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

Zhehao Xu, Lok Ming Lui

TL;DR

This work tackles free-boundary diffeomorphic surface mapping by leveraging Least Squares Quasiconformal (LSQC) energy as a boundary-free, bijectivity-guaranteed foundation. It introduces the Spectral Beltrami Network (SBN) as a differentiable surrogate for LSQC, with a multiscale mesh-spectral architecture that enables gradient-based optimization over Beltrami coefficients and two pinned points via the SBN-Opt framework. The paper proves LSQC properties (existence, uniqueness, similarity-invariance, resolution-independence) and demonstrates through density-equalizing and inconsistent surface registration experiments that SBN-Opt achieves superior distortion control and mapping accuracy compared to conventional numerical solvers. The approach combines theoretical rigor with neural surrogacy to produce explicit, tunable control over boundary geometry and local distortion, offering a scalable tool for surface parameterization, medical imaging, and geometry processing.

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.

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

TL;DR

This work tackles free-boundary diffeomorphic surface mapping by leveraging Least Squares Quasiconformal (LSQC) energy as a boundary-free, bijectivity-guaranteed foundation. It introduces the Spectral Beltrami Network (SBN) as a differentiable surrogate for LSQC, with a multiscale mesh-spectral architecture that enables gradient-based optimization over Beltrami coefficients and two pinned points via the SBN-Opt framework. The paper proves LSQC properties (existence, uniqueness, similarity-invariance, resolution-independence) and demonstrates through density-equalizing and inconsistent surface registration experiments that SBN-Opt achieves superior distortion control and mapping accuracy compared to conventional numerical solvers. The approach combines theoretical rigor with neural surrogacy to produce explicit, tunable control over boundary geometry and local distortion, offering a scalable tool for surface parameterization, medical imaging, and geometry processing.

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.

Paper Structure

This paper contains 23 sections, 6 theorems, 44 equations, 21 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Previous works
  3. Diffeomorphism Optimization Problem
  4. Computational Quasiconformal Geometry
  5. Contributions
  6. Mathematical Background
  7. Quasiconformal Mappings
  8. Least Squares Quasiconformal Energy
  9. Properties of Least Squares Quasiconformal Energy
  10. Spectral Beltrami Network
  11. Encoding input information
  12. Message Passing Mechanism
  13. Mesh Spectral Layer
  14. Model architecture and training
  15. SBN-Opt: A novel optimization framework of the free-boundary diffeomorphism problem
  16. ...and 8 more sections

Key Result

theorem 1

(Measurable Riemann Mapping Theorem) For any function $\mu : U \to \mathbb{C}$ on with bounded essential supremum norm $\|\mu\|_\infty < 1$, there is a quasiconformal map $\phi$ on $\overline{U}$ satisfying the Beltrami equation $\phi_{\overline{z}} = \mu \phi_z$ for almost all $z \in U$. Moreover,

Figures (21)

  • Figure 1: Illustration of the embedding update rule and the hierarchical edge construction.
  • Figure 2: Architecture of Message Passing Block.
  • Figure 3: Architecture of Mesh Spectral Layer
  • Figure 4: Architecture of Spectral Beltrami Network
  • Figure 5: Comparison of SBN, M-MGN, and MSN for typical cases (max BC norm < 0.4). Top row: numerical solution field (left), BC magnitude (center), log gradient magnitude of BCs (right). Bottom row: color-coded nodewise L2 errors for SBN, M-MGN, and MSN vs. reference solution. Format applies to Figs.\ref{['fig:large_mu']} and \ref{['fig:large_distortion']}.
  • ...and 16 more figures

Theorems & Definitions (13)

  • theorem 1
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary
  • proof
  • corollary thmcountercorollary
  • proof
  • proposition thmcounterproposition
  • proof
  • ...and 3 more