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FIGURE 8 Parameters for histogram basis function. (a) Single-material histogram parameters include c, the mean value for the material, and 5, which measures the standard deviation of measurements (see Eq. (2)). (b) Corresponding parameters for a two-material mixture basis function. 50 and 51 affect the slopes of the two-material histogram basis function at either end. For vector-valued data, c and 5 are vectors and are the mean values and standard deviations of the noise for the two constituent materials (see Eq. (3)).

function, p(x), are reasonable. The details of the derivations are in Section 8.

For a single material, the histogram basis function is a Gaussian distribution,

where c is the vector-valued mean, 5 the vector-valued standard deviation, and vi, ci, and si scalar components of v, c, and 5, respectively. This equation is derived by manipulating Eq. (1) evaluated over a region of constant material, where the measurement function, p(x), is a constant value plus additive, normally distributed noise. Because the noise in different channels of multivalued MRI images is not correlated, the general vector-valued normal distribution reduces to this equation with zero covariances.

For mixtures along a boundary between two materials, another equation can be derived similarly:

/double(v;c, 5)= i h((1 - t)ci + tc2 - v; 5)dt• (3) J0

As with the single-material case, this derivation follows from Eq. (1) evaluated over a region where two materials mix. In this case, the band-limiting filter that causes partial-volume effects is approximated with a box filter and an assumption is made that the variance of the additive noise is constant across the region. This basis function is a superposition of normal distributions representing different amounts of the two constituent pure materials. kn is the normal distribution, centered at zero; t is the relative quantity of the second material; c (comprising c1 and c2) the expected values of the two materials; and 5 is the standard deviation of measurements.

The assumption of a box filter affects the shape of the resulting histogram basis function. Similar equations for different filters (triangle, Gaussian, and Hamming) can also be derived, but a box filter is sufficiently accurate in practice and is numerically more efficient.