Volume imaging by tracking sparse topological features in electron micrograph tilt series

The sensitive coherent interference of electron waves arising from a specimen is useful for revealing subtle structural information in electron micrographs, which can be important for minimising dose and for rapid imaging. In general, dynamical diffraction is expected due to the useful strong interactions of electrons with matter, which can create phase contrast that violates the requisite Radon projection assumption for tomography. It is for these reasons that incoherent imaging modalities such as high angle annular dark field have been favoured to date in electron tomography of crystalline specimens, to access a monotonic relationship between specimen thickness and micrograph intensity. Here we use a geometric approach to track topological features that are robust to perturbation of the imaging conditions, to enable 3D reconstructions from electron microscope tilt series under imaging conditions that violate the Radon projection assumption, with an emphasis on phase contrast. Invoking a sparsity assumption, we demonstrate that topological features can be reliably tracked in 3D using a differential geometric form of stereoscopy, to circumvent departures from the projection approximation and reduce noise by effecting segmentation of interest points from the outset. We demonstrate this approach on a variety of different specimen and data types, from polyhedral nanoparticles, to steel dislocation networks, cryo-EM cellular structures and 3D diffuse diffraction of a relaxor ferroelectric.