## 3 Theory and Methodology
### The Proposed Tsallis-Entropy-Based Method
#### 3.2.2 The Proposed Cauchy-TB Method  
The classical Tsallis entropy is defined under the assumption that the intensity distributions of the object and background classes are independent of each other. For each intensity i, after being classified as object or background, it is represented by a single probability in its corresponding class distribution. This probability is then used in the calculation of Tsallis entropy, as illustrated in Equations (8)-(10). In classical information theory, entropy measures the level of uncertainty in a system [59]. However, Figure 6a illustrates that the intensity distributions of objects and the background often overlap. This means that pixels classified as part of the background may actually belong to objects, and vice versa. Essentially, the intensity distributions of objects and backgrounds influence each other. As a result, independently calculating the probabilities of pixels belonging to either the object or background, as shown in Equations (8) and (9), is suboptimal. A better approach is to compute these probabilities using the combined intensity distributions over the entire intensity space. Considering thatthe intensity distributions of background and objects in medical images often contain many outliers, fitting the intensity distribution with a Gaussian distribution may not be suitable. Therefore, we propose replacing the Gaussian distribution with a long-tailed distribution, for example, the Cauchy distribution, as shown in Figure 6b. A long-tailed distribution in statistics is typically characterized by a highly imbalanced distribution, where the majority of observations are concentrated in the front (i.e., the short tail) of the distribution, while the tail part (the long tail) contains relatively few but highly significant observations. The tail of a long-tailed distribution decays more gradually than that of the Gaussian distribution. A defining feature of any long-tailed distribution is that its tail is significantly longer than that of a Gaussian distribution, and the total volume of the tail data may be quite substantial. As a result, outliers in the tail of the distribution can have a significant impact on the overall distribution. This characteristic allows a long-tailed distribution to better accommodate outliers, providing a more robust fit to the intensity of both objects and background. Long-tailed distributions have been widely utilized in previous studies, with one notable application being the t-SNE algorithm proposed by [PERSON] (2016). t-SNE employs Student's t-distribution, which is referred to as the Cauchy distribution when having 1 degree of freedom, to measure the similarity between data points, as opposed to the traditional Gaussian distribution. This choice enables t-SNE to effectively handle rare points or outliers in a long-tailed distribution, thereby preserving the clustering structure of the dataset while appropriately distributing the rare points in the tail. t-SNE performs particularly well in datasets containing a large number of common patterns and a few rare events, making it highly suitable for handling data with long-tailed distributions. Based on the above observation, we propose a new method that combines Tsallis entropy, Bayesian prior probability, and the long-tailed Cauchy distribution; this proposed method is called Cauchy-Tsallis Bayesian Estimation (Cauchy-TB). Let \(p_{o,t}(i)\) and \(p_{b,t}(i)\) describe the probability density functions of pixels classified as part of objects or the background, with intensity i at threshold t, as given by  
\[p_{o,t}(i) =P[\mathcal{F}(s)=i|s\in S_{o,t}] \tag{11}\] \[p_{b,t}(i) =P[\mathcal{F}(s)=i|s\in S_{b,t}]\] (12) \[S =S_{o,t}\cup S_{b,t} \tag{13}\]  
Figure 6: (**a**) The histogram shows two alternating Gaussian distribution curves, with a vertical line indicating a partition at a threshold of t = 50. (**b**) In the tail (orange) of the distribution, the red points represent outliers, which are typically not handled well by a Gaussian distribution (green line).