## 4 Conclusions  
This study presents a novel framework for the deformable registration of 3D point clouds representing brain structures, leveraging Conformal Bayesian Optimization (CBO) to enhance alignment precision and maintain anatomical plausibility. The proposed method uses probabilistic modeling to optimize non-rigid transformations, addressing critical challenges in aligning brain structures with significant anatomical variability. Through the combination of the Hausdorff distance and regularization terms, the framework ensures smooth deformations that respect anatomical constraints, reducing the risk of implausible transformations. The experiments conducted on both the Tosca and Brain Asphyxia databases demonstrate the robustness and versatility of this approach. Results from the Tosca database highlight the model's ability to handle substantial non-rigid deformations, achieving accurate alignments even with significant pose variations. Furthermore, analyzing brain structures from the Brain Asphyxia database underscores the importance of adaptive optimization strategies. The use of CBO enabled the framework to accommodate the anatomical complexities presented by pathological conditions such as perinatal asphyxia, ensuring that transformations remained clinically relevant. Comparative evaluations of different regularization methods revealed the necessity of case-specific configurations to optimize performance. While standard regularization techniques proved sufficient for aligning healthy brain structures, more advanced methods, such as Bending Energy and Total Variation, were critical for managing the more complex deformations observed in asphyxia cases. The modular nature of the proposed framework allows it to adjust to varying anatomical conditions dynamically, enhancing its applicability across diverse clinical scenarios. In addition, integrating CBO into the deformable registration process offers several advantages, including improved alignment accuracy, reduced computational complexity, and better control over deformation smoothness. This framework addresses the key limitations of existing methods by providing an interpretable and adaptable approach that balances alignment precision with anatomical plausibility. The findings of this study demonstrate the potential of CBO-enhanced registration models to support clinical decision-making, particularly in scenarios that require the tracking of anatomical changes over time or across patient populations. Future work will explore the extension of this approach to other imaging modalities and further refine the optimization framework to better address cross-subject anatomical variability. In future work, we plan to integrate elastic transformations controlled by Bayesian latent variable frameworks into our model to enhance the reliability and robustness of the registration process. Bayesian latent variables will enable probabilistic representations of  
Figure 8: Cost of registration processes when the model aligns the point clouds of healthy patients and those of patients with abnormalities in their brain structures caused by perinatal asphyxia. deformation fields, better capturing global and local anatomical variability. Elastic transformations will offer greater adaptability in handling complex deformations, especially in pathological cases, while maintaining anatomical plausibility. We also aim to optimize the computational cost of these models, ensuring efficient training and uncertainty management. This approach will support more reliable alignments across diverse imaging modalities, promoting better clinical outcomes. [PERSON] and [PERSON] conceptualized the methodology, developed the machine learning methods, and prepared the original draft of the manuscript. [PERSON] was responsible for data curation, investigation, and validation and contributed to the methodology. [PERSON] and [PERSON] curated the data and contributed to the original draft's preparation.