Unified segmentation
Introduction
Segmentation of brain images usually takes one of two forms. It can proceed by adopting a tissue classification approach, or by registration with a template. The aim of this paper is to unify these procedures into a single probabilistic framework.
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The first approach rests on tissue classification, whereby voxels are assigned to a tissue class according to their intensities. In order to make these assignments, the intensity distribution of each tissue class needs to be characterised, often from voxels chosen to represent each class. Automatic selection of representative voxels can be achieved by first registering the brain volume to some standard space, and automatically selecting voxels that have a high probability of belonging to each class. A related approach involves modelling the intensity distributions by a mixture of Gaussians, but using tissue probability maps to weigh the classification according to Bayes rule.
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The other approach involves some kind of registration, where a template brain is warped to match the brain volume to be segmented (Collins et al., 1995). This need not involve matching volumes, as methods that are based on matching surfaces (MacDonald et al., 2000, Pitiot et al., 2004) would also fall into this category. These approaches allow regions that are pre-defined on the templates to be overlaid, allowing different structures to be identified automatically.
A paradigm shift is evident in the field of neuroimaging methodology, away from simple sequential processing, towards a more integrated generative modelling approach. An example of such an approach is the recent work by Fischl et al. (2004), with which this paper should be compared. Both papers combine tissue classification, bias correction, and nonlinear warping within the same framework. Although the integrated approaches have some disadvantages, these should be outweighed by more accurate results. The main disadvantage is that the approaches are more complex and therefore more difficult to implement. Implementation time is longer, more expertise is needed and the code becomes less accessible. In addition, the algorithms are more specialised, making it more difficult to mix and match different programs within “pipeline” procedures (Fissell et al., 2003, Rex et al., 2003, Zijdenbos et al., 2002). A perceived disadvantage of these combined models is that execution time is longer than it would be for sequentially applied procedures. For example, optimising two separate models with 100 parameters is likely to be faster than optimising a combined single model with 200 parameters. However, the reason a combined model takes longer to run is because it actually completes the optimisation. There are usually conditional correlations among parameters of the different models, which sequential processing discounts. The advantage of large models is that they are more accurate, making better use of the information available in the data. Scanning time is relatively expensive, but computing time is relatively cheap. Complex models may take longer to run, but they should add value to the raw data.
Many investigators currently use the tools within SPM1 for a technique that has become known as “optimised” voxel-based morphometry (VBM) (Good et al., 2001). VBM performs region-wise volumetric comparisons among populations of subjects. It requires the images to be spatially normalised, segmented into different tissue classes, and smoothed, prior to performing statistical tests. The “optimised” pre-processing strategy involves spatially normalising subjects' brain images to a standard space by matching grey matter in these images to a grey matter reference. The historical motivation behind this approach was to reduce the confounding effects of non-brain (e.g., scalp) structural variability on the registration. Tissue classification in SPM requires the images to be registered with tissue probability maps (Ashburner and Friston, 1997). After registration, these maps represent the prior probability of different tissue classes being found at each location in an image (Evans et al., 1994). Bayes rule can then be used to combine these priors with tissue type probabilities derived from voxel intensities to provide the posterior probability.
This procedure is inherently circular, because the registration requires an initial tissue classification, and the tissue classification requires an initial registration. This circularity is resolved here by combining both components into a single generative model. This model also includes parameters that account for image intensity nonuniformity, although it is now fairly standard to include intensity nonuniformity correction in segmentation (Wells III et al., 1996a) and registration (Friston et al., 1995, Studholme et al., 2004) methods. Estimating the model parameters (for a maximum a posteriori solution) involves alternating among classification, bias correction, and registration steps. This approach provides better results than simple serial applications of each component.
Section snippets
The objective function
In this section, we describe the model and how it is used to define an objective function. In the next section, we will show how this function is used to estimate the parameters of interest. The objective function minimised by the optimum parameters is derived from a mixture of Gaussians model. We show how this objective function can be extended to model smooth intensity nonuniformity. Tissue probability maps are used to assist the classification, and we describe how the objective function
Optimisation
This section describes how the objective function from Eqs. (14), (16) is minimised (i.e., how the model is fitted). There is no closed form solution for finding the parameters, and optimal values for different parameters are dependent upon the values of others. An Iterated Conditional Modes (ICM) approach is used. It begins by assigning starting estimates for the parameters and then iterating until a locally optimal solution is found. Each iteration involves alternating between estimating
Evaluation
Generally, the results of an evaluation are specific only to the data used to evaluate the model. MR images vary a great deal with different subjects, field strengths, scanners, sequencies etc, so a model that is good for one set of data may not be appropriate for another. For example, consider intra-subject brain registration, under the assumption that the brain behaves like a rigid body. If the scanner causes no distortion and computes the pixel sizes and slice thickness of the image volumes
Discussion
This paper illustrates a framework whereby tissue classification, bias correction, and image registration are integrated within the same generative model. Our objective was to explain how this can be done, rather than focus on the details of a specific implementation. The same framework could be used for a more sophisticated implementation. When devising a model, it is useful to think about how that model could be used to generate data. The distribution of randomly generated data should match
Acknowledgment
This work was funded by the Wellcome Trust.
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Commentary
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An MRI-based probabilistic atlas of neuroanatomy
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