## 2 Materials and Methods
### Cell Recognition through Keypoints  
Each stage described above offers a detailed insight into the process by which the SIFT algorithm identifies keypoints and generates precise descriptors of the data. In the present work, the analysis initially focuses on koilocytes, which are typically observed in low-grade lesions. This analysis is illustrated in Figure 6, which exemplifies the cell type in question. This approach not only provides a better understanding of the capabilities of keypoint generation in the context of diagnostic cytology but also explores its applicability in identifying and characterizing cells associated with specific pathological conditions. By using SIFT, a robust and detailed representation of cellular morphological features is achieved, which is crucial for improving the accuracy in the diagnosis and monitoring of diseases. Additionally, the algorithm's invariance to geometric transformations and changes in illumination ensures that the generated descriptors are consistent and reliable, significantly contributing to the robustness and reliability of advanced cytological studies. Once the SIFT algorithm was applied to the image under study, a set of keypoints was identified, reaching a total of 20 in this specific case. Consequently, a set of descriptors was generated, each corresponding to one of the identified keypoints. For this particular example, a set consisting of 20 descriptors was obtained, each characterized by a 128-element vector, as illustrated in Figure 7. This process ensures a detailed and robust representation of the morphological characteristics of the cells, facilitating their analysis and classification in advanced diagnostic contexts. By analyzing the koilocyte, its keypoints are identified, and the corresponding descriptor is generated. This process is fundamental for establishing correspondences later within cervical cytology. Despite variations in image size, the methodology employed ensures the precise detection of the example koilocyte through meticulous keypoint matching. This technique proves effective in identifying the koilocyte at different scales and proportions, demonstrating the method's remarkable robustness in detecting and matching specific features. This approach not only enhances the precision of cytological analysis but also strengthens the applicability of the descriptor for identifying and classifying cells under various pathological conditions, as seen in Figure 8.  
In the framework of this work, the following methodological process is proposed: Starting from a comprehensive set of cells exhibiting traits associated with precursor lesions or potentially cancerous cells, a unique feature descriptor is generated for the set. This step involves a detailed description and analysis of the characteristics present in the entire set of cells to establish a representative descriptor. Figure 6: The image shows a specific segment of a koilocyte, cropped by detecting blobs. This segment was processed using the SIFT algorithm to generate keypoints. Figure 7: The image illustrates the 128-element descriptor corresponding to the 23 keypoints generated from Figure 6.  
Subsequently, a search procedure aims to identify a specific descriptor, denoted as \(d_{1}\), within a set of descriptors, referred to as \(D_{2}\). To facilitate this identification and comparison process, the Euclidean distance is used as a metric to evaluate the distance between descriptors. Thus, considering \(d_{1}\) as the reference descriptor belonging to the set \(D_{1}\), and \(D_{2}\) as the target descriptor set, significant matches are sought. Here, \(n\) represents the dimension of the feature space. The goal is to calculate the Euclidean distance between \(d_{1}\) and each of the descriptors in \(D_{2}\). The algorithm focuses on identifying the \(k\)-nearest neighbors, selecting the \(k\) descriptors within \(D_{2}\) that have the smallest Euclidean distance from \(d_{1}\).