# Positron Emission Tomography Image Segmentation Based on [PERSON]'s Intuitionistic Fuzzy Sets
## 3 PET Image Segmentation with Iterative Thresholding Using A-IFSs  
In this section, we present a general A-IFSs-based multi-level image thresholding that, besides its low computational cost, autonomously determines the thresholds based on the image pixels' gray-levels homogeneity. The method selects iteratively, using the images characteristic, the best threshold value to identify the damaged tissues in PET images. In order to compute the threshold, the proposed algorithm is applied an unspecified number of times to the image using a divide and conquer strategy. First, the algorithm is applied to the original image \(Q\) with its pixels' gray-levels \(\in[0,L-1]\), determining the threshold value \(t_{i}\) corresponding to the smallest entropy of the image \(Q\). This threshold value is then used to create two sub-images: the sub-image with intensities lower than \(t_{i}\) with its pixels' gray-levels \(\in[0,t_{i}-1]\), and the sub-image with intensities greater than \(t_{i}\) with its pixels' gray-levels \(\in[t_{i}+1,L-1]\). Finally, the algorithm is applied to the sub-image, which has a higher amplitude between gray-level entropy values. Each sub-image processed is marked and cannot be processed again. In Figure 1 we represent the computational process resulting from the application of the algorithm to a given image, employing a binary tree where each tree node contains the description of the images' gray-scale and the threshold value obtained through the application of the algorithm to that image. The proposed algorithm consecutively divides the resulting sub-images by means of the threshold value \(t_{i}\) obtained through the application of the general algorithm to each one of them. To enhance the algorithm with the capability to self-stop the process of determining new thresholds and consequently sub-dividing the images, we use the region's homogeneity introduced by the following algorithm. A New Approach to Perform Tumor Delineation in PET Image with Iterative Thresholding Using A-IFSs  
Tumor delineation in PET images is a crucial step because the boundary has to ensure the inclusion of all of the damaged cells, but if the boundary is too large, it will endanger healthy tissues during future treatments [11]. A-IFSs find the best threshold value to split the object from the background, and therefore to refine the threshold to match the small tumor region, we apply the A-IFSs to a selected sub-image until we obtain the pretended pixel gray-scale intensities' homogeneity. Figure 1: Computational process. In our iterative threshold algorithm, we use the homogeneity (\(H\)) value of the sub-image to determine when the algorithm stops searching for another threshold. We define \(H\) as:  
\[H=\sum_{i,j}\frac{p(i,j)}{1+|i-j|} \tag{6}\]  
We also define the entropy amplitude \(C\) with a value that will determine which sub-image will be processed in the next iteration. To determine the value of \(C\) we need to sub-divide the image \(Q_{i}\) into two sub-images \(Q_{i+1}\) with gray-levels \(q(x,y)\in[Low_{i},t_{i}-1]\) and \(Q_{i+2}\) with gray-levels \(q(x,y)\in[t_{i}+1,High_{i}]\) and check if the image \(Q_{i}\) was processed to prevent reprocessing the same image again. Then, for all of the sub-images \(Q_{i+1}\) and \(Q_{i+2}\) the value \(C\) is calculated as:  
\[C_{Q_{i}}=MAX_{Q_{i}}-MIN_{Q_{i}} \tag{7}\]  
with  
\[MAX_{Q_{i}}=max\{\varepsilon_{T}(q)\},\forall q(x,y)\in[Low_{i},High_{i}], \tag{8}\]  
and  
\[MIN_{Q_{i}}=min\{\varepsilon_{T}(q)\},\forall q(x,y)\in[Low_{i},High_{i}], \tag{9}\]  
The sub-image with the highest \(C_{Q_{i}}\) is selected for forward processing. The algorithm stops when \(H\) achieves the desired value. For tumor delineation in PET images, the algorithm stops when \(H\) is greater than or equal to 0.999. This value was obtained experimentally. We tested several numbers of iterations and from \(H\geq\) 0.999 there were no significant improvements in the sub-image homogeneity (Figure 2).