a major role in distinguishing the image objects from its background. In the other hand, separating the images into several different regions by determining multiple threshold values is called multilevel image thresholding. The Kapur entropy thresholding and maximum between-class variance (Otsu) have been widely used in image thresholding. However, these methods are computationally expensive and with increase in level numbers computational complexity increase exponentially. To overcome this problem, this paper presents animal migration optimization algorithm for multilevel thresholding. For evaluating the efficiency of proposed method, various benchmark images are used for carrying out the experiments, and obtained results via animal migration optimization algorithm compared with most popular optimization technique such as Particle Swarm Optimization, Genetic and bacterial foraging algorithm. Experimental results figure out that the proposed method provides better result than the other tested algorithms.

## Keywords

Image segmentation Multilevel thresholding Animal migration optimization Kapur’s function Otsu’s function## 1 Introduction

Segmentation is the fundamental and the basic step in image processing and generally used in partitioning of images into meaningful segments and objects. The aim of segmentation is changing the representation of the images into images that are meaningful and more simpler for interpreting and analyzing [1]. Generally, in image processing applications, there are many differences between grey levels of pixels that belong to the background and those that belong to the foreground (objects). Thresholding is one of the easiest and well-known techniques for distinguishing the image objects from its background. Undoubtedly, image histogram in grey-level images is a fully effective tool for developing algorithm in the field of the image thresholding [2]. Dividing images into different regions (background and object) by calculating a single threshold value is known as bi-level thresholding [3], while separating the images into several different regions by determining multiple threshold values is called multilevel image thresholding [4, 5].

Histogram of grey-level image is regarded as fully effective tool in image segmentation algorithm. The aim of thresholding is determining a proper threshold value in bi-level thresholding and determining proper multiple threshold values in multilevel thresholding. The easiest problem is the bi-level thresholding, where existing peaks represent the different segments, and the valley between the existing peaks represents the threshold. Each pixel in an image which is greater than the threshold value is replaced with a white pixel and considered as an object pixel (foreground). On the other hand, rest of the pixels that are less than threshold value are replaced with black pixels and belong to the background [6]. Moreover, the problem ramp up into more complex by employing multilevel thresholding. Hence the image will be segmented into many regions. Then pixels that are in same range will be placed into a class [7].

In general, determining the exact locations of distinct valleys in an image histogram which include multiple peaks (multimodal histogram) is not easy. A wide variety of algorithms have been proposed in histogram thresholding area [8, 9, 10, 11, 12, 13, 14, 15, 16]; a large number of these algorithms refer to bi-level thresholding and rest of them address the multilevel thresholding. Algorithms that are developed for bi-level thresholding problems (such as Otsu and Entropy) [8, 9] can also be extended to solve multilevel thresholding problems [17, 18, 19].

Optimal thresholds can be determined by analyzing and evaluating the histogram information or by optimizing certain and reliable objective function [20, 21]. In both of bi-level and multilevel thresholding, the aim is determining an accurate threshold or multiple thresholds, respectively.

In the linear search the search process performed by considering all the elements. Hence, with increasing the number of thresholds, computational complexity increases exponentially. This limits the ability of multilevel thresholding algorithms.

Meta-heuristic techniques have been proved to be an efficient tool to solve such problems and they are one of the best ways to yield optimal solutions and also can be helpful to overcome these limitations [22]. PSO, Genetic, Firefly Algorithm (FA), Cuckoo search, Artificial Bee Colony (ABC) and bacterial foraging algorithm (BF), which are the heuristic algorithms, have been applied in multilevel thresholding [23].

Sathya and Kayalvizhi proposed bacterial foraging algorithm, modified bacterial foraging algorithm and amended bacterial foraging algorithm for the minimum variance and maximum entropy criterions [22, 24, 25]. Moreover, Yin [26] proposed a particle swarm optimization for the multilevel minimum cross-entropy criterion; on the other hand, Horng and jiang [2] applied the firefly optimization algorithm to optimize minimum cross-entropy criterion. In addition to the mentioned algorithms, Pal and et al. [27] also proposed a multilevel thresholding algorithm for the minimum variance and maximum entropy criterions using Spider Monkey Optimization Algorithm. Finally, Horng [28] also proposed a multilevel thresholding algorithm based on ABC algorithm to optimize the maximum entropy criterion.

Animal migration optimization (AMO) developed by Xiangtao and et al. is one of the novel heuristic algorithm [29].AMO is inspired by the animal migration behavior, which is a widespread phenomenon that can be seen in most of animal groups. In this study, AMO algorithm is used to solve the problem of multilevel thresholding for image segmentation. In the proposed method Otsu’s and Kapur’s methods are considered as the objective functions. Experimental results are reported and are also compared with some widely used optimization algorithm for multilevel thresholding.

The paper is structured as follows. Section 2 presents problem formulation. Section 3 presents the concepts of animal migration model and algorithm. Section 4 provides the illustration of the proposed method. Section 5 analyzes and compares the results obtained for the proposed approach and other methods. Finally, conclusion is presented in Sect. 6.

## 2 Problem formulation

In this proposed paper, two most popular methods that were used for finding optimal threshold on image histogram are known by the name of Kapur entropy and the between-class variance (Otsu’s) method. Both of these methods separately are used for maximizing an objective function [25].

### 2.1 Entropy criterion method (Kapur’s method)

*H*

_{0}represents partial entropy of histogram for first segment and

*H*

_{1}is for second segment. Kapur’s entropy criterion method can also extended for settling problem of multilevel thresholding. Then, the problem can be turned as a multi-dimensional problem. Determining all of threshold values can be described with the following equation:

### 2.2 Between-class variance method (Otsu’s method)

## 3 Animal migration model and algorithm

- 1.
The animal with the best objective function value is considered as the leader, and its position is preserved in next generation.

- 2.
Size of population is fixed, and a new member will be replaced with the existing animal probability p

_{a}. Therefore, an animal is leaving from the group and a new one will be joining instead of the leaving one.

*j*th component from

*i*th vector defined as:

_{i,j}[0, 1] is a random number with uniformly distribution between 0 and 1,

*i*= 1, …,

*N*and

*j*= 1, …,

*D*. where

*N*denotes the population size.

_{th}index, indices of neighbourhood animal are

*i*− 2,

*i*− 1,

*i*,

*i*+ 1 and i + 2 as it is clear from Fig. 1. The first animal is the neighbor of the last animal and vice versa. So the indices of neighbours of first animal are 1, 2, 3,

*N*and

*N*− 1. When the topology of the neighbourhood has been constructed, one of neighbour will be selected randomly and the individual position will be updated according to this neighbour, as it is formulated in the equation:

where *r*_{1}, *r*_{2} are two randomly selected integers between [1, Population size], and *r*_{1}\( \ne \)*r*_{2}\( \ne \)*i*. The new generated solution *x*_{i}, *G* + 1, will be evaluated to compare with the existing solution *x*_{i, G}, if the fitness value of *x*_{i, G+1} is better than fitness of *X*_{i, G}, the new solution will be *x*_{i, G+1}; otherwise, the accepted solution is *x*_{i, G}. Algorithm 1 show the pseudocode of standard animal migration optimization algorithm.

## 4 Multilevel thresholding based on AMO

*x*= [

*t*

_{1},

*t*

_{2},

*t*

_{3}, \( \ldots \),

*t*

_{D}] will be obtained by maximized the Eqs. (2) or (4). In AMO algorithm both of these objective functions are also applied as the fitness function. Each individual position corresponds to a solution. The number of thresholds determines the dimension number of optimization algorithm. Each dimension of optimization algorithm represents a solution to a threshold. The algorithm starts with a D-dimension population that is initiated randomly. Each dimension is constrained by bounds [0, 255]. So, initialization of

*j*th component from

*i*th vector defined as Eq. (7):

## 5 Experimental results and performance evaluation

Control parameters of the optimization algorithms

Algorithm | Initial values of control parameters |
---|---|

AMO | Neighbourhood length = 5,\( \delta \) is a random number controlled by a Gaussian distribution |

PSO | Cognitive coefficient ( |

GA | Roulette selection, CR = 0.8, Gaussian Mutation scale = 2 shrink = 1 |

BF | Number of chemotactic steps (Nc) = 100, swimming length (Ns) = 3, number of reproduction steps (Nre) = 4, number of elimination of dispersal events (Ned) = 2, step size(Ci) = 5.1, probability of elimination and dispersal (Ped) = 0.25 |

Comparison of best obtained objective and optimal threshold values based on Kapur

Test images | No. of thresholds | Average fitness value | Average optimal threshold values | ||||||
---|---|---|---|---|---|---|---|---|---|

AMO | PSO | GA | BF | AMO | PSO | GA | BF | ||

Cameraman | 2 | | 12.2802 | 12.2802 | 12.2651 | 125,196 | 125,196 | 125,196 | 125,196 |

3 | | 15.3880 | 15.3880 | 15.3765 | 43,101,196 | 44,102,196 | 44,102,196 | 44,103,196 | |

4 | | 18.5497 | 18.5494 | 18.5080 | 42,96,145,196 | 42,96,145,198 | 42,96,145,197 | 42,97,145,197 | |

5 | | 21.2613 | 21.2525 | 21.2230 | 41,96,143,191,221 | 42,96,145,191,222 | 42,90,133,180,214 | 26,66,108,153,201 | |

Lenna | 2 | | | 12.3464 | 12.3344 | 97,164 | 97,164 | 97,164 | 97,164 |

3 | | 15.3184 | 15.3184 | 15.2996 | 82,127,175 | 82,126,175 | 82,126,175 | 83,127,176 | |

4 | | 18.0079 | 18.0061 | 17.9662 | 73,111, 147,183 | 66,101,140,181 | 68,103,141,180 | 68,102,141,180 | |

5 | | 20.6093 | 20.6091 | 20.5042 | 63,94,127,162,193 | 63,94,128,163,194 | 63,94,128,163,194 | 63,95,128,162,194 | |

Livingroom | 2 | | 12.4059 | 12.4059 | 12.3945 | 89,170 | 94,175 | 94,175 | 94,175 |

3 | 15.5210 | 15.4056 | | 15.5262 | 49,107,178 | 60,123,192 | 47,104,175 | 48,104,174 | |

4 | 18.4067 | 18.2453 | | 18.4198 | 46,100,159,215 | 47,102,171,230 | 47,98,149,196 | 47,98,150,194 | |

5 | | 21.0947 | 21.1439 | 21.0683 | 45,93,139,183,233 | 47,98,145,188,236 | 44,87,125,163,199 | 41,86,125,164,199 | |

Hunter | 2 | | 12.0033 | 12.0033 | 11.9948 | 114,179 | 112,179 | 112,179 | 112,179 |

3 | | 14.9748 | 14.9745 | 14.9580 | 86,132,180 | 86,129,180 | 86,131,180 | 87,133,181 | |

4 | | 17.6193 | 17.6581 | 17.6032 | 76,113,149,183 | 77,115,152,187 | 76,114,149,183 | 76,113,148,183 | |

5 | | 20.0197 | 20.1024 | 20.0033 | 73,107,139,171,200 | 72,105,138,170,205 | 68,96,125,154,183 | 67,96,125,156,186 | |

Pepper | 2 | | 12.5852 | 12.5852 | 12.5781 | 76, 146 | 77,147 | 77,147 | 77,147 |

3 | | 15.6238 | 15.6238 | 15.6158 | 61,112,163 | 61,112,164 | 61,112,164 | 62,113,165 | |

4 | | 18.5286 | 18.4608 | 18.4269 | 56,99,144,190 | 52,92,136,182 | 59,105,149,195 | 58,105,150,195 | |

5 | | | 21.1922 | 21.1447 | 45,78,114,154,195 | 45,78,114,154,195 | 45,78,115,154,195 | 43,77,116,156,195 | |

Mandrill | 2 | | | | 12.1114 | 80,143 | 80,143 | 80,143 | 80,143 |

3 | | | 15.1309 | 15.1137 | 56,104,153 | 56,104,153 | 56,104,153 | 57,105,153 | |

4 | 17.8940 | | 17.8936 | 17.8589 | 46,84,122,162 | 45,82,120,161, | 47,85,124,163 | 45,83,121,161 | |

5 | 20.5081 | | 20.5037 | 20.4341 | 40,73,107,139,172 | 40,73,106,139,172 | 42,75,108,140,172 | 38,72,106,141,174 |

Comparison of standard deviation for Kapur based

Test images | No. of thresholds | Standard deviation | |||
---|---|---|---|---|---|

AMO | PSO | GA | BF | ||

Cameraman | 2 | 5.4202e−15 | 3.6450e−15 | 3.6450e−15 | 0.0987 |

3 | 1.6113e−05 | 9.1125e−15 | 9.1125e−15 | 0.0523 | |

4 | 0.000031 | 3.6450e−15 | 4.2827e−04 | 0.0706 | |

5 | 3.0161e−04 | 3.6450e−15 | 0.0232 | 0.0640 | |

Lenna | 2 | 0.0022 | 3.6450e−15 | 3.6450e−15 | 0.1082 |

3 | 0.0117 | 0 | 0 | 0.0985 | |

4 | 0.0072 | 0.0092 | 0.0079 | 0.0876 | |

5 | 3.4142e−04 | 0 | 3.7072e−04 | 0.0884 | |

Livingroom | 2 | 5.4202e−15 | 5.4675e−15 | 5.4675e−15 | 0.0948 |

3 | 0.2346 | 0.2500 | 0.0013 | 0.0906 | |

4 | 0.1465 | 0.0972 | 0.0040 | 0.0654 | |

5 | 0.0202 | 7.2900e−15 | 0.0031 | 0.0541 | |

Hunter | 2 | 1.8067e−15 | 0 | 0 | 0.0819 |

3 | 7.5814e−04 | 0 | 3.4816e−04 | 0.0745 | |

4 | 3.6134e−15 | 0.1208 | 0.0012 | 0.0810 | |

5 | 0.0705 | 0.0836 | 0.0085 | 0.0621 | |

pepper | 2 | 0 | 5.4675e−15 | 5.4675e−15 | 0.0650 |

3 | 9.0597e−05 | 0 | 3.7205e−06 | 0.0435 | |

4 | 0.0145 | 0.0161 | 5.0796e−04 | 0.0290 | |

5 | 3.7449e−15 | 1.0840e−14 | 0.0011 | 0.0275 | |

Mandrill | 2 | 5.4675e−15 | 5.4675e−15 | 5.4675e−15 | 0.1055 |

3 | 9.1125e−15 | 9.1125e−15 | 1.7303e−04 | 0.0874 | |

4 | 0.0021 | 0.0021 | 0.0023 | 0.0663 | |

5 | 4.6559e−04 | 3.6450e−15 | 0.0048 | 0.0585 |

*m*” levels. The best obtained results are bolded in each row. As it is clear from Tables 2 and 5, in most case the best results are obtained by AMO algorithm. Also, Tables 3 and 6 reveal that except BA, most of the calculated standard deviations from other three algorithms are too small. By comparing Tables 3 and 6 it turns out that the obtained standard deviation from Kapur method is lower than Otsu method. Moreover, two well-known performance indicators, peak signal-to-noise ratio (PSNR) and structural-similarity index (SSIM), are used to compare the segmentation results using the multilevel image threshold methods. General form of PSNR and SSIM can be modelled as in Eqs. (8) and (9), respectively.

*M*\( \times \)

*N*, respectively.

*N*indicates number of runs, \( x_{i} \) is fitness value in

*i*th run and \( \mu \) is mean values among all runs.

Comparison of PSNR and SSIM for Kapur based

Test images | Number of thresholds | Standard deviation | |||||||
---|---|---|---|---|---|---|---|---|---|

AMO | PSO | GA | BF | ||||||

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||

Cameraman | 2 | 13.7753 | 0.5146 | 13.7753 | 0.5146 | 13.7753 | 0.5146 | 13.7753 | 0.5146 |

3 | 14.3070 | 0.6019 | 14.4263 | 0.6046 | 14.4263 | 0.6046 | 14.5417 | 0.60701 | |

4 | 20.1555 | 0.6637 | 20.1555 | 0.6637 | 20.1633 | 0.6636 | 20.2167 | 0.66456 | |

5 | 20.1686 | 0.6707 | 20.2799 | 0.6628 | 20.0774 | 0.6480 | 21.4091 | 0.68342 | |

Lenna | 2 | 14.5859 | 0.52382 | 14.5859 | 0.52382 | 14.5859 | 0.52382 | 16.838 | 0.51895 |

3 | 17.2232 | 0.6231 | 15.7096 | 0.62779 | 17.2244 | 0.62506 | 18.9112 | 0.61391 | |

4 | 18.6493 | 0.67944 | 19.4767 | 0.69249 | 18.8699 | 0.67877 | 20.8974 | 0.64346 | |

5 | 19.856 | 0.70838 | 19.7569 | 0.70255 | 19.8211 | 0.70712 | 22.2527 | 0.65467 | |

Livingroom | 2 | 14.631 | 0.49419 | 14.5962 | 0.48312 | 14.5962 | 0.48312 | 14.5962 | 0.48312 |

3 | 17.2887 | 0.60866 | 16.9132 | 0.57559 | 17.1987 | 0.61413 | 17.2845 | 0.61542 | |

4 | 18.2184 | 0.64196 | 10.8862 | 0.38196 | 19.1671 | 0.66974 | 19.22 | 0.66957 | |

5 | 19.8144 | 0.68587 | 19.6189 | 0.6816 | 21.2505 | 0.72583 | 21.0899 | 0.72317 | |

Hunter | 2 | 13.1492 | 0.3477 | 13.2331 | 0.35289 | 13.2331 | 0.35289 | 13.2331 | 0.35289 |

3 | 16.1986 | 0.48296 | 16.2147 | 0.48668 | 16.1971 | 0.48375 | 16.1222 | 0.48013 | |

4 | 17.8826 | 0.56959 | 17.7638 | 0.56322 | 17.894 | 0.56994 | 17.9166 | 0.57187 | |

5 | 18.4828 | 0.59193 | 18.5315 | 0.59075 | 19.2596 | 0.63065 | 19.3138 | 0.63115 | |

Pepper | 2 | 16.4554 | 0.58523 | 16.4435 | 0.58126 | 16.4435 | 0.58126 | 16.4435 | 0.58126 |

3 | 18.3578 | 0.65889 | 18.3578 | 0.65889 | 18.3578 | 0.65889 | 18.3571 | 0.65832 | |

4 | 20.397 | 0.69456 | 20.397 | 0.69456 | 19.1435 | 0.68261 | 19.1194 | 0.68263 | |

5 | 21.3492 | 0.74725 | 21.3492 | 0.74725 | 21.3756 | 0.74773 | 21.3604 | 0.74863 | |

Mandrill | 2 | 16.1789 | 0.61377 | 16.1789 | 0.61377 | 16.1789 | 0.61377 | 16.1789 | 0.61377 |

3 | 19.1587 | 0.72407 | 19.1587 | 0.72407 | 19.1587 | 0.72407 | 19.2053 | 0.72503 | |

4 | 21.0974 | 0.77348 | 21.0733 | 0.77163 | 21.0668 | 0.77425 | 21.0887 | 0.77336 | |

5 | 22.7607 | 0.82299 | 22.7317 | 0.82299 | 22.8556 | 0.825 | 22.5138 | 0.8171 |

## 6 Conclusion

Comparison of best obtained objective and optimal threshold values based on Otsu

Test images | No. of thresholds | Average fitness value | Average optimal threshold values | ||||||
---|---|---|---|---|---|---|---|---|---|

AMO | PSO | GA | BF | AMO | PSO | GA | BF | ||

Cameraman | 2 | | | 3.6039e+03 | 3.5985e+03 | 69,144 | 69,144 | 70,144 | 70,144 |

3 | 3.6765e+03 | | | 3.6735e+03 | 59,119,155 | 56,115,153 | 56,115,153 | 56,115,153 | |

4 | | | 3.7324e+03 | 3.7264e+03 | 41,94,139,169 | 41,94,139,169 | 43,95,139,169 | 42,93,139,169 | |

5 | 3.7617e+03 | 3.7631e+03 | 3.7624e+03 | 3.7544e+03 | 36,85,125,152,175 | 35,82,121,148,172 | 40,85,123,149,172 | 34,83,122,149,172 | |

Lenna | 2 | | | | 1.9563e+03 | 92,152 | 92,152 | 93,151 | 93,151 |

3 | | | 2.1283e+03 | 2.1236e+03 | 79,125,172 | 79,125,172 | 81,127,171 | 81,126,171 | |

4 | | | 2.1918e+03 | 2.1845e+03 | 74,112,145,177 | 74,112,145,177 | 75,114,145,180 | 74,114,145,179 | |

5 | 2.2176e+03 | 2.2178e+03 | 2.2177e+03 | 2.2085e+03 | 73,108,136,160,188 | 73,109,136,160,188 | 71,105,133,157,186 | 70,107,134,160,189 | |

Livingroom | 2 | | | 1.6273e+03 | 1.6196e+03 | 86,145 | 86,145 | 87,145 | 87,145 |

3 | | | 1.7594e+03 | 1.7510e+03 | 73,123,164 | 73,123,164 | 76,123,163 | 75,123,163 | |

4 | | 1.8593e+03 | 1.8281e+03 | 1.8178e+03 | 48,91,132,168 | 48,92,133,168 | 56,97,132,168 | 56,97,132,168 | |

5 | | 1.9150e+03 | 1.8709e+03 | 1.8583e+03 | 47,86,123,151,182 | 47,87,124,152,182 | 51,89,121,148,179 | 45,86,120,146,177 | |

Hunter | 2 | | | 1.9945e+03 | 1.9907e+03 | 87,143 | 87,143 | 86,141 | 86,141 |

3 | | | 2.1153e+03 | 2.1114e+03 | 70,116,158 | 70,116,158 | 71,114,155 | 71,113,155 | |

4 | | 2.2443e+03 | 2.1697e+03 | 2.1653e+03 | 46,83,124,163 | 54,93,133,168 | 67,106,140,172 | 68,106,141,174 | |

5 | | | | 2.1935e+03 | 62,93,122,150,181 | 62,93,122,150,181 | 63,93,122,150,180 | 62,94,123,151,181 | |

Pepper | 2 | | | 2.4695e+03 | 2.4616e+03 | 66,135 | 66,135 | 69,136 | 69,136 |

3 | | | 2.6367e+03 | 2.6339e+03 | 61,118,167 | 61,118,167 | 64,119,166 | 64,119,167 | |

4 | | | 2.6983e+03 | 2.6943e+03 | 53,90,130,175 | 50,88,128,173 | 48,86,126,170 | 48,87,126,169 | |

5 | | | 2.7424e+03 | 2.7364e+03 | 43,79,113,146,177 | 43,79,113,146,177 | 44,80,113,146,177 | 44,78,113,147,178 | |

Mandrill | 2 | | | | 1.3588e+03 | 98,148 | 98,148 | 98,148 | 98,148 |

3 | | | | 1.4472e+03 | 86,124,159 | 86,124,159 | 86,124,159 | 86,124,159 | |

4 | | | | 1.4946e+03 | 73,106,135,165 | 73,106,135,165 | 73,106,135,165 | 73,106,135,165 | |

5 | 1.5282e+03 | | 1.5285e+03 | 1.5164e+03 | 68,98,123,146,171 | 69,100,125,148,172 | 69,100,125,148,172 | 66,96,122,146,171 |

Comparison of standard deviation for Otsu based

Test images | No. of thresholds | Standard deviation | |||
---|---|---|---|---|---|

AMO | PSO | GA | BF | ||

Cameraman | 2 | 1.8501e−12 | 1.8501e−12 | 1.3997e−12 | 44.2376 |

3 | 1.7373 | 53.9406 | 1.8662e−12 | 15.9253 | |

4 | 0.0120 | 9.3312e−13 | 0.1830 | 10.8050 | |

5 | 0.9896 | 4.6656e−13 | 0.8725 | 7.1871 | |

Lenna | 2 | 9.2504e−13 | 9.2504e−13 | 2.3328e−13 | 51.8123 |

3 | 1.3876e−12 | 1.3876e−12 | 0 | 23.8312 | |

4 | 1.8352e−12 | 1.8501e−12 | 4.6656e−13 | 15.3602 | |

5 | 0.3689 | 0.3160 | 0.6278 | 9.8940 | |

Livingroom | 2 | 6.9378e−13 | 6.9378e−13 | 6.9984e−13 | 64.2211 |

3 | 9.2504e−13 | 9.2504e−13 | 0 | 35.7859 | |

4 | 0.1456 | 3.6995 | 0.0154 | 21.6782 | |

5 | 2.3126e−13 | 1.5258 | 0.2647 | 14.1206 | |

Hunter | 2 | 9.2504e−13 | 9.2504e−13 | 2.3328e−13 | 31.9147 |

3 | 9.2504e−13 | 9.2504e−13 | 0 | 14.4766 | |

4 | 12.6234 | 32.1884 | 0.0024 | 7.1271 | |

5 | 0.0940 | 0 | 0.1160 | 6.5958 | |

Pepper | 2 | 9.2504e−13 | 9.2504e−13 | 0 | 71.8040 |

3 | 1.1012e−15 | 0 | 4.6656e−13 | 14.4440 | |

4 | 0.4845 | 1.8383 | 0.0945 | 10.1671 | |

5 | 0.1946 | 0.9984 | 0.1275 | 4.5501 | |

Mandrill | 2 | 4.6656e−13 | 4.6656e−13 | 4.6656e−13 | 73.1514 |

3 | 4.6656e−13 | 4.6656e−13 | 4.6656e−13 | 32.4821 | |

4 | 4.6656e−13 | 4.6656e−13 | 0.0069 | 17.6939 | |

5 | 0.5030 | 4.6656e−13 | 0.0370 | 12.4319 |

Comparison of PSNR and SSIM for Otsu based

Test images | Number of thresholds | Standard deviation | |||||||
---|---|---|---|---|---|---|---|---|---|

AMO | PSO | GA | BF | ||||||

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||

Cameraman | 2 | 17.1732 | 0.59537 | 17.1732 | 0.59537 | 17.2478 | 0.59641 | 17.2478 | 0.59641 |

3 | 20.2363 | 0.63574 | 20.3227 | 0.64408 | 20.3227 | 0.64408 | 20.3227 | 0.64408 | |

4 | 21.5455 | 0.65043 | 21.5455 | 0.65043 | 21.6175 | 0.65091 | 21.4807 | 0.64909 | |

5 | 23.2902 | 0.68322 | 23.3765 | 0.68896 | 23.4499 | 0.68699 | 23.3814 | 0.6886 | |

Lenna | 2 | 15.2802 | 0.54522 | 15.2802 | 0.54522 | 15.2823 | 0.54652 | 15.2823 | 0.54652 |

3 | 17.3742 | 0.63411 | 17.3742 | 0.63411 | 17.3146 | 18.6142 | 17.3146 | 18.6142 | |

4 | 18.6337 | 0.67846 | 18.6337 | 0.67846 | 0.62887 | 0.67729 | 18.6502 | 0.6786 | |

5 | 19.2572 | 0.70155 | 19.2517 | 0.70288 | 19.4237 | 0.69785 | 19.4245 | 0.70323 | |

Livingroom | 2 | 16.0077 | 0.53765 | 16.0077 | 0.53765 | 15.9995 | 0.53677 | 15.9995 | 0.53677 |

3 | 18.1379 | 0.61375 | 18.1379 | 0.61375 | 18.1975 | 0.61634 | 18.1415 | 0.61475 | |

4 | 20.54 | 0.70526 | 20.6179 | 0.70537 | 20.6735 | 0.70428 | 20.6735 | 0.70428 | |

5 | 21.9078 | 0.74681 | 21.9366 | 0.7466 | 22.1618 | 0.75334 | 22.244 | 0.75756 | |

Hunter | 2 | 15.5292 | 0.45068 | 15.5292 | 0.45068 | 15.5917 | 0.45593 | 15.5917 | 0.45593 |

3 | 17.8573 | 0.5603 | 17.8573 | 0.5603 | 17.7883 | 0.56217 | 17.7709 | 0.56207 | |

4 | 20.0522 | 0.6475 | 19.4655 | 0.62044 | 18.7652 | 0.60213 | 18.6814 | 0.59819 | |

5 | 19.8157 | 0.65123 | 19.8157 | 0.65123 | 19.8157 | 0.65123 | 19.8263 | 0.65048 | |

Pepper | 2 | 16.2227 | 0.61328 | 16.2227 | 0.61328 | 16.3234 | 0.60961 | 16.3234 | 0.60961 |

3 | 18.3452 | 0.65895 | 18.3452 | 0.65895 | 18.4696 | 0.65733 | 18.4426 | 0.65726 | |

4 | 20.6145 | 0.702 | 20.6753 | 0.71086 | 20.6804 | 0.71674 | 20.6766 | 0.71482 | |

5 | 22.393 | 0.76129 | 22.393 | 0.76129 | 22.4186 | 0.7591 | 22.3611 | 0.76169 | |

Mandrill | 2 | 15.4335 | 0.58674 | 15.4335 | 0.58674 | 15.4335 | 0.58674 | 15.4335 | 0.58674 |

3 | 17.8513 | 0.67863 | 17.8513 | 0.67863 | 17.8513 | 0.67863 | 17.8513 | 0.67863 | |

4 | 20.4155 | 0.77083 | 20.4155 | 0.77083 | 20.4155 | 0.77083 | 20.4155 | 0.77083 | |

5 | 21.8362 | 0.81077 | 21.6952 | 0.80974 | 21.6952 | 0.80974 | 22.0639 | 0.8137 |

## Notes

### Acknowledgements

The author declares no conflict of interest in this study.

### Funding

We confirm that we do not have a funding source.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no competing interests.

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