The following two sections can be safely skipped on a first reading. They present detailed derivations and information helpful for implementing the algorithm or for creating an analogous one.
This section contains a derivation of the equations that are used to find model histogram parameters and to classify voxel-sized regions. Bayesian probability theory  is employed to derive an expression for the probability that a given histogram was produced by a particular set of parameter values in the model. This "posterior probability" is maximized to estimate the best-fit parameters:
maximize ^(parameters | histogram). (16)
The optimization procedure is used for two purposes:
• Find model histogram parameters. Initially, it is used to estimate parameters of basis functions to fit histograms of the entire dataset This results in a set of basis functions that describes histograms of voxels containing pure materials or pairwise mixtures.
• Classify voxel-sized regions. Subsequently, the optimization procedure is used to fit a weighted sum of the
The posterior probabilities P111 and Pvox share many common terms. In the following derivation they are distinguished only where necessary, using P where their definitions coincide.
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