Related Work

Many researchers have worked on identifying the locations of materials in sampled datasets [5-8]. Clarke et al. [9] give an extensive review of the segmentation of MRI data. However, existing algorithms still do not take full advantage of all the information in sampled images; there remains room for improvement. Many of these algorithms generate artifacts like those shown in Fig. 5, an example of data classified with a maximum-likelihood technique based on sample values. These techniques work well in regions where a voxel contains only a

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FIGURE 5 Discrete, maximum-likelihood (DML) classification of the same brain data shown in Figure 2. This existing method assigns each voxel to a single material class. The class is identified here by its color: gray for gray matter, blue for CSF/fluid, white for white matter, red for muscle. Note the jagged boundaries between materials within the brain and the layer of misclassified white matter outside of the skull. See Section 7 for more detail. See also Plate 17.

FIGURE 5 Discrete, maximum-likelihood (DML) classification of the same brain data shown in Figure 2. This existing method assigns each voxel to a single material class. The class is identified here by its color: gray for gray matter, blue for CSF/fluid, white for white matter, red for muscle. Note the jagged boundaries between materials within the brain and the layer of misclassified white matter outside of the skull. See Section 7 for more detail. See also Plate 17.

single material, but tend to break down at boundaries between materials. In Fig. 5, note the introduction of both stair-step artifacts, as shown between gray matter and white matter within the brain, and thin layers of misclassified voxels, as shown by the white matter between the skull and the skin. Both types of artifacts can be ascribed to the partial-volume effects ignored by the segmentation algorithms and to the assignment of discrete material types to each voxel.

Joliot and Mazoyer [10] present a technique that uses apriori information about brain anatomy to avoid the layers of misclassified voxels. However, this work still produces a classification where each voxel is assigned to a single, discrete material; results continue to exhibit stair-step artifacts. It is also very dependent on brain anatomy information for its accuracy; broader applicability is not clear.

Drebin et al. [11] demonstrate that accounting for mixtures of materials within a voxel can reduce both types of artifacts, and approximate the relative volume of each material represented by a sample as the probability that the sample is that material. Their technique works well for differentiating air, soft tissue, and bone in CT data, but not for differentiating materials in MR data, where the measured data value for one material is often identical to the measured value for a mixture of two other materials.

Windham etal. [12] andKao etal. [13] avoid partial-volume artifacts by taking linear combinations of components of vector measurements. An advantage of their techniques is that the linear operations they perform preserve the partial-volume mixtures within each sample value, and so partial-volume artifacts are not created. A disadvantage is that the linear operations are not as flexible as nonlinear operations, and so either more data must be acquired or classification results will not be as accurate.

Choi et al. [14] and Ney et al. [15] address the partial-volume issue by identifying combinations of materials for each sample value. As with many other approaches to identifying mixtures, these techniques use only a single measurement taken within a voxel to represent its contents. Without the additional information available within each voxel region, these classification algorithms are limited in their accuracy.

Santago and Gage [16] derive a distribution of data values taken on for partial volume mixtures of two materials. The technique described here shares the distribution that they derive. Their application of the distribution, however, fits a histogram of an entire dataset and then quantifies material amounts over the entire volume. In contrast with this work, they represent each voxel with a single measurement for classification purposes, and do not calculate histograms over single voxels.

Wu et al. [17] present an interesting approach to partial-volume imaging that makes similar assumptions about the underlying geometry being measured and about the measurement process. The results of their algorithm are a material assignment for each subvoxel of the dataset. Taken collectively, these multiple subvoxel results provide a measure of the mixtures of materials within a voxel but arrive at it in a very different manner than is done here. This work has been applied to satellite imaging data, and so their results are difficult to compare with medical imaging results, but aspects ofboth may combine well.

Our earlier work gives an overview of the technique presented below in the context of the Human Brain Project [18], gives a complete description [19] and describes an imaging protocol for acquiring MRI data from solids and applies the classification technique to the extraction of a geometric model from MRI data of a human tooth [20] (see Fig. 11).

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