Computing sparse cones with bounded distortion for conformal parameterizations

Abstract

We propose a novel method to generate sparse cone singularities with bounded distortion constraints for conformal parameterizations. It is formulated as minimizing the ℓ 0 -norm of Gaussian curvature of vertices with hard constraints of bounding the distortion that is measured by the ℓ 2 -norm of the log conformal factor. We use the reweighted ℓ 1 -norm to approximate the ℓ 0 -norm and solve each convex weighted ℓ 1 minimization subproblem by the Douglas-Rachford (DR) splitting scheme. To quickly generate sparse cones, we modify DR splitting by weighting the ℓ 2 -norm of the proximal mapping to force the small Gaussian curvature to quickly approach zero. Accordingly, compared with the conventional DR splitting, the modified method performs one to two orders of magnitude faster. Besides, we perform variable substitution of log conformal factors to simplify the computation process for acceleration. Our algorithm is able to bound distortion to compute sparse cone singularities, so that the resulting conformal parameterizations achieve a favorable tradeoff between the area distortion and the number of cones. We demonstrate its effectiveness and feasibility on a large number of models.