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Published July 2017 | public
Journal Article

Variance-minimizing transport plans for inter-surface mapping

Abstract

We introduce an efficient 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-to-fine 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.

Additional Information

© 2017 ACM. The hands, skull and dino models are courtesy of the AIM@Shape repository; the dog, man and gorilla models are from TOSCA; the baby models from SHREC'16; Fig. 20 features a mesh from [Solomon et al. 2016], courtesy of the authors; all other models are from the SHREC'07 dataset. This work has been funded by the European Research Council (ERC grants 257474: "IRON", 339025: "GUDHI" and 340884: "ACROSS"), NSF CCF-1655306 and the German Research Foundation (DFG, Gottfried-Wilhelm-Leibniz Programm). We wish to thank Vladimir Kim, Justin Solomon and Qifeng Chen for providing us with their respective source codes, and the reviewers for their constructive comments. MD gratefully acknowledges the Inria International Chair program and all the members of the TITANE team for their support.

Additional details

Created:
August 19, 2023
Modified:
October 17, 2023