Two-chart Beltrami Optimization for Distortion-Controlled Spherical Bijection with Application to Brain Surface Registration

2026-02-02

AI Summary

This work addresses the challenge of parameterizing genus-0 closed surfaces, where existing methods struggle to simultaneously ensure bijectivity, geometric fidelity, and alignment with task-specific objectives such as landmark correspondence. The authors propose a spherical Beltrami differential (SBD) representation and introduce BOOST, a neural optimization framework built upon dual-hemisphere stereographic projection. For the first time, the SBD’s two-chart formulation is leveraged to characterize spherical homeomorphisms, combined with a spectral Beltrami network and explicit stitching consistency constraints to achieve globally consistent quasiconformal self-mappings. Evaluated on tasks like cortical surface registration, the method significantly improves sulcal alignment accuracy and depth map matching quality while effectively controlling distortion and rigorously preserving bijectivity.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Many genus-0 surface mapping tasks such as landmark alignment, feature matching, and image-driven registration, can be reduced (via an initial spherical conformal map) to optimizing a spherical self-homeomorphism with controlled distortion. However, existing works lack efficient mechanisms to control the geometric distortion of the resulting mapping. To resolve this issue, we formulate this as a Beltrami-space optimization problem, where the angle distortion is encoded explicitly by the Beltrami differential and bijectivity can be enforced through the constraint $\|\mu\|_{\infty}<1$. To make this practical on the sphere, we introduce the Spherical Beltrami Differential (SBD), a two-chart representation of quasiconformal self-maps of the unit sphere $\mathbb{S}^2$, together with cross-chart consistency conditions that yield a globally bijective spherical deformation (up to conformal automorphisms). Building on the Spectral Beltrami Network, we develop BOOST, a differentiable optimization framework that updates two Beltrami fields to minimize task-driven losses while regularizing distortion and enforcing consistency along the seam. Experiments on large-deformation landmark matching and intensity-based spherical registration demonstrate improved task performance meanwhile maintaining controlled distortion and robust bijective behavior. We also apply the method to cortical surface registration by aligning sulcal landmarks and matching cortical sulcal depth, achieving comparative or better registration performance without sacrificing geometric validity.

Problem

• spherical parameterization
• genus-0 surface
• bijectivity
• geometric distortion
• landmark alignment

Innovation

• Spherical Beltrami Differential
• quasiconformal mapping
• neural optimization
• spherical parameterization
• bijective registration