GeodesicEmbedding (GE): A High-Dimensional Embedding Approach for Fast Geodesic Distance Queries

TL;DR

This paper introduces GeodesicEmbedding, a high-dimensional embedding technique that enables rapid geodesic distance queries on meshes by reducing problem complexity and employing cascaded optimization for accuracy.

Contribution

The paper presents a novel high-dimensional embedding method that efficiently approximates geodesic distances by embedding only saddle vertices and refining with cascaded optimization.

Findings

Achieves near-constant time geodesic distance queries.

Outperforms previous methods in speed and accuracy.

Reduces embedding complexity through saddle vertex selection.

Abstract

In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in the original manifold surface. However, directly solving the high-dimensional embedding problem is not feasible due to the large number of variables and the fact that the embedding problem is highly nonlinear. We overcome the challenges with two novel ideas. First, instead of taking all vertices as variables, we embed only the saddle vertices, which greatly reduces the problem complexity. We then compute a local embedding for each non-saddle vertex. Second, to reduce the large approximation error resulting from the purely Euclidean embedding, we propose a cascaded optimization approach that repeatedly introduces additional embedding coordinates with a non-Euclidean function to reduce the approximation residual. Using the precomputation data, our approach can determine the geodesic distance between any two vertices in near-constant time. Computational testing results show that our method is more desirable than previous geodesic distance queries methods.