We propose a novel method for parameterizations of triangle meshes by finding an optimal quasiconformal map that minimizes an energy consisting of a relative entropy term and a quasiconformal term. By prescribing a prior probability measure on a given surface and a reference probability measure on a parameter domain, the relative entropy evaluates the difference between the pushforward of the prior measure and the reference one. The Beltrami coefficient of a quasiconformal map evaluates how far the map is close to an angular-preserving map, i.e., a conformal map. By adjusting parameters of the optimization problem, the optimal map achieves a desired balance between the preservation of measure and the preservation of conformal structure. To optimize the energy functional, we utilize the gradient flow structure of its components. The gradient flow of the relative entropy is the Fokker-Planck equation, and we apply a finite volume method to solve it. Besides, we discretize the Beltrami coefficient as a piecewise constant function and apply the linear Beltrami solver to find a piecewise linear quasiconformal map.

翻译：本文提出一种三角形网格参数化的新方法，通过寻找最优拟共形映射来最小化由相对熵项与拟共形项构成的能量泛函。通过在给定曲面上设定先验概率测度、在参数域上设定参考概率测度，相对熵用于评估先验测度推前映射与参考测度之间的差异。拟共形映射的Beltrami系数衡量该映射接近保角映射（即共形映射）的程度。通过调整优化问题的参数，最优映射可在测度保持与共形结构保持之间达到期望的平衡。为优化该能量泛函，我们利用其各分量的梯度流结构：相对熵的梯度流为Fokker-Planck方程，采用有限体积法进行求解；同时将Beltrami系数离散为分段常数函数，并应用线性Beltrami求解器得到分段线性拟共形映射。