Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration

TL;DR

This paper introduces a novel spectral shape matching method that uses Laplacian eigenfunctions and unsupervised point registration to effectively match articulated shapes despite noise, topology changes, and sampling differences.

Contribution

It proposes a new eigenfunction selection approach for shape alignment, improving robustness and performance in dense shape matching tasks.

Findings

Handles topology changes and shape variations effectively

Outperforms classical methods on challenging datasets

Robust to noise and sampling density differences

Abstract

Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent $K$-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.