On the convergence of EM-like algorithms for image segmentation using Markov random fields Alexis Roche a,c,⇑ , Delphine Ribes a , Meritxell Bach-Cuadra b , Gunnar Krüger a a CIBM-Siemens, Ecole Polytechnique Fédérale (EPFL), CH-1015 Lausanne, Switzerland b Signal Processing Laboratory 5, Ecole Polytechnique Fédérale (EPFL), CH-1015 Lausanne, Switzerland c Computer Vision Laboratory, Eidgenössische Technische Hochschule (ETHZ), CH-8092 Zurich, Switzerland article info Article history: Received 14 December 2010 Received in revised form 20 April 2011 Accepted 4 May 2011 Available online 13 May 2011 Keywords: Segmentation Markov random field Expectation–maximization Mean field Convergence abstract Inference of Markov random field images segmentation models is usually performed using iterative methods which adapt the well-known expectation–maximization (EM) algorithm for independent mix- ture models. However, some of these adaptations are ad hoc and may turn out numerically unstable. In this paper, we review three EM-like variants for Markov random field segmentation and compare their convergence properties both at the theoretical and practical levels. We specifically advocate a numerical scheme involving asynchronous voxel updating, for which general convergence results can be established. Our experiments on brain tissue classification in magnetic resonance images provide evidence that this algorithm may achieve significantly faster convergence than its competitors while yielding at least as good segmentation results. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Nowadays, Markov Random Field (MRF) models are of wide- spread use in a variety of image segmentation tasks as they provide a natural approach to the general problem of partitioning an image into clusters of homogeneous signal value, while controlling the spatial smoothness of the partition. Following several authors (Van Leemput et al., 1999b; Kapur et al., 1999; Zhang et al., 2001; Bach Cuadra et al., 2005; Forbes and Fort, 2007), this paper uses MRF models for the automatic segmentation of brain mag- netic resonance (MR) images into white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF). The focus here is on the numerical convergence of MRF segmentation algorithms. A customary, but restrictive, way of formulating MRF-based im- age segmentation is to search for the labeling with maximum a posteriori (MAP) probability given an image. In his seminal work, Besag (1974) proposed a simple MAP tracking algorithm called iterated conditional modes (ICM), which was later criticized for being prone to convergence to local maxima. A variety of alterna- tive optimization algorithms were proposed, including simulated annealing (Geman and Geman, 1984), graph cuts (Boykov et al., 2001), max-product loopy belief propagation (Weiss and Freeman, 2001), gradient projection descent (Marroquin et al., 2003), or tree reweighted message passing (Wainwright et al., 2005). Some of these methods were recently compared on a variety of benchmark vision problems by Szeliski et al. (2008). However, all MAP estimation methods have in common that they output a pointwise estimate of the label image, hence they overlook estimation uncertainty. A more comprehensive approach is to compute, or at least approximate, the posterior probability dis- tribution of the label image. This strategy is known as approximate inference in the machine learning literature (Bishop, 2006) and has long been implemented in image segmentation using ‘‘EM-like’’ algorithms. A strong motivation for approximate inference in MR tissue classification is to provide clinicians with error bars on ana- tomical measures derived from segmentation results, such as vol- umes of gray or white matter. Such measures are prone to uncertainty due to image noise and partial volume effects, although deterministic models can be used to account for the latter (Van Leemput et al., 2003; Bach Cuadra et al., 2005; Liang and Wang, 2009). ‘‘EM-like’’ refers to the classical expectation–maximization (EM) algorithm for finite mixture models used, e.g., in Wells et al. (1996), Prima et al. (2001), Ashburner and Friston (2005), which assumes spatially independent labels and is therefore not suited to any but the simplest MRF model. When spatial dependencies are involved, the EM algorithm becomes analytically intractable. While Monte Carlo methods can then be used to accurately approximate poster- ior probabilities, but at heavy computational cost (Van Leemput et al., 2003), several authors have proposed analytical adaptations 1361-8415/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.media.2011.05.002 ⇑ Corresponding author at: CIBM-Siemens, Ecole Polytechnique Fédérale (EPFL), CH-1015 Lausanne, Switzerland. E-mail address: alexis.roche@epfl.ch (A. Roche). Medical Image Analysis 15 (2011) 830–839 Contents lists available at ScienceDirect Medical Image Analysis journal homepage: www.elsevier.com/locate/media