Matching LBO eigenspace of non-rigid shapes via high order statistics

TL;DR

This paper proposes a statistical method to match Laplace-Beltrami eigenspaces of non-rigid shapes by aligning their high order statistical properties, improving shape correspondence tasks.

Contribution

It introduces a novel approach using high order statistics to resolve eigenfunction ambiguities for shape matching, enhancing compatibility of spectral embeddings.

Findings

Successfully matches eigenfunctions across shapes

Improves pointwise correspondence accuracy

Resolves sign and permutation ambiguities

Abstract

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes in flat domains while preserving distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Lapalce-Beltrami operator. The Laplace-Beltrami eigenspace preserves the diffusion distance, and is invariant under isometric transformations. However, Laplace-Beltrami eigenfunctions computed independently for different shapes are often incompatible with each other. Applications involving multiple shapes, such as pointwise correspondence, would greatly benefit if their respective eigenfunctions were somehow matched. Here, we introduce a statistical approach for matching eigenfunctions. We consider the values of the eigenfunctions over the manifold as sampling of random variables, and try to match their multivariate distributions. Comparing distributions is done indirectly, using high order statistics. We show that the permutation and sign ambiguities of low order eigenfunctions, can be inferred by minimizing the difference of their third order moments. The sign ambiguities of antisymmetric eigenfunctions can be resolved by exploiting isometric invariant relations between the gradients of the eigenfunctions and the surface normal. We present experiments demonstrating the success of the proposed method applied to feature point correspondence.