Genus-0 Surface Parameterization using Spherical Beltrami Differentials
This work addresses a domain-specific problem in geometry processing and imaging science for tasks such as brain mapping, but it is incremental as it builds on existing methods like the Spectral Beltrami Network.
The paper tackled the problem of spherical surface parameterization for genus-0 surfaces, which involves balancing task objectives, bijectivity, and distortion control, by introducing the Spherical Beltrami Differential and a neural optimization framework called BOOST, resulting in improved task fidelity with controlled distortion and robust bijective behavior in applications like landmark matching and brain cortical surface registration.
Spherical surface parameterization is a fundamental tool in geometry processing and imaging science. For a genus-0 closed surface, many efficient algorithms can map the surface to the sphere; consequently, a broad class of task-driven genus-0 mapping problems can be reduced to constructing a high-quality spherical self-map. However, existing approaches often face a trade-off between satisfying task objectives (e.g., landmark or feature alignment), maintaining bijectivity, and controlling geometric distortion. We introduce the Spherical Beltrami Differential (SBD), a two-chart representation of quasiconformal self-maps of the sphere, and establish its correspondence with spherical homeomorphisms up to conformal automorphisms. Building on the Spectral Beltrami Network (SBN), we propose a neural optimization framework BOOST that optimizes two Beltrami fields on hemispherical stereographic charts and enforces global consistency through explicit seam-aware constraints. Experiments on large-deformation landmark matching and intensity-based spherical registration demonstrate the effectiveness of our proposed framework. We further apply the method to brain cortical surface registration, aligning sulcal landmarks and jointly matching cortical sulci depth maps, showing improved task fidelity with controlled distortion and robust bijective behavior.