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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: Hidden Markov Random Field
Up: Segmentation of Brain MR
Previous: Segmentation of Brain MR
Yongyue Zhang
2000-05-11