## 3 Results
### Brain Asphyxia Database  
In the case of _Brain Asphyxia_, two sets of experiments were conducted: one focused on registering the point clouds of brain structures from healthy patients and another on registering point clouds from patients diagnosed with perinatal asphyxia. In the case of healthy patients, the brain structures, while exhibiting non-rigid changes, tend to maintain volumetric consistency across time points. This consistency means that although slight variations in contours may exist, critical features such as size and shape are preserved, as illustrated in Figure 5. These deformations are typically mild, which makes the registration task less challenging. One potential reason for this uniformity is that the point clouds were extracted using probabilistic atlases, which ensure a high level of alignment. On the other hand, brain structures from patients with perinatal asphyxia exhibit a significant degree of variability in both shape and size. This is due to the severe anatomical changes caused by the condition. This increased variability leads to a more complex and challenging registration task, where the model struggles to converge, resulting in a higher cost function, as shown in Figure 6. A comparative analysis of different regularization terms--such as L2 Norm, Total Variation, Bending Energy, and Smoothness Constraints--reveals their distinct impacts on deformable registration performance. For healthy patients, where deformations are relatively small, the L2 Norm effectively maintains the overall smoothness and alignment accuracy of the registration. It imposes a penalty on large deformations, ensuring that the anatomical consistency of brain structures is preserved. For patients with perinatal asphyxia, where significant deformations are present, the use of more advanced regularization terms like Total Variation and Bending Energy becomes crucial. The Bending Energy regularization helps to maintain the smoothness of large deformations, ensuring anatomically plausible transformations by minimizing sharp or unnatural changes. Total Variation, on the other hand, balances smoothness with the ability to preserve sharp edges or discontinuities in the deformation, which can be critical in pathological cases where certain brain regions undergo more pronounced changes. Figure 7 provides a comparative analysis of the loss function \(\mathcal{L}\) across four different regularization methods--L2 norm, Total Variation (TV), Bending Energy (BE), and Smoothness Constraints (SCs). The comparison is presented for two patient groups: healthy individuals and patients diagnosed with perinatal asphyxia. This analysis highlights how each regularization method influences the deformation field in brain structure registration, emphasizing the variability between patient groups. The Smoothness Constraints regularization assists in controlling local deformations, ensuring that small neighborhoods within the brain's structure undergo consistent transformations without introducing unrealistic distortions. However, its effectiveness diminishes when the variability in the shape and size of the structures is extreme, as seen in perinatal asphyxia cases. This variability explains the more significant performance variance observed for asphyxia patients when using SCs, as illustrated in the figure. These findings further underscore the adaptability of different regularization terms to varying levels of deformation complexity. Standard regularization methods, such as the L2 Norm and Smoothness Constraints, perform well for healthy patients, ensuring smooth transformations and accurate alignment. However, for patients with perinatal asphyxia, more advanced methods like Total Variation and Bending Energy become essential due to their ability to handle complex anatomical alterations. This adaptability highlights the need for case-specific approaches in medical imaging to ensure accurate and anatomically plausible brain structure registrations across diverse clinical scenarios.