Computing Sparse Integer-Constrained Cones for Conformal Parameterizations

Mo Li

doi:10.1145/3528223.3530118

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Computing Sparse Integer-Constrained Cones for Conformal Parameterizations

2022-07-22

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DOI：10.1145/3528223.3530118

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Abstract

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We propose a novel method to generate sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations. Inspired by [Fang et al. 2021; Soliman et al. 2018], the cone computation is formulated as a constrained optimization problem, where the objective is the number of cones measured by the l(0)-norm of Gaussian curvature of vertices, and the constraint is to restrict the cone angles to be multiples of pi/2 and control the distortion while ensuring that the Yamabe equation holds. Besides, the holonomy angles for the non-contractible homology loops are additionally required to be multiples of pi/2 for achieving rotationally seamless conformal parameterizations. The Douglas-Rachford (DR) splitting algorithm is used to solve this challenging optimization problem, and our success relies on two key components. First, replacing each integer constraint with the intersection of a box set and a sphere enables us to manage the subproblems in DR splitting update steps in the continuous domain. Second, a novel solver is developed to optimize the l(0)-norm without any approximation. We demonstrate the effectiveness and feasibility of our algorithm on a data set containing 3885 models. Compared to state-of-the-art methods, our method achieves a better tradeoff between the number of cones and the parameterization distortion.

Keywords:

conformal flattening

cone singularities

integer constraints

l(0)-norm optimization

low distortion