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Abstract: Threshold selection is a critical preprocessing step for image analysis, pattern recognition and computer vision. On the other hand differential evolution (DE) is a heuristic method for solving complex optimization problems, yielding promising results. DE is easy to use, keeps a simple structure and holds acceptable convergence properties and robustness. In this work, a novel automatic image multi-threshold approach based on differential evolution optimization is proposed. Hereby the segmentation process is considered to be similar to an optimization problem. First, the algorithm fills the 1-D histogram of the image using a mix of Gaussian functions whose parameters are calculated using the differential evolution method. Each Gaussian function approximating the histogram represents a pixel class and therefore a threshold point. The proposed approach is not only computationally efficient but also does not require prior assumptions whatsoever about the image. The method is likely to be most useful for applications considering different and perhaps initially unknown image classes. Experimental results demonstrate the algorithm’s ability to perform automatic threshold selection while preserving main features from the original image. [Copyright &y& Elsevier]

Highlights • Αn Unassisted Thresholding (UTH) is presented to perform image segmentation while simultaneously determinating the optimal number of thresholds and their values. • UTH is implemented with NSGA III, MOPSO, and PESA II to evaluate its performance. • A particle with variable length is used to encode threshold values on the population. Abstract Multi-Objective Evolutionary Algorithms (MOEAs) are known to solve problems where two or more conflicting goals are involved. To accomplish it, MOEAs incorporate strategies to determinate optimal trade-offs between each of the objective functions. In this paper, an Unassisted image Thresholding (UTH) methodology is proposed based on MOEAs. UTH takes advantage of the trade-off mechanisms present on MOEAs to perform the image thresholding while simultaneously determinating the number thresholds required to segment each image and the best placement of each threshold along the histogram of the image. The image thresholding problem is commonly addressed as the search for the best possible thresholds able to partition a given image into a finite number of homogeneous classes. Such approach requires the assistance of a designer to determinate the number of threshold values that will properly segment the image. However, as images can vary significantly, the definition of an optimal number of thresholds should be performed for each image. Thus, a methodology able to determinate both the number of thresholds and the best placement of each value contributes to a general segmentation scheme. In the proposed approach, UTH redefines the thresholding problem as a multi-objective task with two conflicting goals. The first goal is the quality of the segmented image, and it is computed as a non-parametric criteria to evaluate candidate threshold points. The second goal is the normalized number of threshold points. Since the number of thresholds is not fixed, a particle encoding the thresholds with variable length is used. The strategy of UTH is coupled with three MOEAs namely NSGA-III, PESA-II and MOPSO using as the non-parametric criteria the Cross Entropy. According to the results, the UTH NSGA-III formulation outperforms UTH-PESA-II and UTH-MOPSO regarding convergence and quality of the resulting image. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

• An efficient method for image thresholding is proposed. • The Harris hawk optimizer is used for image segmentation. • The Harris hawk optimizer is tested over a multidimensional real problem. • The quality of the segmentation results is higher than other algorithms. • The performance of the algorithm is tested on digital mammograms. Segmentation is a crucial phase in image processing because it simplifies the representation of an image and facilitates its analysis. The multilevel thresholding method is more efficient for segmenting digital mammograms compared to the classic bi-level thresholding since it uses a higher number of intensities to represent different regions in the image. In the literature, there are different techniques for multilevel segmentation; however, most of these approaches do not obtain good segmented images. In addition, they are computationally expensive. Recently, statistical criteria such as Otsu, Kapur, and cross-entropy have been utilized in combination with evolutionary and swarm-based strategies to investigate the optimal threshold values for multilevel segmentation. In this paper, an efficient methodology for multilevel segmentation is proposed using the Harris Hawks Optimization (HHO) algorithm and the minimum cross-entropy as a fitness function. To substantiate the results and effectiveness of the HHO-based method, it has been tested over a benchmark set of reference images, with the Berkeley segmentation database, and with medical images of digital mammography. The proposed HHO-based solver is verified based on other comparable optimizers and two machine learning algorithms K -means and the Fuzzy IterAg. The comparisons were performed based on three groups. This first one is to provide evidence of the optimization capabilities of the HHO using the Wilcoxon test, and the second is to verify segmented image quality using the PSNR, SSIM, and FSIM metrics. Then, the third way is to verify the segmented image comparing it with the ground-truth through the metrics PRI, GCE, and VoI. The experimental results, which are validated by statistical analysis, show that the introduced method produces efficient and reliable results in terms of quality, consistency, and accuracy in comparison with the other methods. This HHO-based method presents an improvement over other segmentation approaches that are currently used in the literature. [ABSTRACT FROM AUTHOR]