Mathematics of Electron Tomography

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Abstract

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.

Keywords

Nuisance Parameter Phase Retrieval Reconstruction Operator Contrast Model Algebraic Reconstruction Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The writing of this review chapter would not be possible without the help of several people. Jan Boman at the Department of Mathematics, Stockholm University and Todd Quinto at the Department of Mathematics, Tufts University provided valuable advice and help, especially regarding material in section “Analytic Methods.” Hans Rullgård at Comsol has provided advice and support regarding the usage of the TEM simulation and the TV-regularization softwares used in section “Examples”. Remco Schoenmakers at FEI generously provided the data used in section “Virions and Bacteriophages in Aqueous Buffer” as well as access to the image in Fig. 1. Milos Vulevic and Bernd Reiger in TU Delft provided valuable insight into the detector modeling in subsection on p. 21, Sergej Masich at the Department of Cell and Molecular Biology, Karolinska Institutet helped out in the alignment of the tilt-series in section “Virions and Bacteriophages in Aqueous Buffer” as well as in running the IMOD reconstructions in section “Examples.” Holger Kohr and Alfred Louis at the Department of Mathematics, Saarlands University contributed to the material on the approximate inverse method and phase contrast tomography. Kohr also provided the approximate inverse reconstructions in section “Examples.” Finally, Günther Uhlmann at the Department of Mathematics, Washington State University provided valuable insight into uniqueness and stability issues discussed in section “Electron–Specimen Interaction.” Work on this chapter is financially supported by the Swedish Foundation for Strategic Research.

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Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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