This section describes the process of classifying each voxel. This process is similar to that described in Section 5 for fitting the histogram basis functions to the entire dataset histogram, but now histograms taken over small, voxel-sized regions are being fitted. The previously computed histogram basis functions calculated from the entire dataset histogram are used. The mean vector c and standard deviation s are no longer varied. The only parameters allowed to vary are the relative material volumes (vector avox) and an estimate of the local noise in the local region (vector N) (see Eqs. (6) and (7)).
Over large regions including many voxels, the noise in p(x> is normally distributed, with zero mean; however, for voxel regions the noise mean is generally nonzero. This is because normally distributed noise is added to each sample value, not to each point of p(x>. When the samples are used to reconstruct p(x>, the values p(x> takes on near a particular sample tend to be similar and so have a non-zero mean. The local mean voxel noise value is labeled N. As derived in Section 9, the equation that is minimized, with respect to avox and N, is
2(v; avox- N> = kvox(v - N>- £ ajoxf(v>, (7>
the minimization is subject to the constraints nf
and vector a is the standard deviation of the noise over the entire dataset. For MR data the standard deviations in the signals for different materials are reasonably similar, and a is estimated to be an average of the standard deviations of the histogram basis functions.
With optimal vector avox for a given voxel-sized region and the mean value, vector v, within that region, the amount of each pure material contributed by each mixture to the voxel is estimated. This is the output, estimates of the amount of each pure material in the voxel-sized region:
v contains the mean signature of the portion of the histogram that arises only from regions with partial-volume effects. The algorithm determines how much of each pure component of pairwise mixture materials would be needed to generate v, given the amount of each mixture that avox indicates is in the voxel. tk represents this relative amount for mixture k, with tk = 0 indicating that the mixture contains only the first pure component, tk = 1 indicating that it contains only its second component, and intermediate values of tk indicating intermediate mixtures. The tk values are calculated by minimizing the following equation with respect to t, subject to the constraint 0 < tk < 1:
Vector cka is the mean value for the first pure material component of mixture k, and vector ckb the mean value for the second component. The total amount of each material is the amount of pure material added to the tk-weighted portion of each mixture.
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