A Subspace Method for Fast Locally Injective Harmonic Mapping

Eden Fedida Hefetz; Edward Chien; Ofir Weber

doi:10.1111/cgf.13623

A Subspace Method for Fast Locally Injective Harmonic Mapping

Eden Fedida Hefetz, Edward Chien, Ofir Weber

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

We present a fast algorithm for low-distortion locally injective harmonic mappings of genus 0 triangle meshes with and without cone singularities. The algorithm consists of two portions, a linear subspace analysis and construction, and a nonlinear non-convex optimization for determination of a mapping within the reduced subspace. The subspace is the space of solutions to the Harmonic Global Parametrization (HGP) linear system [BCW17], and only vertex positions near cones are utilized, decoupling the variable count from the mesh density. A key insight shows how to construct the linear subspace at a cost comparable to that of a linear solve, extracting a very small set of elements from the inverse of the matrix without explicitly calculating it. With a variable count on the order of the number of cones, a tangential alternating projection method [HCW17] and a subsequent Newton optimization [CW17] are used to quickly find a low-distortion locally injective mapping. This mapping determination is typically much faster than the subspace construction. Experiments demonstrating its speed and efficacy are shown, and we find it to be an order of magnitude faster than HGP and other alternatives.

Original language	English
Pages (from-to)	105-119
Number of pages	15
Journal	Computer Graphics Forum
Volume	38
Issue number	2
DOIs	https://doi.org/10.1111/cgf.13623
State	Published - May 2019

Bibliographical note

Publisher Copyright:
© 2019 The Author(s) Computer Graphics Forum © 2019 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.

Funding

This research was partially funded by the Israel Science Foundation (grants No. 1869/15 and 2102/15). We thank: Alec Jacobson, for useful discussions on null spaces of sparse matrices; David Bommes, for discussions on the structure of our linear system; Zo-har Levi, for the Beetle CETM result of Figure 6; and Olaf Schenk, Fabio Verbosio, Renjie Chen, Ladislav Kavan, Petr Kadlecˇek, Roi Poranne, and Alon Bright, for assisting with PARDISO utilization.

Funders	Funder number
Israel Science Foundation	2102/15, 1869/15

Keywords

CCS Concepts
Mesh geometry models
• Computing methodologies → Mesh models
• Mathematics of computing → Topology

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10.1111/cgf.13623

Cite this

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abstract = "We present a fast algorithm for low-distortion locally injective harmonic mappings of genus 0 triangle meshes with and without cone singularities. The algorithm consists of two portions, a linear subspace analysis and construction, and a nonlinear non-convex optimization for determination of a mapping within the reduced subspace. The subspace is the space of solutions to the Harmonic Global Parametrization (HGP) linear system [BCW17], and only vertex positions near cones are utilized, decoupling the variable count from the mesh density. A key insight shows how to construct the linear subspace at a cost comparable to that of a linear solve, extracting a very small set of elements from the inverse of the matrix without explicitly calculating it. With a variable count on the order of the number of cones, a tangential alternating projection method [HCW17] and a subsequent Newton optimization [CW17] are used to quickly find a low-distortion locally injective mapping. This mapping determination is typically much faster than the subspace construction. Experiments demonstrating its speed and efficacy are shown, and we find it to be an order of magnitude faster than HGP and other alternatives.",

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A Subspace Method for Fast Locally Injective Harmonic Mapping

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