A neural optimization framework for free-boundary diffeomorphic mapping problems and its applications

2025-11-11

AI Summary

Free-boundary diffeomorphic mapping faces optimization challenges in ensuring local bijectivity under unconstrained boundaries and large deformations. This paper proposes the Spectral Beltrami Network (SBN), the first framework to embed numerical conformal theory into a differentiable neural optimization pipeline. Grounded on the Least-Squares Quasi-Conformal (LSQC) energy, SBN constructs a neural surrogate model leveraging graph Fourier transforms and multi-scale spectral mesh networks, enabling end-to-end gradient-based optimization without manual annotations while explicitly controlling geometric distortion. SBN achieves unconstrained-boundary mapping with guaranteed local bijectivity and supports differentiable rendering and implicit geometric representation. In tasks of density-equalizing mapping and non-uniform surface registration, SBN significantly outperforms conventional methods—yielding higher accuracy and substantially reduced metric distortion.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Free-boundary diffeomorphism optimization is a core ingredient in the surface mapping problem but remains notoriously difficult because the boundary is unconstrained and local bijectivity must be preserved under large deformation. Numerical Least-Squares Quasiconformal (LSQC) theory, with its provable existence, uniqueness, similarity-invariance and resolution-independence, offers an elegant mathematical remedy. However, the conventional numerical algorithm requires landmark conditioning, and cannot be applied into gradient-based optimization. We propose a neural surrogate, the Spectral Beltrami Network (SBN), that embeds LSQC energy into a multiscale mesh-spectral architecture. Next, we propose the SBN guided optimization framework SBN-Opt which optimizes free-boundary diffeomorphism for the problem, with local geometric distortion explicitly controllable. Extensive experiments on density-equalizing maps and inconsistent surface registration demonstrate our SBN-Opt's superiority over traditional numerical algorithms.

Problem

• Optimizing free-boundary diffeomorphic mappings with unconstrained boundaries and local bijectivity
• Developing neural surrogate to overcome limitations of conventional numerical LSQC algorithms
• Controlling local geometric distortion in surface mapping applications like density-equalization

Innovation

• Neural surrogate embedding LSQC energy into mesh-spectral architecture
• SBN-Opt framework optimizing free-boundary diffeomorphism with controllable distortion
• Multiscale spectral architecture enabling gradient-based optimization of mappings