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.

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