## 1 Introduction  
Image segmentation is a process of extracting primitive objects or constituent regions from the image. Image thresholding is the simplest way of performing image segmentation. Image thresholding methods [1; 2; 3; 4; 5; 6; 7] use threshold value for image segmentation. The first step of image thresholding is converting the image into a grayscale image [8]. The intensity level of each pixel in the grayscale image is compared with the threshold value and based on the comparison result, and the pixels are encoded as one of the class labels. For a binary image, the class labels are '0' and '1'. Most studies have tended to focus on infrared image processing [9; 10; 11; 12; 13; 14; 15]. Image segmentation is recognized as being the most important part of infrared image processing. Infrared (IR) imaging is useful in many applications such as military applications [16; 17], industrial applications [18; 19], and medical applications [20; 21]. Thermal infrared cameras can be used to detect the infrared radiation emitted by the objects [22]. Thermal infrared cameras work based on the law of black body radiation. The thermal cameras detect the infrared radiation emitted by the objects and convert the observed radiation into an electronic signal. The infrared images captured by the IR camera show abnormalities in the temperature of the objects [23]. Due to the different characteristics of infrared images compared to visible images, conventional methods developed for visible image thresholding will tend to fail for infrared image thresholding. Image thresholding algorithms need to be developed in accordance with infrared image characteristics. We analyzed the characteristics of the different infrared images via its one-dimensional grayscale distribution properties and using the grayscale distribution properties we have developed a mathematical model to extract an optimal threshold value for infrared image thresholding. Various approaches have been proposed for the thresholding of infrared images [5; 6; 7]. For instance, [PERSON] et al. [24] proposed a method based on the concept of approximating the grayscale histogram. The purpose of the image thresholding in this method is to detect the pedestrian in the infrared images. In this method, the histogram is approximated to Gaussian distribution and the mean and variance values are calculated from the Gaussian function. The threshold values for the infrared image segmentation are defined based on the mean and variance values. This method works well only if the mean of the Gaussian function lies near the gray level 128. This method gives errors in the image thresholding when the mean of the Gaussian distribution is closed to 0 or 255 for eight-bit grayscale images. In addition, the procedure for selecting a value for the variance was not explained in this method. Many iterative-based thresholding algorithms were proposed over the years for infrared image segmentation. Defining a threshold-dependent criterion function based on the information and properties of the grayscale histogram is the common idea in these methods. For instance, [PERSON] et al. [25] proposed an image thresholding method based on the maximum entropy and the bee algorithm. In this method, the criterion function is defined based on the Shannon entropy concept and the bee algorithm is applied as the iterative model in order to select the optimal threshold value for image segmentation. This method was tested on a limited number of infrared images. Another iterative method was proposed by [PERSON] et al. [26] for infrared image thresholding. In this method, the image is converted into an eight-bit grayscale image to obtain the cloud models related to the object and background of the image. The hyper entropy values obtained from the cloud model are used to extract an optimal threshold for infrared image thresholding. This method is facing the over-segmentation problem in case of some infrared images.