Öktem2015_ReferenceWorkEntry_MathematicsOfElectronTomograph.pdf - Mathematics of Electron Tomography Ozan Öktem Contents 1 2 3 4 5 6 7 8 Introduction

# Öktem2015_ReferenceWorkEntry_MathematicsOfElectronTomograph.pdf

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Mathematics of Electron Tomography Ozan Öktem Contents 1 Introduction ................................................................................. 939 2 The Transmission Electron Microscope (TEM) .......................................... 940 Sample Preparation ......................................................................... 942 3 Basic Notation and Definitions ............................................................ 943 4 The Forward Model ........................................................................ 944 Illumination ................................................................................. 945 Electron–Specimen Interaction ............................................................ 946 Optics ....................................................................................... 957 Detection .................................................................................... 962 Forward Operator for Combined Phase and Amplitude Contrast ........................ 966 Forward Operator for Amplitude Contrast Only .......................................... 969 Summary .................................................................................... 970 5 Data Acquisition Geometry ................................................................ 971 Parallel Beam Geometries ................................................................. 972 Examples Relevant for ET ................................................................. 972 6 The Reconstruction Problem in ET ........................................................ 974 Mathematical Formulation ................................................................. 974 Notion of Solution .......................................................................... 976 7 Specific Difficulties in Addressing the Inverse Problem ................................. 977 The Dose Problem .......................................................................... 977 Incomplete Data, Uniqueness, and Stability .............................................. 978 Nuisance Parameters . ...................................................................... 981 8 Data Pre-processing . ....................................................................... 985 Basic Pre-processing . ...................................................................... 985 Alignment .................................................................................. 986 Deconvolving Detector Response ......................................................... 986 Deconvolving Optics PSF .................................................................. 987 Phase Retrieval ............................................................................. 987 O. Öktem Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected] © Springer Science+Business Media New York 2015 O. Scherzer (ed.), Handbook of Mathematical Methods in Imaging , DOI 10.1007/978-1-4939-0790-8_43 937
938 O. Öktem 9 Reconstruction Methods ................................................................... 988 Analytic Methods .......................................................................... 989 Approximative Inverse . .................................................................... 999 Iterative Methods with Early Stopping .................................................... 1001 Variational Methods ........................................................................ 1005 Other Reconstruction Schemes ............................................................ 1013 10 Validation ................................................................................... 1014 11 Examples ................................................................................... 1016 Balls ......................................................................................... 1016 Virions and Bacteriophages in Aqueous Buffer ........................................... 1020 12 Conclusion .................................................................................. 1020 Cross-References . ............................................................................... 1022 References . ....................................................................................... 1022 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. Acronyms ART Algebraic Reconstruction Technique CCD Charged Coupled Device CPMV Cowpea Mosaic Virus CTF Contrast Transfer Function EELS Electron Energy Loss Spectroscopy EDS Energy-Dispersive X-ray Spectroscopy ET Electron (microscopy) Tomography ELT Electron -Tomography FBP Filtered Back-Projection HAADF High-Angle Annular Dark-Field

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