Spectral methods for image clustering

The clustering problem is to assign labels to points in order to group them in a structurally meaningful way. This is often accomplished by defining cluster centroids in the vector space and assigning points to the cluster with the nearest centroid, as the k-means algorithm does. But this approach does not work for clusters that have unusual shapes, particularly if the clusters are interwoven or nested. Spectral clustering is a clustering method based on Kernel PCA (Principal Component Analysis). PCA computes eigenvectors of the covariance matrix of data to find the directions along which the variance is greatest; Kernel PCA computes eigenvectors of the kernel matrix, essentially performing PCA in the feature space defined by the kernel. This can overcome the problems faced by direct centroid-based clustering methods because a suitable kernel can be chosen to project the data into a feature space in which the clusters are spatially distinct. This is essentially the approach taken by spectral clustering algorithms. We use the spectral clustering algorithm given by Ng, Jordan, and Weiss [3]: