2025/11/12 by Zhehao Xu, Xu, Zhehao, Lok Ming Lui +1
Engineering · Mathematics · #30C62 #3D Shape Modeling and Analysis #65D18 #65K10 #68T07 #Analytic and geometric function theory #Complex Variables (math.CV) #Computer Vision and Pattern Recognition (cs.CV) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Graphics (cs.GR) #Machine Learning (cs.LG) #Piezoelectric Actuators and Control
paper · pdf · doi:10.48550/arxiv.2511.11679
openalex publication_date 2025/11/12 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/30
Free-boundary diffeomorphism optimization, an important and widely occurring task in geometric modeling, computer graphics, and biological imaging, requires simultaneously determining a planar target domain and a locally bijective map with well-controlled distortion. We formulate this task through the least-squares quasiconformal (LSQC) operator and establish key structural properties of the LSQC minimizer, including well-posedness under mild conditions, invariance under similarity transformations, and resolution-independent behavior with stability under mesh refinement. We further analyze the sensitivity of the LSQC solution with respect to the Beltrami coefficient, establishing stability and differentiability properties that enable gradient-based optimization over the space of Beltrami coefficients. To make this differentiable formulation practical at scale and to facilitate the optimization process, we introduce the Spectral Beltrami Network (SBN), a multiscale mesh-spectral surrogate that approximates the LSQC solution operator in a single differentiable forward pass. This yields SBN-Opt, an optimization framework that searches over admissible Beltrami coefficients and pinning conditions to solve free-boundary diffeomorphism objectives with explicit distortion control. Extensive experiments on equiareal parameterization and inconsistent surface registration demonstrate consistent improvements over traditional numerical algorithms.