Efficient Computation of Isometry‐Invariant Distances Between Surfaces

Abstract

We present an efficient computational framework for isometry‐invariant comparison of smooth surfaces. We formulate the Gromov–Hausdorff distance as a multidimensional scaling–like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples. For isometry‐invariant comparison of surfaces in the case of partially missing data, we present the partial embedding distance, which is computed using a similar scheme. The main idea is finding a minimum‐distortion mapping from one surface to another, while considering only relevant geodesic distances. We discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency.