This section discusses works in MRI brain tissue classification and nonparametric Markov modeling along with their relationships to the proposed method. It compares and contrasts the proposed strategy, in brief, with the key ideas around which various classification strategies have evolved, including (a) partitioning based on grayscale voxel-intensity data, (b) regularization schemes based on local interactions among class labels, and (c) spatial priors based on probabilistic and anatomical atlases.
Wells et al. [176] present a method that couples tissue classification with inhomogeneity correction based on ML parameter estimation. They use the EM algorithm of Dempster et al. [43] to simultaneously estimate the unknown bias field and the classification. Leemput et al. [93,94] extend this approach by posing the problem in the context of mixture density estimation to estimate the grayscale intensity PDFs for each tissue type. They apply the EM algorithm to estimate these PDFs as well as the bias and, in turn, the classification. Their approach assumes that each tissue-intensity distribution conforms to a parametric Gaussian PDF whose parameters are obtained via the EM algorithm. The proposed method, in contrast to typical EM-based strategies, does not impose any parametric model on the tissue intensities. Instead, it automatically adapts to the data using neighborhood sampling and nonparametric density estimation.
The EM-classification algorithm [176] does not impose any smoothness constraint on the classification and it is therefore susceptible to outliers in the tissue intensities. Some approaches for tissue classification do not explicitly account for noise, but employ image-denoising methods as a preprocessing step [64,100]. Many subsequent works incorporate noise models into the classification without such preprocessing. Several authors [86,76,93,94,121,187] have extended the EM-classification algorithm to incorporate spatial smoothness via Gibbs/Markov priors on the label image. For instance, Kapur et al. [86] use spatially-stationary Gibbs priors to model local interactions between neighboring labels. Typically, these methods modify single-voxel tissue-probabilities based on energies defined on local configurations of classification labels. They assign lower energies to spatially-smooth segmentations, making them more likely. Such strong Markov models, however, can over regularize the fine-structured interfaces, e.g., the one between gray matter and white matter. Hence, it is often necessary to impose additional heuristic constraints [76,93,94]. Ruf et al. [146] extend the EM approach to perform spatial regularization by incorporating the spatial coordinates of the voxels, in addition to their grayscale intensities, in the feature vector.
This tissue-classification work dovetails with the mainstream image-processing literature, which presents a variety of algorithms that rely on MRF models of images [61,16,120,99,161]. Such methods typically involve iterative stochastic-relaxation schemes that compute local image updates based on random sampling from local conditional PDFs. These conditional PDFs on neighborhood configurations define an energy that is progressively reduced. Typically, the methods specify the conditional PDFs in parametric forms, e.g., Gaussian [99]. In this way, they encode a set of probabilistic assumptions (priors) about the geometric/statistical properties of the image data, and thus they are effective only when the data conform sufficiently well to the prior. Furthermore, the previous work on MRI classification models each tissue class with Gaussian-mixture models, which is homogeneous across the image. The proposed method, rather than enforcing a particular Markov prior on the data, learns the relevant Markov statistics nonparametrically from the input data and bases the classification on this adaptive model.
Researchers have also used active contour models [38,36] to impose smoothness constraints for segmentation. These methods typically attempt to minimize the area of the segmentation boundary (smoothness) simultaneously with proper fidelity to the data. These models produce results that can be quite sensitive to the contour parameters that control the influence of the data and the smoothness. Hence, these methods typically require careful manual parameter-tuning. The proposed method, on the other hand, sets its important internal free parameters via data-driven techniques using information-theoretic optimality criteria. As a result, it easily applies to a wide spectrum of data with little parameter tuning.
An important component in MRI brain tissue classification is the correction of intensity inhomogeneities or bias fields. Several approaches propose an approach that couples iterative updates of the class labels with the bias-field correction based on polynomial least-squares fitting [176,69,93]. Although the focus of this chapter is not on inhomogeneity correction, it is compatible with all such schemes. The literature also presents many methods that aim at implicitly dealing with the inhomogeneities in MR data in the classification method itself [184,183,92,136,113]. For instance, Yan and Karp [184] employ an adaptive K-means clustering strategy that, over many iterations, gradually takes the feature-space points from increasingly-local neighborhoods. The initial segmentation uses all points in the image but the final segmentation implicitly accounts for local intensity variations such as those cause by the inhomogeneity field.
More recently, researchers have realized the importance of the nonstationarity of head images in tissue classification--different anatomical structures in the brain represent different image patterns, each possessing unique higher-order/Markov statistics--and several authors introduce global information in the form of anatomical atlases [165,37,142]. Typically, they use atlases in one of two ways. First is to convert the classification problem into a deformable-registration problem between the MR-image and the anatomical brain atlas. Once the registration is done, the method uses the resulting transformation to map the anatomical structure from the atlas onto the data to produce a segmentation based on the labels in the atlas. Several authors use probabilistic atlases, which are generated from ensembles of head images. These atlases encode tissue probabilities (rather than discrete label values) at each voxel, and are used as a prior in the EM estimation described previously [35]. The proposed method uses probabilistic atlases for the initialization, which is important to the success of the algorithm, and can include probabilities from atlases in the posterior estimation.