Surface parameterization via optimization of relative entropy and quasiconformality

TL;DR

The paper addresses triangle-mesh parameterization by balancing density distribution and angular distortion through an energy $E(f)=\mathcal{H}(f_*(\\gamma)|\\nu) + \alpha\|\\mu_f\|_{L^2}^2 + \beta\|\nabla\\mu_f\|_{L^2}^2$, where $\\mu_f$ is the Beltrami coefficient. It advances a flow-based optimization that couples a Fokker-Planck gradient flow for the relative entropy with QC-driven updates via the Beltrami equation, implemented through a finite-volume transport scheme and a linear Beltrami solver; the method is extended to genus-0 and genus-1 surfaces via an appropriate initial conformal structure and torus parameterization. The main contributions are (i) a concrete energy formulation integrating measure preservation and conformal regularization, (ii) a practical, split-flow algorithm with operators $S_{\\Gamma}$, $T_t$, $P_k$, and $R_t$ for iterative refinement, and (iii) detailed discretization strategies for transport and QC maps, plus genus-1 adaptations. The approach yields piecewise-linear, bijective maps with controlled density alignment and bounded angular distortion, enabling improved remeshing and mesh quality for complex surfaces.

Abstract

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.

Figures (8)

• Figure 1: Some maps by specifying different initial and reference probability measures
• Figure 2: The remeshing results given by the maps shown above
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Theorems & Definitions (4)

• Definition 2.1
• Theorem 2.2: Measurable Riemann Mapping Theorem
• Theorem 3.1
• Proof 1