Genus-0 Surface Parameterization using Spherical Beltrami Differentials

TL;DR

This work introduces the Spherical Beltrami Differential (SBD) to represent genus-0 sphere self-maps via two hemispherical charts and proves its correspondence with spherical quasiconformal mappings up to conformal automorphisms. Building on the Spectral Beltrami Network (SBN), it then formulates BOOST, a neural optimization framework that optimizes Beltrami fields on both hemispheres while enforcing seam-aware consistency, enabling large-deformation, bijective spherical parameterizations. The method achieves high task fidelity (e.g., landmark/feature alignment) with controlled distortion and robust bijectivity, demonstrated on cortical surface registration, extreme distortions, and hybrid intensity-landmark matching. By optimizing directly in Beltrami space and using free-boundary, seam-aware constraints, BOOST provides a flexible, mathematically principled approach for diffeomorphic spherical mapping with potential for broad application in neuroimaging and geometry processing.

Abstract

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.

Figures (10)

• Figure 1: Architecture of Spectral Beltrami Network
• Figure 1: Illustration of three issues addressed by the loss functions $\mathcal{L}_{\textbf{bm}},\mathcal{L}_{\textbf{folding}}$ and $\mathcal{L}_{\textbf{bs}}$
• Figure 1: Registration results for brain 32 to brain 35 using six sulcal landmarks. Landmarks are grouped (Group 1: CS, ITS, STS, postCS; Group 2: IFS, SFS) for visual clarity. Columns, from left to right: source/target landmark pairs (green: moving surface, blue: template), followed by results from FLASHchoi2015flash, our method with resampling, and our method with chamfer distance. Red shows deformed sulci, blue shows targets.
• Figure 2: Comparison of induced angle distortion for three methods; FLASHchoi2015flash (Left), our method with landmark resampling (Middle), and our method using chamfer distance (Right), represented by distributions of Beltrami coefficients associated with deformation.
• Figure 3: Registration of brain 8 to brain 18 with 6 sulcal landmarks(for visualization and color conventions, see Fig. \ref{['fig:small_deformation_move32_temp35']}). Note that in this figure, first row is visualization of CS, IFS, SFS and postCS, while the second row is visualization of ITS and STS. The second column are two independent results done by FLASH choi2015flash, respectively.

Theorems & Definitions (16)

• Theorem 4.1
• Definition 4.2
• Definition 4.3
• Proposition 4.4
• Proposition 4.5
• Proposition 4.6
• Proposition 4.7
• Lemma 5.1
• Proof 1
• Lemma 5.2