Inversion of circular means and the wave equation on convex planar domains

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary \partial \Om of a smooth convex bounded domain \Om \subset \R^2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in \Om modulo an explicitly computed smoothing integral operator \K_\Om. For circular and elliptical domains the operator \K_\Om is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on \partial \Om. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.Comment: [14 pages, 2 figures

Deep learning versus ℓ1\ell^1-minimization for compressed sensing photoacoustic tomography

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and $\ell^1$-minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint $\ell^1$-minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements the $\ell^1$-minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).Comment: This work has been presented at the Joint Photoacoustics Session with the 2018 IEEE International Ultrasonics Symposium Kobe, October 22-25, 201

Quantitative photoacoustic imaging in radiative transport regime

The objective of quantitative photoacoustic tomography (QPAT) is to reconstruct optical and thermodynamic properties of heterogeneous media from data of absorbed energy distribution inside the media. There have been extensive theoretical and computational studies on the inverse problem in QPAT, however, mostly in the diffusive regime. We present in this work some numerical reconstruction algorithms for multi-source QPAT in the radiative transport regime with energy data collected at either single or multiple wavelengths. We show that when the medium to be probed is non-scattering, explicit reconstruction schemes can be derived to reconstruct the absorption and the Gruneisen coefficients. When data at multiple wavelengths are utilized, we can reconstruct simultaneously the absorption, scattering and Gruneisen coefficients. We show by numerical simulations that the reconstructions are stable.Comment: 40 pages, 13 figure

Reconstructing Functions on the Sphere from Circular Means

The present thesis considers the problem of reconstructing a function f that is defined on the d-dimensional unit sphere from its mean values along hyperplane sections. In case of the two-dimensional sphere, these plane sections are circles. In many tomographic applications, however, only limited data is available. Therefore, one is interested in the reconstruction of the function f from its mean values with respect to only some subfamily of all hyperplane sections of the sphere. Compared with the full data case, the limited data problem is more challenging and raises several questions. The first one is the injectivity, i.e., can any function be uniquely reconstructed from the available data? Further issues are the stability of the reconstruction, which is closely connected with a description of the range, as well as the demand for actual inversion methods or algorithms. We provide a detailed coverage and answers of these questions for different families of hyperplane sections of the sphere such as vertical slices, sections with hyperplanes through a common point and also incomplete great circles. Such reconstruction problems arise in various practical applications like Compton camera imaging, magnetic resonance imaging, photoacoustic tomography, Radar imaging or seismic imaging. Furthermore, we apply our findings about spherical means to the cone-beam transform and prove its singular value decomposition.Die vorliegende Arbeit beschäftigt sich mit dem Problem der Rekonstruktion einer Funktion f, die auf der d-dimensionalen Einheitssphäre definiert ist, anhand ihrer Mittelwerte entlang von Schnitten mit Hyperebenen. Im Fall d=2 sind diese Schnitte genau die Kreise auf der Sphäre. In vielen tomografischen Anwendungen sind aber nur eingeschränkte Daten verfügbar. Deshalb besteht das Interesse an der Rekonstruktion der Funktion f nur anhand der Mittelwerte bestimmter Familien von Hyperebenen-Schnitten der Sphäre. Verglichen mit dem Fall vollständiger Daten birgt dieses Problem mehrere Herausforderungen und Fragen. Die erste ist die Injektivität, also können alle Funktionen anhand der gegebenen Daten eindeutig rekonstruiert werden? Weitere Punkte sind die die Frage nach der Stabilität der Rekonstruktion, welche eng mit einer Beschreibung der Bildmenge verbunden ist, sowie der praktische Bedarf an Rekonstruktionsmethoden und -algorithmen. Diese Arbeit gibt einen detaillierten Überblick und Antworten auf diese Fragen für verschiedene Familien von Hyperebenen-Schnitten, angefangen von vertikalen Schnitten über Schnitte mit Hyperebenen durch einen festen Punkt sowie Kreisbögen. Solche Rekonstruktionsprobleme treten in diversen Anwendungen auf wie der Bildgebung mittels Compton-Kamera, Magnetresonanztomografie, fotoakustischen Tomografie, Radar-Bildgebung sowie der Tomografie seismischer Wellen. Weiterhin nutzen wir unsere Ergebnisse über sphärische Mittelwerte, um eine Singulärwertzerlegung für die Kegelstrahltomografie zu zeigen

Study of generalized radon transforms and applications in compton scattering tomography

This thesis is concerned with the study of new modalities of Compton scattering tomography which are a relevant alternative with current imaging techniques. Such a study requires powerful mathematical tools. Then I, first, extended the known properties of the classical Radon transform to larger manifolds of curves. In particular, I established the analytical inversion formulas for solving the associated image reconstruction problem. Due to these inversion properties and a numerical study of involved processes, the theoretical feasibility of the proposed modalities in Compton scattering tomography could be established. In a second time, I established an iterative algorithm to correct the attenuation factor in the studied modalities (GIPC). Finally, I proposed the first bimodality based on the scattered radiation. This new system, akin to the SPECT-CT scan, combines two modalities of Compton scattering tomography. The simulation results show the interest of such a future system.Diese Arbeit konzentriert sich auf die Untersuchung neuer Modelle in der Compton - Streutomographie, die eine relevante Alternative oder Ergänzung aktueller bildgebender Verfahren darstellen. Da eine mathematische Untersuchung dafür benötigt ist, habe ich zuerst die Eigenschaften der Radontransformation auf eine größere Familie von Kurven erweitert. Insbesondere habe ich die analytischen Umkehrformeln zur Bildrekonstruktion etabliert. Dank dieser Inversionseigenschaften und der numerischen Untersuchung der beteiligten Prozessen, hat die theoretische Machbarkeit der Compton-Streutomographie Modelle bewiesen. Dann habe ich einen iterativen Algorithmus (GIPC) hergeleitet, um den Dämpfungsfaktor in den Modalitäten zu korrigieren. Schließlich habe ich die erste Bimodalität auf der Grundlage der Streustrahlung aufgestellt. Dieses neue System kombiniert zwei Methoden der Compton-Streutomographie ähnlich des SPECT-CT Bilder. Die Simulationsergebnisse zeigen dass in Zukunft großes Interesse an einen solchen System existiert