Images generated with a transmission electron microscope (TEM) can reveal information up to the scale of individual atoms. However, the information contained in a TEM image is blurred by aberrations and the partial coherence of the electron beam. Furthermore, the images correspond to the squared amplitude of the image plane electron wave and are thus missing valuable information about the phase. Exit wave reconstruction attempts to solve these problems by reconstructing the electron wave at the exit plane of the specimen, the so-called exit wave, from a series of images recorded with varying focus of the objective lens. This introduces the additional problem of aligning the image series, which is crucial for a successful reconstruction of the exit wave. One possible approach to reconstructing the exit wave involves the minimization of a least squares functional, which is implemented by the well-known MIMAP and MAL algorithms. The MIMAP and MAL algorithms solve the registration problem by alternatingly optimizing the exit wave and the registration. In this thesis, a novel objective functional $E_\sigma$ for the joint optimization of the exit wave and the registration is proposed. The properties of the forward model of TEM image simulation, which is given by a weighted autocorrelation of the exit wave, are investigated on the basis of the weighted cross-correlation and the novel notion of $\star$-separable weights. The most important weight functions (commonly called transmission cross-coefficients, TCCs) for TEM image simulation are analyzed and integrated into the present framework. The results regarding the forward model are then used for the analysis of the inverse problem. It is shown that the data term of $E_\sigma$ is not coercive for $\star$-separable TCCs, which in particular implies that the MAL functional is not coercive. One of the main results is the existence of minimizers of the objective functional $E_\sigma$, which is shown with the direct method. Additionally, it is shown that the objective functional is not convex in general. These results are complemented by a numerical analysis, which includes the discretization of the objective functional and the treatment of several problems regarding the numerical minimization of $E_\sigma$. A novel preconditioner for the exit wave is proposed, showing a reduction of the number of iterations for a given residual energy. The least squares sum in the data term of the objective functional is usually calculated by summing the squared differences of the simulated and experimental images over the same domain for each image. A novel method for the dynamic adjustment of these domains based on the current estimate for the registration is proposed, which allows to use the full image data for the reconstruction while at the same time avoiding the need for a continuation of the images. Numerical experiments are presented that evaluate the utility of the preconditioner and compare the alternating optimization approach with the joint optimization of the exit wave and the registration. Finally, a numerical experiment shows the result of reconstructing the exit wave for a real image series.

Dissertation, RWTH Aachen University, 2020; Aachen 1 Online-Ressource (VIII, 148 Seiten) : Illustrationen, Diagramme (2020). = Dissertation, RWTH Aachen University, 2020

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