Image Segmentation Using Maximum Spanning Tree on Affinity Matrix

The objective of image segmentation and clustering is to extract meaningful regions out of an image. A graph theoretic approach, as an alternative to the gradient-based methods, is usually based on the eigenvectors of an affinity matrix [3,4,5,6,7]. The theoretical foundation of this development is the Spectral Graph Theory [1], through which the combinatorial features of a graph can be revealed by its spectra. This characteristic can be applied into graph partitioning and preconditioning. The typical eigendecompostion based segmentation work is called the normalized cuts [6]. The normalized cut measure incorporates the local similarity within cluster and total dissimilarity between clusters. The minimization of this measure is to solve the Rayleigh quotient, a generalized eigenvalue solving problem. However, solving a standard eigenvalue problem for all eigenvectors has exponential complexity and is very time consuming. Shi [6] made use of the sparsity of the affinity matrix and introduced the Lanczos method to simplify the computation of eigenvalues. In this paper, we give a new image segmentation algorithm using the maximum spanning tree [2]. Our method works on affinity matrix and addresses the physical meanings of an affinity matrix. Instead of computations of eigenvalues and eigenvectors, we proved that the image segmentation could be transformed into an optimization problem: finding the maximum spanning tree of the graph with image pixels as vertices and pairwise similarities as weights. Section 2 describes the related theory and Section 3 gives the experimental results on synthetic and real data to illustrate the performance of this algorithm. Finally, we draw a conclusion and discuss future work in Section 4.