## 3 Results  
### Tosca Database  
To validate the deformable registration model, the performance of the model was mainly analyzed using synthetic databases, such as the Tosca database. In this context, a series of experiments were initiated that allowed us to identify how the cost function weights the deformations applied to the point clouds. Figure 2 shows an example of the initial correspondence process applied to the point clouds to calculate the deformation \(\mathcal{T}\). The plots on the right side of each subfigure illustrate the initial correspondences between the fixed (red) and moving (blue) point clouds. Close alignment between the red and blue points indicates successful registration, while more significant gaps suggest areas with alignment discrepancies. In our first experiment, we applied the proposed deformable registration method to the Tosca dataset, which consists of non-rigid 3D models of various subjects, such as horses and cats. Figure 2 illustrates the registration process between a fixed point cloud (red) and a moving point cloud (blue). The right side of each subfigure presents the point correspondences established during the initial registration phase. Figure 2a shows the deformable registration of a horse model, where both the fixed and moving point clouds exhibit significant variations in pose. The corresponding points identified between the clouds (shown on the right) highlight the complexity of aligning the limbs and head, which exhibit distinct positional changes. However, the method's robustness is evident as it can initialize correspondences even when significant structural differences exist between poses. Figure 2b presents a similar process for a cat model, which also underwent non-rigid deformations. The initial correspondences are visualized, indicating a  
Figure 1: Proposed approach for the deformable registration of point clouds. In this context, the proposed registration approach based on Bayesian optimization allows for optimizing the number of initial correspondences in the registration process and uses a cost function capable of quantifying the degree of deformation applied. robust alignment of the major body components, such as the torso and limbs. The method's versatility is demonstrated as it effectively handles variations in pose, with the established correspondences maintaining consistency despite local deformations. The results show that even in highly non-rigid cases, such as those seen in the Tosca dataset, our method successfully aligns the point clouds in an anatomically plausible manner, outperforming conventional rigid alignment techniques. To validate the plausibility of this approach in the Tosca database, we analyzed the behavior of the deformable model for point clouds with variations in pose (Figure 1(a)) and for point clouds representing different quadrupeds (Figure 1(b)). Simultaneously, we examined the cost function in Figure 3. Here, \"lower cost\" refers to the value of the cost function \(\mathcal{L}(\mathcal{P},\mathcal{Q},\mathcal{T})\), which combines geometric distance with a regularization term to ensure smooth and anatomically plausible deformations. A lower cost indicates a better alignment with minimal deformation, achieving a more accurate and natural registration. For both experiments, we observed that point clouds from the same subjects generally result in a lower cost, reflecting more effective alignment than those from different subjects. The results shown in Figure 3(a),b illustrate the effectiveness and limitations of our deformable registration approach that uses Bayesian optimization. By using a cost function that combines the Hausdorff distance with a regularization term, we achieve a quantifiable measure of alignment quality, where lower values indicate more accurate alignment and anatomically plausible deformations. Figure 3(a) shows that the method successfully aligns point clouds with significant pose variations, demonstrating its robustness in handling non-rigid deformations.