## 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 |
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Pages (from-to) | 105-119 |

Number of pages | 15 |

Journal | Computer Graphics Forum |

Volume | 38 |

Issue number | 2 |

DOIs | |

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 |
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Israel Science Foundation | 2102/15, 1869/15 |

## Keywords

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