## 3 Results
### Tosca Database
This capability is particularly useful in medical imaging scenarios where anatomical structures exhibit significant but plausible variations over time or due to patient movement. However, as shown in Figure 3(b), this method encounters challenges when applied to point clouds with fundamentally different anatomies, resulting in anatomically implausible registrations. This highlights the critical importance of incorporating anatomical constraints into the registration process. This method's ability to preserve structural consistency in plausible cases underscores its potential for applications in clin  
Figure 2: An example of the deformable registration process in the Tosca database. The figure represents the initial correspondence between the fixed point cloud (red) and the moving point cloud (blue). The plot on the right illustrates the matching points between the two clouds, with red points indicating the fixed positions and blue points indicating the corresponding positions in the moving point cloud. ical settings, particularly when tracking changes in brain structures over time or within a population. The results from Figure 3 provide a quantitative analysis of the registration loss, comparing alignments between point clouds from the same subject in different poses versus point clouds from different subjects. The lower loss values observed in the same group indicate that the deformable registration framework, enhanced by Bayesian optimization, performs well in cases where anatomical consistency is preserved, as demonstrated in Figure 2. The use of the Hausdorff distance combined with regularization effectively captures the quality of the alignment, showing minimal deformation and high anatomical plausibility when the same subject is being registered across varying poses. In contrast, the higher loss values in the different-shapes group, as seen in Figure 4b, reflect the model's  
Figure 4: An example of a deformable registration process in the Tosca database. The figure shows a representation of the registration process in point clouds from figures that have the same anatomy and different poses and from figures with different anatomies. Red and blue correspond to fixed and moving respectively. Figure 3: Comparison of the registration process costs: aligning point clouds from the same subject in different poses versus aligning point clouds from different subjects. limitations when aligning point clouds with fundamentally different anatomies. These higher loss values indicate that the method struggles to maintain anatomical fidelity when confronted with subjects that deviate significantly in shape and structure, leading to implausible deformations. The results further emphasize the importance of incorporating anatomical constraints and highlight that while the method excels in non-rigid but plausible scenarios, it requires further refinement to handle more complex cases involving diverse anatomies. Compared to well-known approaches in deformable registration, our framework demonstrates distinct advantages in handling anatomical variability. While methods like VoxelMorph [13] leverage convolutional neural networks (CNNs) for efficient and unsupervised image registration, their reliance on pixel-based transformations often leads to limitations in preserving anatomical structure, particularly in cases with high variability or intricate anatomical details. Large Deformation Diffeomorphic Metric Mapping (LDDMM) [11], a widely used technique in medical imaging due to its diffeomorphic properties, excels at handling complex deformations but is computationally intensive, making it less practical for clinical scenarios that require rapid processing. DeepReg [24] integrates deep learning with regularization to improve anatomical accuracy, but its performance can be sensitive to initial conditions, and it may converge to suboptimal solutions when faced with significant shape variations, such as those encountered in neuroanatomical structures affected by perinatal asphyxia.