Lightweight preprocessing and fast query of geodesic distance via proximity graph

Abstract

Computing geodesic distance on a mesh surface S efficiently and accurately is a central task in numerous computer graphics applications. In order to deal with high-resolution mesh surfaces, a lightweight preprocessing is a proper choice to make a balance between query accuracy and speed. In the preprocessing stage, we build a proximity graph G with regard to a set of sample points and keep the exact geodesic distance between any pair of nearby sample points. In the query stage, given two query points s and t, we augment the proximity graph G by adding s and t on-the-fly, and then use the shortest path between s and t on the augmented proximity graph to approximate the exact geodesic path between s and t. We establish an empirical relationship between the number of samples and expected accuracy (measured in relative error), which facilitates fast and accurate query of geodesic distance with a lightweight processing cost. We exhibit the uses of the new approach in two applications—real-time computation of discrete exponential map for texture mapping and interactive design of spline curves on surfaces.