British Journal of Radiology (2004) 77, S114-S125
© 2004 British Institute of Radiology
doi: 10.1259/bjr/11445826
Voxel based analysis of tissue volume from MRI data
N A Thacker, PhD
D C Williamson, PhD
and
M Pokric, PhD
Imaging Science and Biomedical Engineering, Stopford Building, University of Manchester, Oxford Road, Manchester M13 9PT, UK
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Abstract
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There are many techniques available for the analysis of MRI data. Often these methods are presented as completed algorithms, which specify what processing must be performed, but they are rarely presented in a way which makes clear the assumptions that must hold in order that these algorithms will provide valid results. The aim of this review article is to relate the common forms of algorithms and to explain the assumptions behind them. This is done in the context of the use of quantitative statistical methods, which we understand to be the only self-consistent method for any data analysis. We hope that this will go some way towards helping with the choice of which algorithm to use for particular analysis tasks.
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Introduction
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When attempting to segment data in an image we often start from a preconception based upon our own perceptive abilities. In order to make progress we must be able to identify a computational goal. The most direct piece of information that can be obtained from an image is the conditional probability, P(C|D), that interpretations C is made given the data D. Given this probability for each pixel in an image, it becomes possible to identify different tissues or locate the boundaries between them. However, probabilities do not provide definitive answers. The segmentation of data contained in an image is a problem that has no perfect solution. In order to ensure optimal segmentation of a data set by a computer program, the program algorithm should be based on well-founded statistical principles.
Many segmentation methods seem to be based around calculations, which, superficially appear to have nothing in common with either probability theory or statistics. This is rarely the case and in order to compare different segmentation schemes it becomes necessary to relate each method to its underlying statistical models. If the algorithm chosen to determine these probabilities is appropriate to the statistical characteristics of the image then the structures identified in this manner can be consider as an optimal description. That is to say that all the useful information pertaining to the segmentation problem has been extracted from the given data. Determining whether an algorithm is appropriate amounts to verifying whether the assumptions underlying the statistical framework employed are valid. To make such an assessment requires that the assumptions themselves are known. Some areas of algorithmic research in image processing are fundamentally based on matching assumptions to data sets and though it is possible to develop successful algorithms in a trial and error manner, it is far more efficient to apply a hierarchical statistical methodology and systematically test the effects of each assumption on the results.
As a consequence of the wide variety of imaging modalities (Figure 1
), even within the field of MR, and the varied forms of information required from a data set, it is quite clear that there is no "silver bullet" algorithm that can be expected to work on all image data. In fact, for a particular method to work, it should only be applied to data sets that conform to the assumptions inherent within the algorithm itself. In general, the range of applicability of an algorithm will decrease as the number of assumptions increases. This observation suggest that algorithms developed by minimizing the number of assumptions made about the data will always perform better over a wider range of data types. However, the most simple algorithms often make the most assumptions regarding the expected behaviour of the data, though poorly designed algorithms may be both complex and still make many assumptions. The balance between algorithmic complexity and the validity of data assumptions is of prime importance during the development of a new technique.
The various approaches employed in tissue segmentation algorithms that are used to identify tissue regions and tissue boundaries are described in the following sections. Technical information, for the reader who is interested in the mathematics, is given in the appendices.
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Understanding image segmentation
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The first assumption that we will make about the image formation process is that to a very good approximation the grey-level values in an image are generated through a linear process. In other words, the signal in a pixel due to a particular tissue is directly proportional to the relative fraction of that tissue in the imaging voxel [1]. The details of this approximation are given in Appendix A. The second assumption made is that the instance of voxels in an image that contain more than two distinct tissues is negligible. A number of authors have investigated partial volume approximations [2–6] and in general conclude that while it is necessary to include partial volumes, the two-tissue only assumption is quite satisfactory. In addition to these assumptions about the image formation process, we will assume that the image noise is zero mean Gaussian, also known as white noise. In effect this assumptions describes the shape of the expected distribution of grey-level value around the ideal or noiseless value for a pure tissue or tissue mixture.
From these assumptions it follows that a particular grey level corresponds to a given tissue mixture, which in turn lends justification to rather simple methods for tissue boundary identification such as grey-level thresholding. In this approach, pixel label assignment is made according to the grey level falling above or below a specified grey level threshold. A 50% fraction of two tissues defines the most likely location of the actual tissue boundary. Such an approach is used as the basis for many visualization methods that require the identification of surfaces, e.g. "marching cubes" [7].
Such an algorithm provides an adequate methods of identifying tissue boundaries provided that the boundary between the same two tissues are always required and that there are no other image formation processes (see field inhomgeneity below) that make the tissue boundaries non unique. Unfortunately, the majority of images do not conform to these rather stringent conditions.
The assumption that a data set is composed solely from two tissues is the most likely source of erroneous segmentation using the threshold approach. As an example Figure 2a shows the results of thresholding the image in Figure 1b
. The threshold was set in an attempt to identify voxels that contain grey matter. It is quite clear that regions close to the skull have been identified in addition to grey matter.
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Figure 2. Common methods for segmentations and boundary identification in a normal brain, (a) thresholding, (b) edge detection, (c) linear solution and (d) Bayesian estimation.
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A technique for boundary location, which will work with multiple tissues, is based closely on the way that humans perceive image data. The idea involves identifying the boundaries between otherwise homogeneous regions. This is achieved by locating the position of maximum contrast, or the "edges". So-called edge detectors are more common in the field of machine vision than medical image analysis [8] and are based on computing the local spatial derivatives of an image after smoothing. Taking the peak of this derivative defines the maximum transition point between the tissues and this corresponds with the boundary point for symmetrical step edges where the labelling probabilities for each tissue are 50%, see Appendix B for details of a simple edge detector. Complete extractions of such boundaries would facilitate the unambiguous labelling of all regions in an image. The edge detector method works well on data that has well defined, sharp boundaries. Unfortunately, the more slowly varying partial volume regions that commonly occur in MR data can undermine the sharp boundary assumption. The issue is a particular problem for thick slices where as many as 40% of voxels may be affected [6]. Typical results obtained by applying an edge detector to the image in Figure 1b are shown in Figure 2b. It is evident that not all the tissue boundaries have been located; in particular those between grey matter and white matter. This failure occurs because edge detectors are, in essence, a hypothesis test and the ability to identify edges is a function of the local contrast to noise ratio (CNR). In the brain the partial volume process significantly reduces the CNR for any boundary, which is not close to being perpendicular to the plane of acquisition, leaving gaps in the boundary.
Partial volume effects of multiple tissues can be compensated for but overcoming CNR limitations may require more information than is present in a single image. This additional information can be provided by the user, one technique "livewire" has now been implemented in several software packages. This technique aids manual mark-up of tissue boundaries by attempting to link through detected edge locations, but leaves the user to selected between possible routes via a mouse interaction. Typical results from a commercially available package are shown in Figure 3.
Other segmentation techniques have been developed which incorporate information from a specific pair of imaging sequences [9]. In fact, the general solution for the proportion of each of N tissues within a voxel is an exercise in linear algebra and requires N–1 images, see Appendix C. An approach that incorporates data from multiple images through the linear equations will provide unbiased estimates of tissue proportion [10]. Typical results using a three-tissue model applied to the data in the images show in Figures 1a and b are shown in Figure 2c. It is important to note that the data has been skull stripped, i.e. all pixels outside the brain have their grey levels set to zero. This is because the method can only provide accurate estimates for tissues that are explicitly included in the model, in this example grey matter (GM), white matter (WM) and cerebrospinal fluid (CSF). In effect this limitation means that the method cannot deal with unexpected behaviour, in this case the data outside the skull. However, from a medical point of view this also amounts to restricting applicable data sets to ones that contain only health tissue. Furthermore, the assumption of a pure linear model is equivalent to the statistical assumption of noiseless data. Consequently, tissue estimates obtained using this method will be noisy, with a typical random uniform Gaussian noise level of around 15%. Such errors result in tissue proportional estimates that fall outside the range 0–100%. Dealing with both these problems requires an approach to the data analysis that is more firmly rooted in statistics.
The application of probability theory to the data modelling provides the most direct route. In this approach a likelihood model is constructed for each tissue component present in the data (Figure 4). A common implementation, available in most image analysis packages, models only the pure tissue distributions. These distributions generate probabilities that correspond to the most likely tissue label given the data. The major flaw in this implementation is the absence of partial volume contributions in the model. In order to model the partial volume a suitable model distribution must be chosen. Modelling of partial volumes using additional Gaussian distributions turns out to be both inappropriate and unstable [2]. In addition, to take into account the image formation process it turns out that it is not sufficient to generate only pure tissue labelling probabilities, rather the probabilities of most likely tissue volume fraction must be computed [5]. This subtle change, which might be viewed as being at odds with the approach taken in a majority of papers in the area, is entirely consistent with earlier observations regarding segmentation on the basis of tissue volume and entirely necessary if partial volume effects are to be dealt with correctly.
The parameters in the probability density model must be determined from the data using an optimization algorithm to minimize the difference between the model and the data. The SIMPLEX algorithm [11] and the expectation maximization [12] (EM) are commonly used for this purpose. Estimation of relative volumetric tissue proportions can then be made by direct use of Bayes theory, see Appendix D. The probabilities obtained are precisely those described in the introduction. If the derived model of the data is an accurate representation of the data then the results obtained using this technique is the most probable tissue volume fraction within each voxel for data comprised of tissues with the same prior proportions. This is the optimal solution to the problem of segmenting data on the basis of voxel grey level in terms of the minimal error on the volume estimates. This technique will give accurate volume estimates for data comprising multiple tissues (N) in a single image, provided that the CNR between pairs of tissues is sufficiently large. This is in contrast to the previous solution of linear equations method, which always requires N–1 images. The method can be extended to include data from multiple images. The inclusion of the additional independent data removes ambiguities between tissues where the CNR is small. This extension, however, requires a slightly more complicated analysis in order to determine all the parameters in the multidimensional model. The technique can be extended further to deal with pathological tissues. Such tissues can be incorporated through the introduction of an additional category to the model that accounts for infrequently occurring data [13], see Appendix E. Typical results for the segmentation of GM using this technique are shown in Figure 2d. Here the data from all the images in Figure 1 have been used to determine the parameters in a multidimensional model. The model itself takes into account six different tissue types including, fat, muscle, bone/air, as well as GM, WM and CSF. Unlike the solution of linear equation approach, this method does not require a preliminary skull-stripping step to acquire a reasonably unambiguous segmentation. Clearly there has been some erroneous pixel labelling around the skull, this occurs because within the four-dimensional data space some ambiguities between tissue still remain. That is to say, some tissues have similar grey level values in all the four images.
Bayesian approaches may be further extended to make use of prior knowledge regarding the expected spatial location of tissues, i.e. the fact that GM should not be found outside the brain, or correlations between structures. This is simply introduced by including additional prior probability terms and can result in improvements in the appearance of segmentation. The use of Markov random fields as a method of modelling spatial distributions has become quite popular. However, the common use of Gibbs distributions, a methodology taken from theoretical physics, is quite inappropriate for medical data analysis. Poole [14] describes the most appropriate approach, which is based upon quantitative use of probability theory. His method involves predicting the most likely interpretation of a central voxel given its neighbours based upon a statistical example of sample data. An approximation, suitable for MR data, based upon characterizing the local structure using the grey level slope is described by Williamson et al [15], see Appendix F. This approach can be thought of as combining the information available from the edge detection methods with that available in the raw grey level data. In an axial slice through human leg (Figure 5a) three tissues are readily visible. The darkest is cortical bone, the intermediate is muscle and the brightest tissue is fat. Fatty tissue is evident on the outside of the leg, in the centre of the bone (marrow) and in a phantom containing vegetable oil. The results of segmenting muscle using only the grey level information are presented in Figure 5b. The segmentation has performed reasonably well in identifying regions of muscle. However, it is quite clear that some pixels at the boundaries between air and fat and cortical bone and fat have grey level values within the range expected for muscle. Consequently the grey level segmentation cannot distinguish these from muscle. In contrast, inclusion of the local slope information provides a method of distinguishing between pure muscle tissue and partial volume effects (Figure 5c). Voxels that previously could not be correctly labelled based upon the grey level information can now be unambiguously attributed to the correct tissue component. Similar results were obtained for brain images [15] in which partial volumes from bone and fat can produce grey-levels that are consistent with GM. Figure 6a shows a typical slice through a human brain using an inversion recovery (IR) sequence. The results of segmenting GM using only the grey level information are shown in Figure 6b a thin rim of mislabelled pixels can be seen close to the edge of the brain and surrounding the ventricles. These are attributable to partial volume effects. The results from the GM segmentation using both grey level information and edge information are shown in Figure 6c. In this image the previously mislabelled pixels are no longer attributed to pure grey matter.
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Figure 6. (a) Tissue segmentation in the normal brain, (b) without and (c) with the use of local slope information.
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The commonly used technique of introducing additional prior terms can produce biases in any quantitative measurement and great care needs to be taken [16]. This is particularly the case when the prior terms are based on a "normal atlas". In such cases it is possible that the segmentation becomes driven by the prior terms rather than the data and biases will be introduced into any estimation of spatial distribution of tissue. A more complete description of this and other problems with the practical application of Bayesian methods is given by Bromiley et al [17].
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Dealing with data inhomogeneities
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In the previous section the issues of noise, partial volumes and pathological data have been discussed, especially with respect to the breakdown of methodological assumptions. In this section we explore the breakdown in the assumption of a pure linear model for the image formation process due to spatially varying nominal tissue data values. This can occur for one of two reasons. First, the tissue itself has physical properties that genuinely vary depending on location. Second, the values appear to vary due to inhomogeneities in the measurement system. This latter problem is exemplified in MR data acquired using a surface coil. Examples are shown in Figures 7a and b. It can be corrected if we can determine the spatially varying sensitivity, or gain, of the system, which can be used to create a multiplicative correction image. The problem may be exacerbated if the shimming across a sample volume of interest (VOI) is poor, resulting in inhomogeneities in static magnetic field. Correcting for such field distortions is a difficult thing to achieve reliably, particularly in the presence of partial volume effects. In general there is no substitute for minimizing the problem by using a well-shimmed, homogeneous acquisition method.
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Figure 7. Correction of field inhomogeneity on the assumption of homogeneous tissue regions in the shoulder and around the orbit.
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Of the algorithms discussed in the previous section, only the edge detector methods could conceivably deal with this problem without modification. This is purely because the edge detectors operate at a local level in the image. However, as mentioned previously the edge detectors rely on very good CNR characteristics, which may be somewhat diminished in "low signal" regions of the image. Thus a method for gain correction may still be necessary in some circumstances.
There are two general approaches to determining a gain correction. The first involves building a low parameter model for the expected gain correction into the solution for the label probabilities. These parameters can then be adjusted via an optimization process that minimizes the variance of the pure tissues [18]. This approach has a number of drawbacks. First, the correct parametric function for a given correction image may not be known and it would be easy to assume a functional form that is an inappropriate description of the true data characteristics. Second, the approach cannot work well if there are regions of tissue in the data that are not described by the underlying tissue model. This effectively excludes images that contain pathological tissues. Finally, the determination of the statistically optimal set of parameters can only be achieved through an iterative mechanism. Such iterative processes performed on data sets of typical size are generally slow and unreliable.
An alternative approach can be developed from the assumption that the image is composed of homogeneous regions of tissue separated by two tissue partial volume boundaries. Provided that partial volume regions can be detected, using a local contrast or derivative estimate, the local relative gain change across a voxel in a uniform region can be estimated. Applying this to images from a 3D volume data set will generate two relative gain change estimates, an image of in-slice, horizontal changes and a single value relating interslice, vertical changes. The local specific signal and noise characteristics will result in some regions in which no estimates are obtained and others where the relative gain estimates will be more accurate than others. However, spatial gain variations are expected to be locally smooth. This can be incorporated directly into the correction methodology. By smoothing the local gain change by an amount less than the expected level of smoothness in the inhomogeneity we can fill in the missing data and increase the accuracy and stability of the local estimates. This must be carried out using an appropriate statistical calculation, which also takes into account measurement accuracy. The relative horizontal and vertical gain changes obtained can subsequently be used to compute, via integration, an estimate of the original gain variation in the image. Figures 7c and d show the results of applying such a method to the images in Figures 7a and b. The details of this method are given in Appendix G.
In comparison with the iterative parametric approach, this technique is simple, fast, and reliable and does not have to assume a particular form for the gain variation [19]. In addition to these advantages, it is also capable of dealing with pathological data provided those regions are homogeneous or have high spatial derivatives and are therefore excluded as image discontinuities. For large regions of slight inhomogeneity there will be a distortion of the local estimated gain variation due to the use of the regularization term 
reg.
As with any other data analysis task, correction of field inhomogeneities will only work reliably if the statistical characteristics of the data conforms to the assumptions made during the algorithm design. Unfortunately, this is not an easy thing to determine automatically and inevitably there will be some classes of images, which cannot be corrected, whether this is due to problems with the data acquisition or due to the anatomy of the subject. It is possible to perform some validation of the correction process by checking whether the output data have been "improved" in some way. This is achievable using measures based loosely upon concepts of information. It is important to note, however, that we do not believe that this approach is an appropriate measure with which to design the correction process [20].
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Conclusions
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In this paper a number of common approaches to voxel based MRI segmentation have been presented. The process of algorithm design has been discussed with particular focus on the importance of the underlying assumptions and the validity of those assumptions given a particular data set. The examples and discussion highlight the fact that segmentation goes beyond simply assigning tissue labels to pixels, but is in fact related to identifying the underlying most probable image formation mechanism, e.g. tissue volume estimates.
The overall strategy for selecting a segmentation method based on the characteristics of the image data is summarized in Figure 8. Answers to each of the questions in bubbles determine the appropriate approach for a given data set. One component of algorithm evaluation can be carried out at a theoretical level, based on what variablities must be accounted for in the data model. The diagram specifies the tests that should be carried out in order to confirm the suitability of an algorithmic approach. Clearly, more formal evaluation is also necessary for any subset of algorithms that were subsequently identified as suitable. This formal testing establishes the best method for a given data set.
The path of dotted lines shows the most general approach to tissue segmentation that deals with all the most commonly occurring problems associated with typical MR data sets. It includes both gain correction and Bayes estimation of conditional probabilities. In general this path can be implemented as a quick and reliable algorithm and as such is likely to be the preferred description of general image segmentation method. However, it is important to note that there is no single method for tissue segmentation that is generally applicable to all data sets. In some cases simple thresholding may be adequate, yet in other even the most refined and sophisticated methods may fail. With this in mind it is apparent that while algorithms need to be designed with great care, the data should be acquired with equal care and attention. The problem posed by field inhomogeneity with respect to quantitative analysis highlights just this point.
The problems of selecting and integrating prior knowledge into the Bayesian formalism were touched upon in the discussion. Since the arbitrary introduction of multiple prior terms may lead to biases in the volumetric estimates obtained, caution is advised. How much of the information from edge boundaries, shape and expected location can be legitimately combined in order to perform the quantitative task remains a highly active area of research in the image processing community. All of the analysis techniques discussed in this paper are available within the TINA freeware package distributed from our website [21].
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Appendix A: Image formation in MR
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The expression for the signal intensity of a pure tissue in an inversion recovery spin-echo (IRSE) sequence follows directly from the Bloch equation
where N(H) is the spin density, TE is the echo time, TR is the repetition time, TI is the inversion time and T1 and T2 are the usual characteristic relaxation times for the tissue.
It is clear from this expression that the signal intensity has a linear dependence on the spin density. Since the spin intensity has a linear dependence it is possible to write the grey level, g, within a voxel as a linear combination of partial volume contributions,
where PN is the partial volume contribution from the Nth tissue component whose mean tissue grey level is GN. The partial volumes in a single voxel being normalized as
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Appendix B: A simple edge detector
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Detection of edge boundaries in two dimensions is performed in a manner similar to the following:- Convolution of the image with a spatial noise reduction filter, e.g. a Gaussian,
- Calculation of the local image gradients
- Calculation of the edge strength,
- Identification of edges as those voxels with an edge strength above a statistical threshold E(x,y)>k, and less than no more than two of its neighbours, which implies simple linear connectivity.
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Appendix C: Direct estimation of partial volume fraction using linear algebra
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The three linear equations for a grey level values in two images and the total proportion constraint can be solved for each of three tissues, e.g. grey matter, white matter and cerebrospinal fluid, (g,w,c), within each voxel
and
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Appendix D: Bayes classification of grey level values for three tissues
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The total probability of getting a particular set of grey level values (g) within a region of the image comprising three tissues (1–3) and two sets of partial volumes (12, 21 and 23 and 32) can be written as
The probability distributions of the pure tissues are modelled as Gaussian distributions with means µ1–µ3 and equal standard deviations 
. The partial volume distributions are modelled as a convolution of a triangular distribution and a zero mean Gaussian with standard deviation, . The full set of parameters fn, µ1–µ3 and are determined by minimizing the difference between the model distribution and a histogram of grey level occurrence constructed from the image data.
At the given grey level g, the separate likelihood components for each of the tissues can be written as:
Bayes theory can now be used to compute the conditional probability of a tissue given the grey level g,
The inclusion of the partial volume terms as demonstrated here means that the P(n|g) is no longer simply the probability of the label of the voxel. It is now an estimate of the mean volumetric contribution to the formation of the signal in a voxel with grey level g.
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Appendix E: Multispectral modelling with an unknown tissue
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For multispectral data with n images, there is no longer a single grey level value, but rather a data vector describing all the grey levels associated with a particular tissue, 
As a result, the tissue distributions discussed in Appendix D are multivariate distributions
Where 
is the mean tissue grey level vector, Ct is the covariance matrix and 
t is chosen to give unit normalization. Partial volume distributions can be modelled along the line between two pure tissues with mean vectors 
.
with
and Pts(h) is the one dimensional partial volume distribution described in Appendix D. Parameters for the model can be iteratively estimated by taking weighted averages over the selected volume V using a method generally referred to as expectation maximization (EM)
Unknown tissues are included in the Bayesian formulation by including a fixed extra term, fo in 
to deal with outlying data points.
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Appendix F: Bayesian tissue classification with local grey-level slope prior terms
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In this method, the distribution of grey levels in an image is modelled in the same manner as Appendix D. In addition to this the local grey level slope is determined in the x and y directions and the square root of sum of the square of these values calculated. The distribution of the slope at a given grey level g is modelled as the sum of a Rayleigh distribution, representing local slope due to noise in regions of uniform grey level and a Rician distribution describing the slope distribution for each pair of boundary tissues, i.e. for an image with three tissues and two boundary pairs
with
Here, I0(x) is the modified zero-th order Bessel function of the first kind, is the standard deviation of the Gaussian noise in the original images and A(g) is the slope-image pixel value in the absence of noise, and the Rayleigh distribution is defined as
Combining the grey level distribution with the slope distribution gives the following likelihoods
Ptot(g,s) is given as the sum of the likelihoods at a given pair grey level and slope values. The corresponding Bayesian conditional probability is given as
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Appendix G: Gain correction using local slope
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Calculation of a smooth correction image can be carried out as follows;
- Estimation of local image noise
- Estimation of local image relative grey level slopes
and variances 
.
- Maximum likelihood estimation of smoothed local derivative using statistical averaging with a stability term for missing data reg (points with large slopes relative to the noise) that assumes no image slope 0reg.
- Integration of these derivatives along any path L from l0 to l=(x,y) can be written as
defining the starting point as unity gain gives the relative gain factor to that point F(x,y)
The regularization for missing data requires the iteration of the algorithm a few times for data with large regions of indeterminate slope.
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Acknowledgments
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One of us (DCW) would like to thank the Medical Research Council for the award of a special training fellowship in Bioinformatics. Voxar Ltd., UK provided the Livewire data. This document was also prepared using resources from the EU under the Performance Characterisation in Computer Vision (PCCV) project (IST 1999 14159).
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- http://www.tina-vision.net
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Abstract
Introduction
