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
2011277999 1486632395 2003967703 2015245929
The Expectation-Maximization (EM) algorithm is an iterative approach to maximum likelihood parameter estimation. Jordan and Jacobs (1993) recently proposed an EM algorithm for the mixture of experts architecture of Jacobs, Jordan, Nowlan and Hinton (1991) and the hierarchical mixture of experts architecture of Jordan and Jacobs (1992). They showed empirically that the EM algorithm for these architectures yields significantly faster convergence than gradient ascent. In the current paper we provide a theoretical analysis of this algorithm. We show that the algorithm can be regarded as a variable metric algorithm with its searching direction having a positive projection on the gradient of the log likelihood. We also analyze the convergence of the algorithm and provide an explicit expression for the convergence rate. In addition, we describe an acceleration technique that yields a significant speedup in simulation experiments. 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. This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution. This paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data. The adjective "unsupervised" is justified by two properties of the algorithm: 1) it is capable of selecting the number of components and 2) unlike the standard expectation-maximization (EM) algorithm, it does not require careful initialization. The proposed method also avoids another drawback of EM for mixture fitting: the possibility of convergence toward a singular estimate at the boundary of the parameter space. The novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models; instead, we seamlessly integrate estimation and model selection in a single algorithm. Our technique can be applied to any type of parametric mixture model for which it is possible to write an EM algorithm; in this paper, we illustrate it with experiments involving Gaussian mixtures. These experiments testify for the good performance of our approach.