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

Applications of microlocal analysis to some hyperbolic inverse problems

This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when imaged using synthetic aperture radar. Finally, based on a joint work with Hanming Zhou, we show the analytic microlocal regularity of a class of analytic generalized Radon transforms, using this to show injectivity and stability for a generic class of generalized Radon transforms defined on analytic Riemannian manifolds

Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.Comment: 88 pages, 8 figure

Iterative CT reconstruction from few projections for the nondestructive post irradiation examination of nuclear fuel assemblies

The core components (e.g. fuel assemblies, spacer grids, control rods) of the nuclear reactors encounter harsh environment due to high temperature, physical stress, and a tremendous level of radiation. The integrity of these elements is crucial for safe operation of the nuclear power plants. The Post Irradiation Examination (PIE) can reveal information about the integrity of the elements during normal operations and off‐normal events. Computed tomography (CT) is a tool for evaluating the structural integrity of elements non-destructively. CT requires many projections to be acquired from different view angles after which a mathematical algorithm is adopted for reconstruction. Obtaining many projections is laborious and expensive in nuclear industries. Reconstructions from a small number of projections are explored to achieve faster and cost-efficient PIE. Classical reconstruction algorithms (e.g. filtered back projection) cannot offer stable reconstructions from few projections and create severe streaking artifacts. In this thesis, conventional algorithms are reviewed, and new algorithms are developed for reconstructions of the nuclear fuel assemblies using few projections. CT reconstruction from few projections falls into two categories: the sparse-view CT and the limited-angle CT or tomosynthesis. Iterative reconstruction algorithms are developed for both cases in the field of compressed sensing (CS). The performance of the algorithms is assessed using simulated projections and validated through real projections. The thesis also describes the systematic strategy towards establishing the conditions of reconstructions and finds the optimal imaging parameters for reconstructions of the fuel assemblies from few projections. --Abstract, page iii

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Discrete Singular Convolution and Its Application to the Analysis of Plates with Internal Supports. Part 1: Theory and Algorithm

10.1002/nme.526International Journal for Numerical Methods in Engineering558913-946IJNM