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

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 $|μ|_{\infty}<1$. 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.

💡 Research Summary

This paper introduces a novel framework for optimizing spherical self‑homeomorphisms while explicitly controlling geometric distortion and guaranteeing bijectivity. Traditional spherical mapping approaches either rely on a single chart—leading to severe pole distortion—or directly optimize vertex positions, which creates a highly non‑convex landscape and offers no intrinsic folding protection. To overcome these limitations, the authors propose a two‑chart representation of quasiconformal maps on the unit sphere, called the Spherical Beltrami Differential (SBD). Each chart (a stereographic projection of the northern or southern hemisphere) carries its own Beltrami field μ, and cross‑chart consistency constraints enforce that the μ values match along the seam. By maintaining the supremum norm ‖μ‖∞ < 1, local orientation preservation and global bijectivity are ensured.

Reconstruction of the map from μ is achieved with the Spectral Beltrami Network (SBN), a differentiable solver based on the least‑squares quasiconformal (LSQC) energy. Building on SBN, the authors develop BOOST (Beltrami Optimization On Spherical Topology), a gradient‑based optimization framework that updates the two Beltrami fields simultaneously. BOOST minimizes a composite loss consisting of (i) task‑specific objectives (e.g., landmark alignment, intensity similarity), (ii) a distortion regularizer penalizing ‖μ‖², (iii) seam‑matching and seam‑smoothness terms to guarantee a seamless transition between charts, and (iv) anti‑folding penalties that prevent triangle inversion. Because all components are differentiable, the method integrates naturally into modern deep‑learning pipelines and scales to large datasets.

Experiments cover three representative scenarios. First, large‑deformation landmark matching demonstrates significantly lower alignment error and virtually no folding compared with existing spherical parameterization techniques. Second, intensity‑driven spherical registration shows improved normalized cross‑correlation and mean‑squared error over methods such as Spherical Demons and MSM, while keeping angle and area distortion tightly bounded. Third, cortical surface registration aligns sulcal depth maps and sulcal landmarks on brain meshes; BOOST achieves comparable or superior registration accuracy to state‑of‑the‑art pipelines, with markedly reduced distortion near the poles thanks to the two‑chart design.

Overall, the work provides a principled way to optimize spherical maps in the space of distortion variables rather than vertex coordinates, thereby eliminating the numerical instability and folding issues of prior approaches. The integration of a differentiable quasiconformal solver with explicit distortion control opens new possibilities for a wide range of spherical‑based applications in medical imaging, computer vision, and geometry processing.