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.

翻译：自由边界微分同胚优化是几何建模、计算机图形学和生物成像领域中一个重要且广泛存在的任务，它需要同时确定一个平面目标域和一个具有良好可控畸变的局部双射映射。我们通过最小二乘拟共形（LSQC）算子来形式化这一任务，并建立了LSQC极小化器的关键结构性质，包括温和条件下的适定性、相似变换下的不变性，以及网格细化下具有稳定性的分辨率无关行为。我们进一步分析了LSQC解关于Beltrami系数的敏感性，建立了稳定性和可微性性质，从而支持在Beltrami系数空间上进行基于梯度的优化。为了使这一可微形式化方法在大规模场景下实用并促进优化过程，我们引入了谱Beltrami网络（SBN），这是一种多尺度网格-谱代理模型，可在单次可微前向传播中近似LSQC解算子。由此产生了SBN-Opt，一个在容许的Beltrami系数和钉扎条件上进行搜索的优化框架，用于求解具有显式畸变控制的自由边界微分同胚目标。在等面积参数化和非一致曲面配准上的大量实验表明，该方法相对于传统数值算法取得了持续改进。