Graph-spectral methods for computer vision

This thesis describes a family of graph-spectral methods for computer vision that exploit the properties of the first eigenvector of the adjacency matrix of a weighted graph. The algorithms are applied to segmentation and grouping, shape-from-shading and graphmatching. In Chapter 3, we cast the problem of grouping into an evidence combining setting where the number of clusters is determined by the modes of the adjacency matrix. With the number of clusters to hand, we model the grouping process using two sets of variables. These are the cluster memberships and the pairwise affinities or link-weights for the nodes of a graph. From a simple probability distribution for these parameters, we show how they may be estimated using the apparatus of the expectation-maximisation (EM) algorithm. The new method is demonstrated on the problems of line-segment grouping and gray-scale image segmentation. The method is shown to outperform a non-iterative eigenclustering method. In Chapter 4, we present a more direct graph-spectral method for segmentation and grouping by developing an iterative maximum likelihood framework for perceptual clustering. Here, we focuss in more detail on the likelihood function that results from the Bernoulli model. Rather than using the EM algorithm to estimate an updated link-weight matrix, we show how a modal decomposition technique can be used to remove noisy linkweights. We establish the relationship between the modes of the link-weight matrix and the maxima of the log-likelihood function. The matrix analysis of the log-likelihood function results in a method that is faster to converge and which gives better defined clusters. The method is evaluated on gray-scale image segmentation, line-segment grouping and motion analysis. In Chapter 5, we turn our attention to the problem of recovering the 3D representation of an object from its shading information. To do this, we develop a graph-spectral method for shape-from-shading. We commence by characterising the field of surface normals using a transition matrix whose elements are computed from the sectional curvature i between different image locations. With the transition matrix to hand, we use a graph seriation method to define a curvature minimising surface integration path for the purposes of height reconstruction. We extend the surface height recovery algorithm to perform surface normal adjustment. We do this by fitting quadric patches to the height data. A performance study of the resulting shape-from-shading algorithm is presented. We perform experiments on real-world imagery and compare our results to those obtained using two alternative shape-from-shading algorithms. Finally, in Chapter 6, we show how the eigenstructure of the adjacency matrix can be used for purposes of graph edit distance computation. We make use of the graph seriation method developed in Chapter 5 to convert the adjacency matrix into a string or sequence order. We pose the problem of graph-matching as a maximum a posteriori probability alignment of the seriation sequences for pairs of graphs. We model this probability by making use of a compatibility co-efficient for the edge alignment of the sequences and the probability of individual node correspondences. With these ingredients, we compute the edit distance by finding the sequence of string edit operations which minimise the cost of the path traversing the edit lattice. The edit costs are defined in terms of the a posteriori probability of visiting a site on the lattice. We demonstrate the utility of the method for purposes of graph clustering.