Abstract

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 a nity 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 unied 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 di erent eigenvector segmentation algorithms. We illustrate our analysis with results on real and synthetic images. Human perceiving a scene can often easily segment it into coherent segments or groups. There has been a tremendous amount of e ort devoted to achieving the same level of performance in computer vision. In many cases, this is done by associating with each pixel a feature vector (e.g. color, motion, texture, position) and using a clustering or grouping algorithm on these feature vectors. Perhaps the cleanest approach to segmenting points in feature space is based on mixture models in which one assumes the data were generated by multiple processes and estimates the parameters of the processes and the number of components in the mixture. The assignment of points to clusters can then be easily performed by calculating the posterior probability ofa point belonging to a cluster. Despite the elegance of this approach, the estimation process leads to a notoriously di cult optimization. The frequently used EM algorithm [3] often converges to a local maximum that depends on the initial conditions. Recently, anumber of authors [11, 10, 8, 9, 2] have suggested alternative segmentation methods that are based on eigenvectors of the (possibly normalized) \a nity matrix". Figure 1a shows two clusters of points and gure 1b shows the a nity matrix de ned by:

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