David H. Laidlaw 1 Introduction 195

Biown University 1.1 A Partial-Volume Classification Approach Using Voxel Histograms • 1.2 Related Work

Kurt W. Fleischer 2 °verview 198

„. . . .. .. 2.1 Problem Statement • 2.2 Definitions • 2.3 Assumptions • 2.4 Sketch of Derivation

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3 Normalized Histograms 200

Alan H. Barr 3.1 Computing Voxel Histograms

Cahfornia institute of 4 Histogram Basis Functions for Pure Materials and Mixtures 201

5 Estimating Histogram Basis Function Parameters 201

6 Classification 202

7 Results 203

8 Derivation of Histogram Basis Functions 205

8.1 Pure Materials • 8.2 Mixtures

9 Derivation of Classification Parameter Estimation 206

9.1 Definitions • 9.2 °ptimization • 9.3 Derivation of the Posterior Probability, P(a, c, s, N\h)

10.1 Mixtures of Three or More Materials • 10.2 Mixtures of Materials Within an °bject • 10.3 Benefits of Vector-Valued Data • 10.4 Partial Mixtures • 10.5 Nonuniform Spatial Intensities • 10.6 Quantitative Comparison with °ther Algorithms • 10.7 Implementation

11 Conclusions 209

The distribution of different material types can be identified in volumetric datasets such as those produced with magnetic resonance imaging (MRI) or computed tomography (CT). By allowing mixtures of materials and treating voxels as regions, the technique presented in this chapter reduces errors that other classification techniques can create along boundaries between materials and is particularly useful for creating accurate geometric models and renderings from volume data. It also has the potential to make more accurate volume measurements and to segment noisy, low-resolution data well.

There are two unusual aspects to the approach. First, it uses the assumption that, because of partial-volume effects, or blurring, voxels can contain more than one material, e.g., both muscle and fat; it computes the relative proportion of each material in the voxels. Second, the approach incorporates

Based on "Partial-Volume Bayesian Classification of Material Mixtures in MR Volume Data Using Voxel Histograms" by David H. Laidlaw, Kurt W. Fleischer, and Alan H. Barr, which appeared in IEEE Transactions on Medical Imaging, Vol. 17, No. 1, pp. 74-86. © 1998 IEEE.

information from neighboring voxels into the classification process by reconstructing a continuous function, p(x), from the samples and then looking at the distribution of values that p(x) takes on within the region of a voxel. This distribution of values is represented by a histogram taken over the region of the voxel; the mixture of materials that those values measure is identified within the voxel using a probabilistic Bayesian approach that matches the histogram by finding the mixture of materials within each voxel most likely to have created the histogram. The size of regions that are classified is chosen to match the spacing of the samples because the spacing is intrinsically related to the minimum feature size that the reconstructed continuous function can represent.

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