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This content will become publicly available on August 1, 2024

Title: Variational quasi-harmonic maps for computing diffeomorphisms

Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation ofvariational quasi-harmonic mapsto obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps.

 
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Award ID(s):
1838071
NSF-PAR ID:
10483958
Author(s) / Creator(s):
; ;
Publisher / Repository:
Association for Computing Machinery
Date Published:
Journal Name:
ACM Transactions on Graphics
Volume:
42
Issue:
4
ISSN:
0730-0301
Page Range / eLocation ID:
1 to 26
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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