Maximum pseudolikelihood estimation with Markov random fields in the segmentation of brain magnetic resonance images

Abstract

Having an accurate model of the tissue structure of the brain is useful in studying the development andprogression of neurodegenerative diseases like dementia. Brain magnetic resonance images (MRI) can beused to create such models, by detecting the tissue boundaries in the image and classifying each voxel asa particular tissue. This is task known as image segmentation.Many segmentation methods use a mixture-Markov random field probabilistic model for the image inten-sities, which can then be used to determine the most likely segmentation of the image. This model consistsof normal distribution for the image intensities of each tissue, and a Markov random field (MRF) for theprior distribution of tissue labels. The purpose of the MRF is to incorporate spatial dependence betweenthe labels of neighbouring voxels, adding smoothness to the segmentation to remove noise.In this thesis, we develop and validate methods to perform automatic tissue segmentation of brain MRI.Wespecifically focus on the MRF component of the image model. This is used to model spatial dependencebetween neighbouring tissue labels, which has the effect of spatial regularisation on the segmentations.This thesis begins with a general introduction to mixture models, Markov random fields, and the combinedmixture-MRF model as it applies to image segmentation. Estimation of the tissue intensity parametersand also of the segmentation using Expectation-Maximisation (EM) is explained, as well as effectiveapproximations required to accommodate the MRF’s intractable normalising constant.First, the homogeneous Potts MRF is introduced. It is used ubiquitously for MRI segmentation. ThePotts model has one parameter that controls the strength of the MRF compared to the normal intensityprobabilities, and hence the smoothness of the resulting segmentation. In the literature and in practice,this parameter is almost always fixed to a value chosen by manual tuning or with the use of trainingdata. When no training data is available, selection of an appropriate parameter value is subjective andcan affect the accuracy of the segmentation. We propose use of the maximum pseudolikelihood estimator(Besag, 1986) to automatically determine the value of this parameter and show how to incorporate it intothe EM algorithm. The proposed method adaptively determines the amount of spatial regularisation ona per-image basis, without needing training data or an anatomical atlas. The maximum pseudolikelihoodestimator (MPLE) is statistically consistent. It is also computationally tractable, involving only univar-iate maximisation of a concave function, and a straightforward extension of EM. The proposed method isdemonstrated on real brain MRI and compared to various existing methods that require manual specifi-cation of the smoothing parameter. It is also compared to the least-squares method of Derin and Elliott(1987) which has previously been used to automatically determine the smoothing parameter by Van Leem-put et al. (1999b). The MPLE produces segmentations that are comparable or significantly more accuratethan these.Next, the image model is extended to use the non-homogeneous Potts MRF, which has not been studiedin detail for tissue segmentation. While the homogeneous Potts MRF has one parameter that controlsglobal smoothness, the non-homogeneous MRF has multiple pairwise parameters that allow differentsmoothness constraints depending on the specific neighbouring tissues. The MRF additionally has unaryparameters allowing for tissue-specific prior information to be incorporated. The role of each of theseparameters is studied in isolation and together. The previously proposed MPLE is applied to this MRF toautomatically determine the parameters. The method is applied to real brain images. Model selection usingpseudolikelihood information criterion (Forbes and Peyrard, 2003) suggests that the MRF with smoothingparameters but without unary parameters is favoured. However, segmentation accuracy suggests that thenon-homogeneous Potts MRF (with various combinations of unary and smoothing parameters) is not morebeneficial than the homogeneous Potts MRF. A review of similar MRFs in the literature suggests thatthe use of prior anatomical knowledge is required to constrain the parameters of the non-homogeneousPotts MRF to tailor it for brain segmentation. Leaving all parameters free to be estimated can lead tooversmoothing, particularly if a given tissue boundary is relatively rare compared to others.Finally, the image model is extended to consider anisotropic MRFs. Based on the Potts MRF, these allowfor smoothing that can incorporate local features of the image in addition to the tissue labels. Drawingfrom the principles of Perona-Malik diffusion (Perona and Malik, 1990), a model is designed and proposedto smooth the segmentation tangentially along a detected edge but not across it, with strength proportionalto the detected edge strength. Similar anisotropic MRFs have been used for tissue segmentation before,but are discriminative models requiring training data and different solution methods. The proposed modelis generative and may be estimated using Expectation-Maximisation and maximum pseudolikelihood es-timation, thus requiring no training. The model MRF and two variants are applied to brain MRI, and theirsegmentation accuracy compared to the homogeneous Potts MRF. The two supplementary MRFs under-perform the homogeneous Potts MRF but demonstrate that the anisotropy is being appropriately applied.The proposed MRF significantly outperforms the homogeneous Potts MRF and demonstrates anisotropicsmoothing as intended. Suggestions are made to further improve the framework and MRF to make betteruse of the local image structure.In summary the thesis comprises two main directions of research. First, automatic determination of MRFparameters in the mixture-Markov random field framework may be achieved in a computationally tractablemanner using maximum pseudolikelihood, avoiding poor segmentations due to manual specification ofthe spatial parameter. Second, different MRFs allow for finer control of smoothing on a tissue-specific oreven more local neighbourhood-specific level, and when properly specified may improve segmentationaccuracy.