Towards a solution of the inverse X-ray diffraction tomography challenge: theory and iterative algorithm for...

The theoretical framework and a joint quasi-Newton-Levenberg-Marquardt-simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field function f_Ctpd(r - r₀) = h · u(r - r₀) due to a Coulomb-type point defect (Ctpd) located at a point r₀ within a crystal [h is the diffraction vector and u(r - r₀) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function {\cal F}\{ {\cal P} \} in a χ² sense in order to recover the function f_Ctpd(r - r₀) [{\cal P} is the parameter vector that characterizes the 3D function f_Ctpd(r - r₀) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi-Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function {\cal F} \{ {\cal P} \} to tend towards the global minimum even if the vector {\cal P} in the search is initially chosen rather a long way from the true one.