A Subspace Method for Fast Locally Injective Harmonic Mapping

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 mappi...

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Bibliographic Details
Published in:Computer graphics forum Vol. 38; no. 2; pp. 105 - 119
Main Authors: Hefetz, Eden Fedida, Chien, Edward, Weber, Ofir
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.05.2019
Subjects:
Algorithms
CCS Concepts
Computational geometry
Computing methodologies → Mesh models
Cones
Construction costs
Convexity
Decoupling
Distortion
Mapping
Mathematical analysis
Mathematics of computing → Topology
Mesh geometry models
Optimization
Parameterization
Singularities
Subspace methods
Subspaces
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary: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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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.13623