Efficient recovery algorithms in electron crystallography

This project is concerned with solving the crystal structures of nanomaterials, from electron diffraction experiments. The applications range from structural chemistry and structural biology to materials science. Using the recently developed Automated Diffraction Tomography (ADT) approach to electron crystallography, cross-sections of the 3D diffraction volume can be observed by passing an electron beam through a thin specimen and then collecting the corresponding 2D diffraction patterns via CCD. By rotating the specimen in incremental steps, the diffraction volume is sampled by these 2D slices. However, the amount of sampling data is limited physically by the experimental apparatus and practically by the number of static slices which can be collected. Moreover, this sampling of reciprocal space means that some areas are high sampled, those close to the centre of the diffraction volume, while others are very sparsely sampled, those at the outermost parts of the diffraction volume or not sampled at all. Extrapolation techniques are used to approximate where the diffraction peaks maxima occur, and based on these positions intensity data for the structure solution of the observed crystal structure is extracted. In this project, we want to construct efficient numerical schemes for the optimal approximation of 3D diffraction data. The idea is to exploit the strong structure of the unknown density function, as a sparse union of 3D ellipsoids. We aim at the efficient recovery of these ellipsoids from their 2D cross-sections in the observed diffraction patterns.