Geodesics using Waves: Computing Distances using Wave Propagation

TL;DR

This paper introduces a fast and accurate wave-based method for approximating geodesic distances on manifold meshes by solving two linear systems, enabling efficient computation with minimal pre-processing.

Contribution

The paper presents a novel wave propagation approach that efficiently computes approximate geodesic distances using only two linear systems, one of which is solved once.

Findings

Method completes in a few seconds for 300-400 iterations.

High accuracy of geodesic distance approximation.

Requires only one pre-factorization of linear systems.

Abstract

In this paper, we present a new method for computing approximate geodesic distances. We introduce the wave method for approximating geodesic distances from a point on a manifold mesh. Our method involves the solution of two linear systems of equations. One system of equations is solved repeatedly to propagate the wave on the entire mesh, and one system is solved once after wave propagation is complete in order to compute the approximate geodesic distances up to an additive constant. However, these systems need to be pre-factored only once, and can be solved efficiently at each iteration. All of our tests required approximately between 300 and 400 iterations, which were completed in a few seconds. Therefore, this method can approximate geodesic distances quickly, and the approximation is highly accurate.