13 Partial Volume Segmentation with Voxel Histograms TABLE 6 Definitions of terms used in the derivations
Term Dimension Definition nm Scalar Number of pure materials
«f Scalar Number of pure materials and mixtures nv Scalar Dimension of measurement (feature space)
a «f Relative volume of each mixture and material within the region c «f x nv Mean of material measurements for each material s «f x nv Standard deviation of material measurements (chosen by procedure discussed in Section 5) for each material
N «v Mean value of noise over the region
_p1_6 Scalars Arbitrary constants
^(v) R«v -^R Histogram of an entire dataset fovox(v) R«v -^R Histogram of a tiny, voxel-sized region fo if£j£i a = 1 P(a) = < 0 if a;- <0 or a;- > 1. (20)
[ (constant) otherwise
A different assumption is used for P(c, s) depending on which fit is being done (^alJ or fovox). For fitting ^aIl(v), all values of c, s are considered equally likely:
For fitting fovox, the means and standard deviations, c, s, are fixed at c0, s0 (the values determined by the earlier fit to the entire data set):
For a small region, it is assumed that the mean noise vector, N, has normal distribution with standard deviation a:
For a large region, the mean noise vector, N, should be very close to zero; hence, Pdl(N) will be a delta function centered at N = 0.
The likelihood, P(fo|a, c, s, N), is approximated by analogy to a discrete normal distribution. q(v) is defined as the difference between the "expected" or "mean" histogram for particular a, c, s, N and a given histogram fo(v):
«f q(v; a, c, s, N) = fc(v - N)- ^ ajf(v; c, s). (24)
Now a normal-distribution-like function is created. w(v) is analogous to the standard deviation of q at each point of feature space:
Note that the denominator of Eq. (18) is a constant normalization of the numerator:
P(fc)= J P(a, c, s, N)P(fo|a, c, s, N)dadcdsdN (26) = P4. (27)
Using the approximations just discussed, we can calculate the following expression for the posterior probability:
For fitting fo111, the mean noise is assumed to be zero, so maximizing Eq. (28) is equivalent to minimizing SaVl to find the free parameters (a11, c, s),
subject to P(aal') = 0. Because both P(aalI) and P111 (c, s) are constant valued in that region, they are not included.
For fitting fovox, the parameters c and s are fixed, so maximizing Eq. (28) is equivalent to minimizing Svox to find the free parameters (avox, N),
As stated in Eq. (6), Section 6, Eq. (30) is minimized to estimate relative material volumes, avox, and the mean noise vector, N.
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