A neural optimization framework for free-boundary diffeomorphic mapping problems and its applications
This work addresses a core challenge in computational geometry and medical imaging for surface mapping, offering a novel neural approach to optimize diffeomorphisms with controllable distortion, though it appears incremental as it builds on existing LSQC theory.
The paper tackled the problem of free-boundary diffeomorphic mapping, which is difficult due to unconstrained boundaries and bijectivity requirements, by proposing a neural optimization framework called SBN-Opt that embeds LSQC energy into a spectral network, resulting in superior performance over traditional numerical algorithms in experiments on density-equalizing maps and surface registration.
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.