Florian Boßmann

Nada Sissouno, Florian Boßmann, Frank Filbir et al.

Measurements achieved with ptychographic imaging are a special case of diffraction measurements. They are generated by illuminating small parts of a sample with, e.g., a focused X-ray beam. By shifting the sample, a set of far-field diffraction patterns of the whole sample are then obtained. From a mathematical point of view those measurements are the squared modulus of the windowed Fourier transform of the sample. Thus, we have a phase retrieval problem for local Fourier measurements. A direct solver for this problem was introduced by Iwen, Viswanathan and Wang in 2016 and improved by Iwen, Preskitt, Saab and Viswanathan in 2018. Motivated by the applied perspective of ptychographic imaging, we present a generalization of this method and compare the different versions in numerical experiments. The new method proposed herein turns out to be more stable, particularly in the case of missing data.

Florian Boßmann

Sparse data approximation has become a popular research topic in signal processing. However, in most cases only a single measurement vector (SMV) is considered. In applications, the multiple measurement vector (MMV) case is more usual, i.e., the sparse approximation problem has to be solved for several data vectors coming from closely related measurements. Thus, there is an unknown inter-vector correlation between the data vectors. Using SMV methods typically does not return the best approximation result as the correlation is ignored. In the past few years several algorithms for the MMV case have been designed to overcome this problem. Most of these techniques focus on the approximation quality while quite strong assumptions to the type of inter-vector correlation are made. While we still want to find a sparse approximation, our focus lies on preserving (possibly complex) structures in the data. Structural knowledge is of interest in many applications. It can give information about e.g., type, form, number or size of objects of interest. This may even be more useful than information given by the non-zero amplitudes itself. Moreover, it allows efficient post processing of the data. We numerically compare our new approach with other techniques and demonstrate its benefits in two applications.

Florian Boßmann, Tomas Sauer, Nada Sissouno

Mathematical methods of image inpainting involve the discretization of given continuous models. We present a method that avoids the standard pointwise discretization by modeling known variational approaches, in particular total variation (TV), using a finite dimensional spline space. Besides the analysis of the resulting model, we present a numerical implementation based on the alternating method of multipliers. We compare the results numerically with classical TV inpainting and give examples of applications.