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Introduction

MRI is an advanced medical imaging technique providing rich information about the human soft tissue anatomy. It has several advantages over other imaging techniques enabling it to provide 3-dimensional data with high contrast between soft tissues. However, the amount of data is far too much for manual analysis/interpretation, and this has been one of the biggest obstacles in the effective use of MRI. For this reason, automatic or semi-automatic techniques of computer-aided image analysis are necessary. Segmentation of MR images into different tissue classes, especially gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF), is an important task. Brain MR images have a number of features, especially the following: Firstly, they are statistically simple: MR Images are theoretically piecewise constant with a small number of classes. Secondly, they have relatively high contrast between different tissues. Unlike many other medical imaging modalities, the contrast in an MR image depends strongly upon the way the image is acquired. By adding RF or gradient pulses, and by carefully choosing relaxation timings, it is possible to highlight different components in the object being imaged and produce high-contrast images. These two features facilitate segmentation. On the other hand, ideal imaging conditions never realised in practice. The piecewise-constant property is degraded considerably by electronic noise, the bias field (intensity inhomogeneities in the RF field) and the partial-volume effect (multiple tissue class occupation within a voxel), all of which cause classes to overlap in the image intensity histogram. Moreover, MR images are not always high-contrast. Many T2-weighted and proton density images have low contrast between gray matter and white matter. Therefore, it is important to take advantage of useful data while at the same time overcoming potential difficulties. A wide variety of approaches have been proposed for brain MR image segmentation. These can be roughly divided into two categories: structural and statistical. Structural methods are based on the spatial properties of the image, such as edges and regions. Various edge detection algorithms have been applied to extract boundaries between different brain tissues [5,10,8]. However such algorithms are vulnerable to artifacts and noise. Region growing [6,34] is another popular structural approach. In this approach, one begins by dividing an image into small regions, which can be considered as ``seeds''. Then, all boundaries between adjacent regions are examined. Strong boundaries (in terms of certain specific properties) are kept, while weak boundaries are rejected and the adjacent regions merged. The process is carried out iteratively until no boundaries are weak enough to be rejected. This method is employed in [9] to extract brain surfaces. The segmentation tool in the commercial software package ANALYZE [28] is also based on this idea. However, as concluded by Clarke et al. in [8] critically, the performance of the method depends on seed selection and whether the regions are well defined, and therefore is also not considered robust. Starting from a totally different viewpoint, statistical methods label pixels according to probability values, which are determined based on the intensity distribution of the image. In their simplest form, thresholding-based methods are always chosen for scenes containing solid objects resting on a background with intensities well separated from the objects. However, as noted above, this generally is not effective for brain MR images. Therefore, thresholding-based methods are unlikely to produce reliable results [23,7,18]. Most statistical approaches rely on certain assumptions or models of the probability distribution function of the image intensities and its associated class labels, which can both be considered random variables. Let X and Y be two random variables for the class label and the pixel intensity, respectively, and x and y be typical instances. The class-conditional density function is p(y|x). Statistical approaches attempt to solve the problem of estimating the associated class label x, given only the intensity y for each pixel. Such an estimation problem is necessarily formulated from an established criterion. Maximum a posteriori (MAP) or maximum likelihood (ML) principles are two such examples. But before those criteria can be assessed, the formula for the density function p(y) has to be chosen carefully [4]. Many statistical segmentation methods differ in terms of models of p(y). Depending on whether a specific functional form for the density model is assumed, a statistical approach can either be parametric or non-parametric. Both have been widely used in segmentation of brain MR images. In non-parametric methods, the density model p(y) relies entirely on the data itself, i.e. no prior assumption is made about the functional form of the distribution but a large number of correctly labelled training points are required in advance. One of the most widely used non-parametric methods is K-Nearest-Neighbours (K-NN). One starts by choosing a fixed K, which is the number of nearest neighbours to find in the neighbourhood of any unlabelled pixel in the y space. Then a certain distance measure between pairs of points is applied to determine their relationship [3,15,24]. The Parzen-window method is another example of non-parametric methods, in which the intensity density function is modelled using a Parzen-window distribution. Such a distribution can be obtained by centering a small Gaussian around each training point [17,14]. Non-parametric methods are adaptive, but suffer from the difficulty of obtaining a large number of training points, which can be tedious and a heavy burden even for experienced people. Clearly, such methods are not fully automatic. Unlike non-parametric approaches, parametric approaches rely on an explicit functional form of the intensity density function. For brain MR images, the only method developed to date is based on the finite mixture (FM) model, in particular the finite Gaussian mixture (FGM) model when the Gaussian likelihood distribution is assumed [31,13,14]. FM models have a number of elegant features and are mathematically simple. However, being a histogram-based model, the FM has an intrinsic limitation - spatial information is not taken into account because all the data points are considered to be independent samples drawn from a population. Such a limitation causes the FM model to work only on well-defined images with low level of noise; unfortunately, this is often not the case due to artifacts such as the partial volume effect and bias field distortion. Under such conditions, FM model-based methods produce unreliable results. In order to address this problem, we develop a hidden Markov random field (HMRF) model, which is a stochastic process generated by a Markov random field whose state sequence cannot be observed directly but which can be observed through a field of observations. The importance of the HMRF model derives from Markov random field (MRF) theory, in which the spatial information of an image is encoded through contextual constraints of neighbouring pixels. By imposing such constraints, we expect neighbouring pixels to have the same class labels (in the case of piecewise constant images) or similar intensities (in the case of piecewise continuous images). This is achieved through characterizing mutual influences among pixels using conditional MRF distributions. To apply the HMRF model, an expectation-maximization (EM) algorithm is also derived. We show that by incorporating both the HMRF model and the EM algorithm into a mathematically sound HMRF-EM framework, an accurate and robust segmentation approach can be achieved, which is demonstrated through experiments on both simulated images and real data, and comparison made with the FM-EM framework. Being a flexible approach, the HMRF-EM can be easily combined with other techniques to improve the segmentation performance. As an example, we show how the bias field correction algorithm of Guillemaud and Brady [13] is incorporated into it. Although MRF modelling and its application in image segmentation have been investigated by many other researchers [12,2,22], only in recently years has MRF theory become popular in MR image segmentation. But most reported methods use MRF only as a general prior in an FM model-based parametric approach to build the MAP estimation. They either lack a proper parameter estimation step to fit the FM model [14,16] or the parameter estimation procedure they use, such as ML or EM [33,27,20], suffers from the limitation of the FM model mentioned above. In general, although an MRF prior can improve the performance, the FM assumption is still a big limitation. In this paper, segmentation is treated as a statistical model-based problem with three steps: (i)model selection; (ii)model fitting; (iii)classification. The HMRF model is presented in Section 2; it has all the advantages of the FM model while at the same time being more robust because of the MRF neighbourhood relationships. The HMRF-EM framework is presented in Section 5, which contains both the model fitting and classification steps. It enables adaptive and reliable automatic segmentation. The framework is easily extensible by combining other techniques, such as bias field correction, as shown in Section 7.
next up previous
Next: Hidden Markov Random Field Up: Segmentation of Brain MR Previous: Segmentation of Brain MR
Yongyue Zhang
2000-05-11