Computing Voxel Histograms

Histograms can be calculated in constant-sized rectangular "bins," sized such that the width of a bin is smaller than the standard deviation of the noise within the dataset. This ensures that significant features are not lost in the histogram.

The bins are first initialized to zero. Each voxel is subdivided into subvoxels, usually four for 2D data or eight for 3D data, and p(x) and its derivative evaluated at the center of each subvoxel. p(x) is interpolated from the discrete data using a tricubic B-spline basis [22] that approximates a Gaussian. Thus, function and derivative evaluations can be made not only at sample locations, but anywhere between samples as well. From the function value and the derivative, Eq. (1) is used to calculate the contribution of a linear approximation of p(x) over the subvoxel to each histogram bin, accumulating the contributions from all subvoxels. This provides a more accurate histogram than would be obtained by evaluating only the function values at the same number of points.

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