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

arXiv: https://arxiv.org/abs/2602.01589 · HTML: https://mlsanity.com/paper/2602.01589/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 41 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 12 of 12)

- [Spherical U-Net on Cortical Surfaces: Methods and Applications](https://mlsanity.com/paper/1904.00906/) (2019)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [TEMPO: Feature-Endowed Teichm"uller Extremal Mappings of Point Clouds](https://mlsanity.com/paper/1511.06624/) (2015)
- [Cortical surface registration using unsupervised learning](https://mlsanity.com/paper/2004.04617/) (2020)
- [Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing](https://mlsanity.com/paper/1508.07569/) (2015)
- [Spherical Density-Equalizing Map for Genus-0 Closed Surfaces](https://mlsanity.com/paper/2401.11795/) (2024)
- [A Deep-Discrete Learning Framework for Spherical Surface Registration](https://mlsanity.com/paper/2203.12999/) (2022)
- [Automatic Landmark Detection and Registration of Brain Cortical Surfaces via Quasi-Conformal Geometry and Convolutional Neural Networks](https://mlsanity.com/paper/2208.07010/) (2022)
- [Computing quasiconformal folds](https://mlsanity.com/paper/1804.03936/) (2018)
- [SUGAR: Spherical Ultrafast Graph Attention Framework for Cortical Surface Registration](https://mlsanity.com/paper/2307.00511/) (2023)
- [A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints](https://mlsanity.com/paper/2308.02393/) (2023)
- [Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization](https://mlsanity.com/paper/2511.11679/) (2025)

## Abstract (arXiv)

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|∞<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 mathbbS2, 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.