Normalized Histograms

This section presents the equation for a normalized histogram of a sampled dataset over a region. This equation will be used as a building block in several later sections, with regions that vary from the size of a single voxel to the size of the entire dataset. It will also be used to derive basis functions that model histograms over regions containing single materials and regions containing mixtures of materials.

For a given region in spatial coordinates, specified by M, the histogram hM(v) specifies the relative portion of that region where p(x) = v, as shown in Fig. 4. Because a dataset can be treated as a continuous function over space, histograms, hM(v) : R"v—R, are also continuous functions:

II Segmentation Real-World Object

Whole Data Set,, Histogram,(v)

FIGURE 7 The classification process. MR data is collected, and a histogram of the entire dataset, halI(v), is calculated and used to determine parameters of histogram-fitting basis functions. One basis function represents each pure material and one represents each mixture in the dataset. Histograms are then calculated for each voxel-sized region, hvox( v), and used to identify the most likely mixture of materials for that region. The result is a sampled dataset of material densities within each voxel.

Material Densities

FIGURE 7 The classification process. MR data is collected, and a histogram of the entire dataset, halI(v), is calculated and used to determine parameters of histogram-fitting basis functions. One basis function represents each pure material and one represents each mixture in the dataset. Histograms are then calculated for each voxel-sized region, hvox( v), and used to identify the most likely mixture of materials for that region. The result is a sampled dataset of material densities within each voxel.

Equation (1) is the continuous analogue of a discrete histogram. M(x) is nonzero within the region of interest and integrates to 1. M(x) is set constant in the region of interest, making every spatial point contribute equally to the histogram hM(v), but M(x) can be considered a weighting function that takes on values other than 0 and 1 to more smoothly transition between adjacent regions. Note also that hM(v) integrates to 1, which means that it can be treated as a probability density function, or PDF. 5 is the Dirac delta function.

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