The sensitive coherent interference of electron waves is useful for revealing subtle information in electron micrographs, which can be important for minimising dose and for rapid imaging. For weakly scattering specimens, certain approximation paradigms can reliably provide interpretable results for experiments such as inline holography, contrast transfer function inversion and ptychography. In general, significant dynamical diffraction is expected due to the useful strong interactions of electrons with matter, creating 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 in electron tomography of crystalline specimens. For specimens such as thin crystals in zone-axis orientations or small nanoparticles, multiple scattering can nonetheless be accurately forward-simulated from assumed models and refined. To this end, recent advances in atomic resolution electron tomography have been made using different approaches to the inverse problem with constraints based upon a priori structural knowledge.
But are there ways to harness useful electron phase contrast information to reconstruct three-dimensional (3D) morphology, without recourse to weak scattering approximations or model refinement?
Here we present 3D reconstructions for experimental electron microscope tilt series ranging from polyhedral nanoparticles, to steel dislocation networks, cryo-EM cellular structures and 3D diffuse diffraction of a complex ferroelectric. Invoking a sparsity assumption, we will demonstrate that robust topological features can be tracked in 3D using a differential geometric form of stereoscopy, to circumvent significant departures from the projection approximation and reduce noise as form of compressed sensing.