Popular clustering methods for segmentation include K-Means and Fuzzy C-Means, as noted in [5; 6]. Graph-based methods [7; 8] leverage graph theory for segmentation, while watershed segmentation [9] separates overlapping objects based on pixel similarities. Classical machine learning methods such as PCA [10], LDA [11], LBP [12], HOG [13], GLCM [14], and frequency domain analysis [15] are employed to extract shape, texture, and structural features from images. These features are utilized with classifiers such as SVM [16] and K-means [14] in tasks such as classification or clustering. Deep learning methods, especially CNNs [17] such as U-Net [18], are highly effective in medical image segmentation. U-Net uses data augmentation to expand datasets and improve the model's generalization capability. However, CNNs encounter difficulties in capturing global dependencies. To address this, researchers apply Transformer architectures [19], known for their self-attention mechanism, to various image processing tasks. Recently, SAM [20], a universal model for image segmentation, has shown promising results spanning various image types. Due to significant grayscale differences among various biological tissues and organs in medical images, thresholding has therefore become a popular technique for object segmentation that balances efficiency and accuracy. The key difficulty is to find the optimal threshold that distinguishes the foreground from the background in the image. Entropy measures the amount of information in images. The higher the entropy is, the more information the image has. Essentially, the goal of the maximum entropy algorithm is to find an optimal threshold that maximizes the sum of entropy between the background and foreground regions. [PERSON]'s maximum entropy method [21] can achieve good results even when the image histogram is not bimodal. However, similar to the Otsu method [22], it does not consider spatial information and is susceptible to noises. [PERSON] [23] developed a class-uncertainty-based framework called minimum homogeneity- and uncertainty-based energy (MHUE) for optimum threshold selection. [PERSON] [24] further improved the MHUE algorithm by integrating the histogram-based feature with the spatial contextual feature, which demonstrates excellent performance in medical image segmentation. In 2004, [PERSON] [25] first proposed a method based on Tsallis entropy. Subsequently, [PERSON] [26] employed 2D Tsallis entropy for a multi-thresholding algorithm. More recently, methods such as Masi entropy [27] have been continuously introduced. Entropy-based thresholding methods are widely used due to their simplicity and applicability in various scenarios. However, they face challenges when dealing with pixel intensity distributions that contain overlapping components and outliers, as illustrated in Figure 1. This study introduces a novel method, called Cauchy-TB, to address these challenges. By integrating Bayesian prior probability estimation and the Cauchy distribution into the classical Tsallis entropy framework (Figure 2), we aim at improving the robustness and accuracy of segmentation results. Experimental results demonstrate the significant advantages of Cauchy-TB on two publicly available medical brain image datasets. The contributions of this paper to the literature can be summarized as follows:  
1. We introduce a novel Tsallis-entropy-based thresholding method for medical image segmentation by incorporating Bayesian prior probability estimation to effectively handle overlaps in intensity distributions between two classes;
2. We integrate the Cauchy distribution, recognized for its heavy-tailed properties, to improve the framework's robustness to outliers;
3. We further develop this approach for multi-class segmentation and determine the optimal threshold by optimizing the information measure function. The structure of this paper is outlined as follows: Section 2 provides a review of related work, and Section 3 explains the principles of our method in detail.