Three-dimensional (3D) electron tomography (ET) is used to understand the structure and properties of samples, for applications in chemistry, materials science, and biology. By illuminating the sample at many tilt angles using an electron probe and modelling the image formation model, 3D information can be reconstructed at a resolution beyond the optical diffraction limit. However, as samples become thicker and more scattering, simple image formation models assuming projections or single scattering are no longer valid, causing the reconstruction quality to degrade. In this work, we develop a framework that takes the non-linear image formation process into account by modelling multiple-scattering events between the electron probe and the sample. First, the general acquisition and inverse model to recover multiple-scattering samples is introduced. We mathematically derive both the forward multi-slice scattering method as well as the gradient calculations in order to solve the inverse problem with optimization. As well, with the addition of regularization, the framework is robust against low dose tomography applications. Second, we demonstrate in simulation the validity of our method by varying different experimental parameters such as tilt angles, defocus values and dosage. Next, we test our ET framework experimentally on a multiple-scattering Montemorillonite clay, a 2D material submerged in aqueous solution and vitrified under cryogenic temperature. The results demonstrate the ability to observe the electric double layer (EDL) of this material for the first time. Last but not least, because modern electron detectors have large pixel counts and current imaging applications require large volume reconstructions, we developed a distributed computing method that can be directly applied to our framework for seeing multiple-scattering samples. Instead of solving for the 3D sample on a single computer node, we utilize tens or hundreds of nodes on a compute cluster simultaneously, with each node solving for part of the volume. As a result, both high resolution sample features and macroscopic sample topology can be visualized at the same time.




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