Convolution/deconvolution of generalized Gaussian kernels with applications to proton/photon physics and electron capture of charged particles

Scatter processes of photons lead to blurring of images produced by CT (computed tomography) or CBCT (cone beam computed tomography). Multiple scatter is described by, at least, one Gaussian kernel. In various tasks, this approximation is crude; we need two/three Gaussian kernels to account for long-range tails (Landau tails), which appear in Molière scatter of protons, energy straggling and electron capture of charged particles passing through matter and Compton scatter. The ideal image (source function) is subjected to Gaussian convolution to yield a blurred image. The inverse problem is to obtain the source image from a detected image. Deconvolution methods of linear combinations of two/three Gaussian kernels with different parameters s0, s1, s2 can be derived via an inhomogeneous Fredholm integral equation of second kind (IFIE2) and Liouville - Neumann series (LNS) to provide the source function ρ. Scatter functions s0, s1, s2 are best determined by Monte-Carlo. An advantage of LNS is given, if the scatter functions s0, s1, s2 depend on coordinates. The convergence criterion can always be satisfied with regard to the above mentioned cases. A generalization is given by an analysis of the Dirac equation and Fermi-Dirac statistics leading to Landau tails applied to Bethe-Bloch equation (BBE) of charged particles and electron capture.