A Subspace Method for Fast Locally Injective Harmonic Mapping

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.

Inversion-free geometric mapping construction: A survey

“…In Ref. [135], an algorithm based on Ref. [134] is presented for low-distortion locally injective harmonic mappings.…”

Locally supported tangential vector, n‐vector, and tensor fields

“…Reduced systems have been proposed for the simulation of fluids [TLP06, LMH∗15, CSK18], elastic solids and shells [BJ05, AKJ08, YLX∗15, BEH18], fluid‐solid interaction [LJF16, BSEH19], example‐based elastic material [ZZM15], motion planning [BdSP09, HSvTP12, PM18], clothing [HTC∗14], and hair [CZZ14]. In the context of mesh processing, subspace methods have been introduced for surface modeling [HSL∗06, HSvTP11, JBK∗12, WJBK15], shape interpolation [vTSSH15, vRESH16], injective mappings [HCW19], motion processing [BvTH16], and spectral mesh processing [NBH18]. The goal of this paper is to explore subspace constructions for tangential vector, n ‐vector, and tensor fields on surfaces and the use of subspace methods for the design and processing of tangential fields.…”

An Efficient Hybrid Optimization Strategy for Surface Reconstruction

“…The first step, i.e. the mapping, aims at retrieving a suitable map of the 3D tessellation over a simplified 2D domain (generally a unit square or a unit circle) [1][2][3][4][5]. The mapping of the TS should preserve some fundamental information of the 3D shape of the tessellation to ensure a smooth surface approximation.…”

“…In [15], Floater developed a general mapping method that has been used (and extended) in other studies [3,4,16,17]. The mapping method proposed by Floater, also called shape preserving method (SPM), is a fast procedure, based on a linear system of equations, capable of preserving the shape of the original TS.…”