Derivation of Histogram Basis Functions

This section derives parameterized model histograms that are used as basis functions, f, for fitting histograms of data. Two where c is the constant expected value of a measurement of the pure material, and s is the standard deviation of additive, normally distributed noise.

The basis function used to fit the histogram of the measurements of a pure material is

./single(v; c, s)= j '(x)^(Psingle(x; c> s) — v)dx

= kn(v; s)* J ¡M(x)S(c — v)dx = kn(v; s) * ^¿(c — v) J &(x)dx

exp — öE

Thus, jsingle(v; c, s) is a Gaussian distribution with mean c and standard deviation s. v{, c,-, and s{ are scalar components of v, c, and s. The noise is assumed to be independent in each element of vector-valued data, which for MRI appears to be reasonable.

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