<title>Bayesian image reconstruction with a hyperellipsoidal posterior in x-ray fiber diffraction</title>

Bayesian image reconstruction from partial image and aliased spectral intensity data

“…However, the general symmetry case leads to more complicated distributions for the data that result in more complicated estimators. We are currently investigating this case [18], [19]. Finally, in some important fiber diffraction problems, the molecules are oriented parallel and randomly rotated, but do not form crystallites [10], [12].…”

“…The quantity given by (18), is a measure of the overall magnitude of the missing part of the electron density. This parameter must be estimated from the intensity data and the known part of the structure, represented by in order to calculate the MMSE estimate derived above.…”

“…The real part and, referring to (16), is therefore distributed with zero mean and variance Since the are independent, and assuming that the variance of the 's with respect to is small (i.e., the atoms are approximately of the same atomic number), the central limit theorem may be applied to (7) (provided that is large), so that (17) where is defined by (18) The same distribution applies to Equation (17) may be used to obtain and The real and imaginary parts are independent [16], so that (19) is a function of the amplitude but is independent of the phase and may be obtained as (20) Note that, referring to (8) and (18), is a function of in general. However, since the atomic electron densities are, to a very good approximation, spherically symmetric, so too are the atomic scattering factors in reciprocal space.…”

“…This corresponds to "space group P1" in crystallographic parlance [2], and is the case considered here. Symmetry relationships often exist between groups of atoms in the unit cell, in which case the estimation problem is more complicated and is considered elsewhere [18], [19].…”