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May 26, 2023

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This survey starts with a brief description of the scientific relevance of electron tomography in life sciences followed by a survey of image formation models. In the latter, the scattering of electrons against a specimen is modeled by the Schrödinger equation, and the image formation model is completed by adding a description of the transmission electron microscope optics and detector. Electron tomography can then be phrased as an inverse scattering problem and attention is now turned to describing mathematical approaches for solving that reconstruction problem. This part starts out by explaining challenges associated with the aforementioned inverse problem, such as the extremely low signal-to-noise ratio in the data and the severe ill-posedness due to incomplete data, which naturally brings up the issue of choosing a regularization method for reconstruction. Here, the review surveys both methods that have been developed, as well as pointing to new promising approaches. Some of the regularization methods are also tested on simulated and experimental data. As a final note, this is not a traditional mathematical review in the sense that focus here is on the application to electron tomography rather than on describing mathematical techniques that underly proofs of key theorems.