Normalized cut (Ncut) is one of the popular image segmentation models in which the parametric similarity matrix should be given in advance. In this paper, we propose a normalized cut segmentation algorithm with an adaptive similarity measure and spatial regularization. The statistical Parzen-Rosenblatt window, expectation maximum (EM) method and the regularization technique in partial differential equation (PDE) are unified in a proposed variational framework. The introduced EM strategy makes our method can adaptively update the similarity matrix, and it can produce a better classification criterion than the existing Ncut based methods. While the regularization can guarantee the proposed algorithm has a robust performance under noise and it can also rectify the similarity measure with a priori spatial location. In addition, the existence theorem of the proposed model is given in the paper. Compared with some existing spectral clustering methods such as the traditional Ncut algorithm and the variational based Chan-Vese model, the numerical experiments show that our methods can provide promising segmentation results.
Expectation maximum (EM) algorithm @cite_46 is a classical method for parameters estimation of mixture models. It is used when the data has some missing values, or when optimizing the likelihood function is analytically intractable @cite_46 . EM algorithm has great applicability because of its stable convergence and convenient implementation.
@cite_46
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We describe the maximum-likelihood parameter estimation problem and how the ExpectationMaximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor.