Acknowledgements......Page 6
Contents......Page 7
List of Figures......Page 9
I.1. Motivation......Page 10
I.2.1 Topologies......Page 13
I.2.2 Normed Vector Spaces......Page 14
I.2.3 Measure Theory......Page 16
I.2.4 Convex Analysis......Page 20
II.1.1 Existence and Stability......Page 23
II.1.2 Convergence......Page 26
II.2.1 Preliminaries......Page 33
II.2.2 Parameter Choices for Vanishing Noise......Page 35
II.2.3 Convergence rates......Page 38
III.1.1 Classical Norms......Page 46
III.1.2 Subnorms......Page 51
III.2.2 Basic Properties......Page 56
III.2.3 Continuity Results......Page 59
III.2.4 Applicability as a Discrepancy......Page 64
IV.1. Regularisation with Norms and Closed Operators......Page 69
IV.2.1 Symmetric Tensor Fields......Page 72
IV.2.2 Tensor Fields of Bounded Deformation......Page 78
IV.3.1 Basic Properties......Page 82
IV.3.2 Topological Properties......Page 87
IV.3.3 Total Generalised Variation of Vector-Valued Functions......Page 88
IV.4. TGV Regularisation in a Linear Setting......Page 92
V.1.1 Deriving the Radon Transform......Page 95
V.1.2 Analytical Properties......Page 97
V.1.3 Filtered Backprojection......Page 101
V.2.1 Continuous Tikhonov Problem for STEM Tomography Reconstruction......Page 104
V.2.2 Discretisation Scheme......Page 105
V.2.3 Primal-Dual Optimisation Algorithm......Page 111
V.2.4 STEM Tomography Reconstruction Algorithm......Page 112
V.3. Discussion of Numerical Results......Page 116
V.3.1 Preprocessing......Page 117
V.3.2 Synthetic Experiments......Page 123
V.3.3 Reconstruction of Single-Data HAADF Signals......Page 126
V.3.4 STEM Multi-Spectral Reconstructions......Page 130
Summary......Page 134
Bibliography......Page 137