Vol.:(0123456789) 1 3 Evol. Intel. DOI 10.1007/s12065-017-0152-y RESEARCH PAPER Multilevel image thresholding using entropy of histogram and recently developed population-based metaheuristic algorithms Seyed Jalaleddin Mousavirad 1  · Hossein Ebrahimpour-Komleh 1   Received: 30 December 2016 / Revised: 25 April 2017 / Accepted: 30 May 2017 © Springer-Verlag GmbH Germany 2017 methods to compare P-metaheuristic algorithms. Eventu- ally, to create a more reliable result, another objective func- tion was evaluated based on Cross Entropy. Keywords Kapur’s entropy · Cross entropy · Metaheuristic algorithms · Image thresholding · Statistical test 1 Introduction Image segmentation is one of the most important steps in images analysis. It is the process of dividing an image into some meaningful regions. Pixels inside the same region have similar properties. Image segmentation can be regarded as a preprocessing step in many image pro- cessing and computer-vision-based applications [1–4]. In recent years, various image segmentation methods have been proposed such as fuzzy c-mean and its variants [5–7], deep convolutional neural networks [8], and graph cut [9]. Among the existing image segmentation methods, image thresholding is one of the most popular techniques due to its superiority in simplicity, robustness, and efciency [10]. Image thresholding works based on the existing infor- mation in the image histogram. The histogram shows the distribution of pixel values. In this method, images could be segmented into diferent regions using one or more threshold values. Image thresholding is widely used in many applications such as food quality [11], satellite image processing [12], character recognition [13], and medical imaging [14]. Image thresholding techniques are divided into two types of parametric and non-parametric approaches. Para- metric approaches assume that each region has a statistical distribution, which endeavors to fnd an approximation of Abstract Multilevel thresholding is one of the most broadly used approaches to image segmentation. However, the traditional techniques of multilevel thresholding are time-consuming, especially when the number of the thresh- old values is high. Thus, population-based metaheuris- tic (P-metaheuristic) algorithms can be used to overcome this limitation. P-metaheuristic algorithms are a type of optimization algorithms, which improve a set of solutions using an iterative process. For this purpose, image thresh- olding problem should be seen as an optimization prob- lem. This paper proposes multilevel image thresholding for image segmentation using several recently presented P-metaheuristic algorithms, including whale optimization algorithm, grey wolf optimizer, cuckoo optimization algo- rithm, biogeography-based optimization, teaching–learn- ing-based optimization, gravitational search algorithm, imperialist competitive algorithm, and cuckoo search. Kapur’s entropy is used as the objective function. To con- duct a more comprehensive comparison, the mentioned P-metaheuristic algorithms were compared with fve oth- ers. Several experiments were conducted on 12 benchmark images to compare the algorithms regarding objective func- tion value, peak signal to noise ratio (PSNR), feature simi- larity index (FSIM), structural similarity index (SSIM), and stability. In addition, Friedman test and Wilcoxon signed rank test were carried out as the nonparametric statistical * Seyed Jalaleddin Mousavirad jalalmoosavirad@gmail.com; mousavirad@grad.kashanu.ac.ir * Hossein Ebrahimpour-Komleh ebrahimpour@kashanu.ac.ir 1 Department of Computer and Electrical Engineering, University of Kashan, Kashan, Iran