Alignment using Spectral Clusters

Abstract

This paper describes a hierarchical spectral method for the correspondence matching of point-sets. Conventional spectral methods for correspondence matching are notoriously susceptible to differences in the relational structure of the point-sets under consideration. In this paper we demonstrate how the method can be rendered robust to structural differences by adopting a hierarchical approach. We show how the point-clusters associated with the most significant spectral modes can be used to locate correspondences when significant contamination is present. Spectral graph theory is a term applied to a family of techniques that aim to characterise the global structural properties of graphs using the eigenvalues and eigenvectors of the adjacency matrix [1]. Although the subject has found widespread use in a number of areas including structural chemistry and routeing theory, there have been relatively few applications in the computer vision literature. The reason for this is that although elegant, spectral graph representations are notoriously susceptible to the effect of structural error. In other words, spectral graph theory can furnish very efficient methods for characterising exact relational structures, but soon breaks down when there are spurious nodes and edges in the graphs under study. There are several concrete examples in the pattern analysis literature. Umeyama has an eigendecomposition method that recovers the permutation matrix that maximises the correlation or overlap of the adjacency matrices for graphs of the same size [13]. Horaud and Sossa [5] have adopted a purely structural approach to the recognition of linedrawings. Their representation is based on the immanantal polynomials for the Laplacian matrix of the line-connectivity graph. By comparing the coefficients of the polynomials, they are able to index into a large data-base of line-drawings. Shapiro and Brady [11] have developed a method which draws on a representation which uses weighted edges. They commence from a weighted adjacency matrix (or proximity matrix) which is obtained using a Gaussian function of the distances between pairs of points. The eigen-vectors of the adjacency matrix can be viewed as the basis vectors of an orthogonal transformation on the original point identities. In other words, the components of the eigenvectors represent mixing angles for the transformed points. Matching between different point-sets is effected by comparing the pattern of eigenvectors in different images. Finally, a number of authors have used spectral methods to perform pairwise clustering on image data. Shi and Malik [12] use the second eigenvalue to segment grey-scale images by performing an eigen-decomposition on a matrix of pairwise attribute differences. Inoue and Urahama [6] have shown how the sequential extraction of eigen-modes can be used to cluster pairwise