Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910017274249216 |
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| author | Xu, Zhehao Lui, Lok Ming |
| author_facet | Xu, Zhehao Lui, Lok Ming |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_11679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization Xu, Zhehao Lui, Lok Ming Machine Learning Computer Vision and Pattern Recognition Graphics Complex Variables Differential Geometry 30C62, 65D18, 65K10, 68T07 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. |
| title | Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization |
| topic | Machine Learning Computer Vision and Pattern Recognition Graphics Complex Variables Differential Geometry 30C62, 65D18, 65K10, 68T07 |
| url | https://arxiv.org/abs/2511.11679 |