Bayesian 3D X-Ray Reconstruction From Incomplete Noisy Data

Bayesian image and pattern reconstruction from incomplete and noisy data

“…The latter means that the form of a given functional dictates the need to support certain properties and structures of restored images, such as structures of clusters or planes. The features of four types of a priori functionals named in accordance with the supported structures were tested in practice, i.e., (I) is the volumetric smoothing, (II) is the support for clusters, (III) refers to the support for planes, and (IV) is the support for phases [17].…”

“…In the case of radiation sources, it is natural to assume that the total number of absorbed photons M in the reconstructed object is known and each photon has an equal probability of being absorbed in any of the voxels in which the image of the reconstruction object is divided. These assump tions lead to the entropy, which can be written in the following form: (6) Then, the total variational procedure amounts to min imizing the following functional:…”

“…In functional (6), due to the second factor, the second minimum corre sponds to the vanishing logarithm of the object param eter. Recall that functional (6) was obtained within the Boltzmann statistics. In addition, functional (6) can be used when the number of phase components in the object is not limited and not a priori unknown.…”

“…Calculation of the probability P(p) as a measure of the quality of the problem (optional). 6. The calculation of a posteriori probability P(μ|p) as an unknown probability distribution.…”

“…However, if viewing angles are limited and/or the number of projections is small, the problem becomes very poorly posed (see, e.g., [5,6]). The lack of X ray source power [7] and the presence of back ground noise lead to incorrectness in the inversion problem.…”