## 3 Results
### Brain Asphyxia Database
Figures 5 and 6 show the deformable registration of point clouds representing brain structures from the Brain Asphyxia database. Figure 5 shows the registration process in healthy patients, focusing on two brain structures: the left white matter and the left thalamus. These examples highlight the algorithm's ability to align brain structures with minimal deformation, ensuring smooth transformations and accurate alignment. In contrast, Figure 6 presents the difference in registration between the brain structures of healthy patients (red) and those affected by perinatal asphyxia (blue). The comparison reveals a higher degree of structural variability in the asphyxia group, requiring more complex deformation fields to achieve alignment. Despite this complexity, the framework successfully aligns the brain structures, showcasing the robustness of this approach. The benefit of incorporating probabilistic optimization through Conformal Bayesian Optimization (CBO)  
Figure 5: Example of the deformable registration of point clouds belonging to brain structures from the _Brain Asphyxia_ database. This case shows the registration process in patients with healthy brain structures. Figure 6: Example of deformable registration of point clouds belonging to brain structures from the _Brain Asphyxia_ database. This case shows the difference in the registration process between healthy patients’ brain structures (red) and those affected by perinatal asphyxia (blue). is evident in these results. CBO enables the framework to iteratively learn plausible shape deformations, balancing local flexibility with global alignment. The probabilistic nature of CBO ensures that transformations adhere to anatomical constraints, reducing the risk of producing unrealistic deformations. This is particularly valuable in cases with significant anatomical variability, such as perinatal asphyxia, where traditional optimization techniques might struggle to maintain anatomical plausibility. The framework's modular design allows it to adapt to the specific anatomical characteristics of each patient group. In healthy patients, simpler deformations suffice, while in the asphyxia group, the optimization dynamically adjusts the deformation parameters to account for more substantial changes in brain structures. These results demonstrate the effectiveness of the proposed approach in addressing varying levels of complexity, ensuring both accuracy and clinical relevance in brain structure registration. Figure 8 compares the registration costs of aligning brain structures from healthy patients and those affected by perinatal asphyxia. The cost metric combines the Hausdorff distance and regularization terms, reflecting alignment precision and deformation smoothness. As depicted, the registration cost is consistently higher for the asphyxia group, indicating the increased complexity associated with aligning brain structures that exhibit significant anatomical variability. The elevated costs observed in the asphyxia group reflect the more significant structural deformations required to achieve alignment. Brain structures affected by perinatal asphyxia often exhibit altered shapes and sizes, making the registration task more challenging. In contrast, healthy patients display more consistent anatomical features across different scans, lowering registration costs. This contrast highlights the importance of advanced optimization methods, such as Conformal Bayesian Optimization (CBO), in adapting deformation models to varying anatomical conditions. The results further demonstrate the benefit of incorporating regularization into the loss function. While achieving accurate alignment is crucial, the regularization term ensures that the deformations remain anatomically plausible. The balance between alignment precision and regularization promotes meaningful transformations for both patient groups, minimizing the risk of overfitting or producing unrealistic deformations.