We present a method for locally injective seamless parametrization of triangular mesh surfaces of arbitrary genus, with or without boundaries, given desired cone points and rational holonomy angles (multiples of 2π/q for some positive integer q). The basis of the method is an elegant generalization of Tutte's "spring embedding theorem" to this setting. The surface is cut to a disk and a harmonic system with appropriate rotation constraints is solved, resulting in a harmonic global parametrization (HGP) method. We show a remarkable result: that if the triangles adjacent to the cones and boundary are positively oriented, and the correct cone and turning angles are induced, then the resulting map is guaranteed to be locally injective. Guided by this result, we solve the linear system by convex optimization, imposing convexification frames on only the boundary and cone triangles, and minimizing a Laplacian energy to achieve harmonicity. We compare HGP to state-of-the-art methods and see that it is the most robust, and is significantly faster than methods with comparable robustness.
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This work is supported by the Israel Science Foundation, under grants 1869/15 and 2102/15. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from firstname.lastname@example.org. © 2017 Copyright held by the owner/author(s). Publication rights licensed to ACM. 0730-0301/2017/7-ART89 $15.00 DOI: http://dx.doi.org/10.1145/3072959.3073646
This research was partially funded by the Israel Science Foundation (grants No. 1869/15 and 2102/15). Additional help was provided by several individuals: Eden Fedida, for assistance with figures; Zohar Levi, for formatting of benchmark input data; and Alec Jacobson, for useful discussions on null spaces of sparse matrices.
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- Discrete harmonic
- Global parametrization
- Injective maps
- Mesh parametrization