Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration

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.

Inexact Matching of Large and Sparse Graphs Using Laplacian Eigenvectors

“…The top row of figure 5 shows the result of many-to-many inexact matching obtained with the method described in this paper. The bottom row shows the result of one-to-one rigid point registration obtained with a variant of the EM algorithm [13] initialized from the matches shown on the top row.…”

“…A detailed solution is described in [13]. If matrices S and P are not correctly estimated, matrix R belongs to the orthogonal group, i.e., rotations and reflections.…”

Hierarchical Matching of Non-rigid Shapes

“…[17] and Mateus et al . [21] suggested alternative isometry-invariant shape representations, obtained by using eigendecomposition of discrete Laplace operators. The Global Point Signature (GPS) suggested by Rustamov [31] for shape comparison employs the discrete LaplaceBeltrami operator, which, at least theoretically, captures the shape's geometry more faithfully.…”

Groupwise Spectral Log-Demons Framework for Atlas Construction

“…We now describe a new update scheme based on spectral correspondence [21,11,18,14,13] that will enable the construction of atlases with large deformations. Let us first consider I Ω , the portion of an image I bounded by a contour Ω.…”