The continuous Procrustes distance between two surfaces

TL;DR

This paper introduces the continuous Procrustes distance for 3D surfaces, providing a true metric and an efficient approximation algorithm based on conformal maps, improving shape comparison beyond discrete landmark points.

Contribution

The paper proposes the continuous Procrustes distance as a new shape metric and develops an efficient algorithm for its approximation using conformal maps.

Findings

Proves the continuous Procrustes distance is a true metric for 2D surfaces in 3D.

Shows that for small distances, the optimal map can be approximated by a conformal map.

Provides an efficient computational method for shape comparison using the new distance.

Abstract

The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be homologous (i.e. corresponding to each other), as far as possible, and the Procrustes distance is then computed as the infimum, over all Euclidean transformations R, of the sum of the squared Euclidean distances between R x_j and x'_j, and the correspondences x_j <-> x'_j are picked in an optimal way. The discrete Procrustes distance has the drawback that each shape is represented by only a finite number of points, which may not capture all the geometric aspects of interest; a need has been expressed for alternatives that are still easy to compute. We propose in this paper the concept of continuous Procrustes distance, and prove that it provides a true metric for two-dimensional surfaces embedded in three dimensions. The continuous Procrustes distance leads to a hard optimization problem over the group of area-preserving diffeomorphisms. One of the core observations of our paper is that for small continuous Procrustes distances, the global optimum of the Procrustes distance can be uniformly approximated by a conformal map. This observation leads to an efficient algorithm to calculate approximations to this new distance.