Föcke L (2022)

**Publication Language:** English

**Publication Type:** Thesis

**Publication year:** 2022

**URI:** https://opus4.kobv.de/opus4-fau/files/18550/DissertationFoecke.pdf

In this thesis, we consider inverse problems that either include a randomized data acquisition process or are tackled by randomized reconstruction methods.

In the first part of this thesis, we reiterate on mathematical concepts required for randomized measurement or reconstruction processes. This includes an overview of mathematical preliminaries, e.g., function spaces, variational calculus, and the formulation of inverse problems. Furthermore, we consider a series of optimization methods, which we will use for the inverse problems presented later in this work. In particular, we discuss the concepts of randomized Kaczmarz approaches and introduce a weighting of the randomization process, which includes structural properties of currently considered problems.

For the second part of this work, we discuss a novel imaging modality called *Magnetorelaxometry Imaging* (MRX), which uses magnetic nanoparticles as a contrast agent. This imaging modality forms one of the main pillars of this thesis. In the course of this section, we introduce a mathematical model, which eventually leads to the formulation of the inverse problem of MRX. Additionally, we introduce an approximation of one part of the data acquisition sequence and formulate a mathematical foundation thereof.

In the last part of this thesis, we evaluate the introduced concepts numerically. For this purpose we consider three imaging modalities. The previously introduced MRX modality does not include any random processes: consequently, we apply a randomized reconstruction approach in this case. However, due to the novelty of the modality, the system's properties are not well understood yet. We thus use *Computerized Tomography* (CT) or X-Ray Tomography as reference for the proposed randomized reconstruction methods. Before we translate these results to MRX, we first consider an imaging modality called *Single Detector Imaging* (SDI). This method uses a randomized measurement process to generate a high resolution image with only a single sensor available. Finally, we consider the numerical implementation of the MRX imaging modality. Here, we analyze its structural properties, study its effects on the reconstruction process and evaluate the randomized reconstruction approaches for MRX.

**APA:**

Föcke, L. (2022). *Inverse Problems for Random Measurements* (Dissertation).

**MLA:**

Föcke, Lea. *Inverse Problems for Random Measurements.* Dissertation, 2022.

**BibTeX:** Download