WESD - Weighted Spectral Distance for Measuring Shape Dissimilarity

TL;DR

WESD is a new mathematically grounded spectral distance based on Laplace eigenvalues, designed to measure shape dissimilarity effectively, with proven properties and demonstrated benefits in vision and medical imaging.

Contribution

The paper introduces WESD, a novel shape distance derived from heat-trace analysis of Laplace eigenvalues, with theoretical guarantees and practical advantages.

Findings

WESD converges over all eigenvalues and can be approximated with finite eigenvalues.

WESD is a pseudometric and can be scaled to [0,1).

Experiments show WESD's effectiveness in vision and medical image analysis.

Abstract

This article presents a new distance for measuring shape dissimilarity between objects. Recent publications introduced the use of eigenvalues of the Laplace operator as compact shape descriptors. Here, we revisit the eigenvalues to define a proper distance, called Weighted Spectral Distance (WESD), for quantifying shape dissimilarity. The definition of WESD is derived through analysing the heat-trace. This analysis provides the proposed distance an intuitive meaning and mathematically links it to the intrinsic geometry of objects. We analyse the resulting distance definition, present and prove its important theoretical properties. Some of these properties include: i) WESD is defined over the entire sequence of eigenvalues yet it is guaranteed to converge, ii) it is a pseudometric, iii) it is accurately approximated with a finite number of eigenvalues, and iv) it can be mapped to the [0,1) interval. Lastly, experiments conducted on synthetic and real objects are presented. These experiments highlight the practical benefits of WESD for applications in vision and medical image analysis.