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Segmentation using eigenvectors: a unifying view

Automatic grouping and segmentation of images remains a challenging problem in computer vision. Recently, a number of authors have demonstrated good performance on this task using methods that are based on eigenvectors of the affinity matrix. These approaches are extremely attractive in that they are based on simple eigendecomposition algorithms whose stability is well understood. Nevertheless, the use of eigendecompositions in the context of segmentation is far from well understood. In this paper we give a unified treatment of these algorithms, and show the close connections between them while highlighting their distinguishing features. We then prove results on eigenvectors of block matrices that allow us to analyze the performance of these algorithms in simple grouping settings. Finally, we use our analysis to motivate a variation on the existing methods that combines aspects from different eigenvector segmentation algorithms. We illustrate our analysis with results on real and synthetic images.