Partial Mixtures

Note that the histograms, hvox(v), for some voxel-sized regions are not ideally matched by a linear sum of basis functions. There are two possible sources of this mismatch.

The first source is the assumption that within a small region there is still normally distributed noise. N models the fact that the noise no longer averages to zero, but there is no attempt to model the change in shape of the distribution as the region size shrinks.

The second source is related. A small region may not contain the full range of values that the mixture of materials can produce. The range of values is dependent on the bandwidth of the sampling kernel function. As a result, the histogram over that small region is not modeled ideally by a linear combination of pure material and mixture distributions. Other model histogram basis functions with additional parameters can better match histograms [18,19]. Modeling the histogram shape as a function of the distance of a voxel from a boundary between materials is likely to address both of these effects and give a result with a physical interpretation that will make geometric model extraction more justifiable and the resulting models more accurate.

These two effects weight the optimization process such that it tends to make N much larger than expected. As a result, experience shows that setting w(v) to approximately 30 times the maximum value in hvox(v) gives good classification results. Smaller values tend to allow N to move too much, and larger values hold it constant. Without these problems w(v) should take on values equal to some small percentage of the maximum of hvox(v).

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