Florian Boßmann

Davide Bianchi, Florian Bossmann, Wenlong Wang et al.

We introduce a data-adaptive inversion method that integrates classical or deep learning-based approaches with iterative graph Laplacian regularization, specifically targeting acoustic impedance inversion - a critical task in seismic exploration. Our method initiates from an impedance estimate derived using either traditional inversion techniques or neural network-based methods. This initial estimate guides the construction of a graph Laplacian operator, effectively capturing structural characteristics of the impedance profile. Utilizing a Tikhonov-inspired variational framework with this graph-informed prior, our approach iteratively updates and refines the impedance estimate while continuously recalibrating the graph Laplacian. This iterative refinement shows rapid convergence, increased accuracy, and enhanced robustness to noise compared to initial reconstructions alone. Extensive validation performed on synthetic and real seismic datasets across varying noise levels confirms the effectiveness of our method. Performance evaluations include four initial inversion methods: two classical techniques and two neural networks - previously established in the literature.

Florian Bossmann, Jianwei Ma, Wenze wu

Identifying objects in given data is a task frequently encountered in many applications. Finding vehicles or persons in video data, tracking seismic waves in geophysical exploration data, or predicting a storm front movement from meteorological measurements are only some of the possible applications. In many cases, the object of interest changes its form or position from one measurement to another. For example, vehicles in a video may change its position or angle to the camera in each frame. Seismic waves can change its arrival time, frequency, or intensity depending on the sensor position. Storm fronts can change its form and position over time. This complicates the identification and tracking as the algorithm needs to deal with the changing object over the given measurements. In a previous work, the authors presented a new algorithm to solve this problem - Object reconstruction using K-approximation (ORKA). The algorithm can solve the problem at hand but suffers from two disadvantages. On the one hand, the reconstructed object movement is bound to a grid that depends on the data resolution. On the other hand, the complexity of the algorithm increases exponentially with the resolution. We overcome both disadvantages by introducing an iterative strategy that uses a resampling method to create multiple resolutions of the data. In each iteration the resolution is increased to reconstruct more details of the object of interest. This way, we can even go beyond the original resolution by artificially upsampling the data. We give error bounds and a complexity analysis of the new method. Furthermore, we analyze its performance in several numerical experiments as well as on real data. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required.

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.