Variance-Minimizing Transport Plans for Inter-surface Mapping

Manish Mandad 1, 2 David Cohen-Steiner 3 Leif Kobbelt 2 Pierre Alliez 1 Mathieu Desbrun 1, 4
1 TITANE - Geometric Modeling of 3D Environments
CRISAM - Inria Sophia Antipolis - Méditerranée
3 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We introduce an effcient computational method for generating dense and low distortion maps between two arbitrary surfaces of same genus. Instead of relying on semantic correspondences or surface parameterization, we directly optimize a variance-minimizing transport plan between two input surfaces that defines an as-conformal-as-possible inter-surface map satisfying a user-prescribed bound on area distortion. The transport plan is computed via two alternating convex optimizations, and is shown to minimize a generalized Dirichlet energy of both the map and its inverse. Computational efficiency is achieved through a coarse-tone approach in diffusion geometry, with Sinkhorn iterations modified to enforce bounded area distortion. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly, with little to no user interaction.
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Manish Mandad, David Cohen-Steiner, Leif Kobbelt, Pierre Alliez, Mathieu Desbrun. Variance-Minimizing Transport Plans for Inter-surface Mapping. ACM Transactions on Graphics, Association for Computing Machinery, 2017, 36, pp.14. ⟨10.1145/3072959.3073671⟩. ⟨hal-01519006⟩

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