Retrieval of object information by inverse problems in electron diffraction

The imaging of crystal defects by high-resolution transmission electron microscopy or with the help of the electron diffraction contrast technique is well known and routinely used. However, a direct and phenomenological analysis of electron micrographs is mostly not possible, but requires the application of image simulation and matching techniques, The trial-and-error matching technique is the indirect solution to the direct scattering problem applied to analyse the nature of the object under investigation. Alternatively, inverse problems as direct solutions of electron scattering equations can be deduced using either an invertible linearized eigenvalue system or a discretized form of the diffraction equations. This analysis is based on the knowledge of the complex electron wave at the exit plane of an object reconstructed for the surrounding of single reflections by electron holography or other wave reconstruction techniques. in principle, it enables directly the retrieval of the local thickness and orientation of a sample as well as the refinement of potential coefficients or the determination of the atomic displacements, caused by a crystal lattice defect, relative to the atom positions of the perfect lattice. Considering especially the sample orientation as perturbation the solution is given by a generalized and regularized Moore-Penrose inverse, where the resulting numerical algorithms imply ill-posed inverse problems.