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

Spherical radon transforms and mathematical problems of thermoacoustic tomography

The spherical Radon transform (SRT) integrates a function over the set of all spheres with a given set of centers. Such transforms play an important role in some newly developing types of tomography as well as in several areas of mathematics including approximation theory, integral geometry, inverse problems for PDEs, etc. In Chapter I we give a brief description of thermoacoustic tomography (TAT or TCT) and introduce the SRT. In Chapter II we consider the injectivity problem for SRT. A major breakthrough in the 2D case was made several years ago by M. Agranovsky and E. T. Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, which would be less restrictive in more general situations. We provide some new results obtained by PDE techniques that essentially involve only the finite speed of propagation and domain dependence for the wave equation. In Chapter III we consider the transform that integrates a function supported in the unit disk on the plane over circles centered at the boundary of this disk. As is common for transforms of the Radon type, its range has an in finite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. In Chapter IV we investigate implementation of the recently discovered exact backprojection type inversion formulas for the case of spherical acquisition in 3D and approximate inversion formulas in 2D. A numerical simulation of the data acquisition with subsequent reconstructions is made for the Defrise phantom as well as for some other phantoms. Both full and partial scan situations are considered

Mathematics and Algorithms in Tomography

This was the ninth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, emission tomography, X-ray CT, and vector tomography along with a wide range of mathematical analysis

Towards quantitative computed tomography

Computed tomography is introduced along with an overview of its diverse applications in many scientific endeavours. A unified approach for the treatment of scattering from linear scalar wave motion is introduced. The assumptions under which wave motion within a medium can be characterised by concourses of rays are presented along with comment on the validity of these assumptions. Early and conventional theory applied for modelling the behaviour of rays, within media for which ray assumptions are valid, are reviewed. A new computerised method is described for reconstruction of a refractive index distribution from time-of-flight measurements of radiation/waves passing through the distribution and taken on a known boundary surrounding it. The reconstruction method, aimed at solving the bent-ray computed tomography (CT) problem, is based on a novel ray description which doesn't require the ray paths to be known. This allows the refractive index to be found by iterative solution of a set of linear equations, rather than through the computationally intensive procedure of ray tracing, which normally accompanies iterative solutions to problems of this type. The preliminary results show that this method is capable of handling appreciable spatial refractive index variations in large bodies. A review containing theory and techniques for image reconstruction from projections is presented, along with their historical development. The mathematical derivation of a recently developed reconstruction technique, the method of linograms is considered. An idea, termed the plethora of views idea, which aims to improve quantitative CT image reconstruction, is introduced. The theoretical foundation for this is the idea that when presented with a plethora of projections, by which is meant a number greater than that required to reconstruct the known region of support of an image, so that the permissible reconstruction region can be extended, then the intensity of the reconstructed distribution should be negligible throughout the extended region. Any reconstruction within the extended region, that departs from what would be termed negligible, is deduced to have been caused by imperfections of the projections. The implicit expectation of novel schemes which are presented for improving CT image reconstruction, is that contributions within the extended region can be utilised to ameliorate the effects of the imperfections on the reconstruction where the distribution is known to be contained. Preliminary experimental results are reported for an iterative algorithm proposed to correct a plethora of X-ray CT projection data containing imperfections. An extended definition is presented for the consistency of projections, termed spatial consistency, that incorporates the region with which the projection data is consistent. Using this definition and an associated definition, spatial inconsistency, an original technique is proposed and reported on for the recovery of inconsistencies that are contained in the projection data over a narrow range of angles

Improved compressed sensing algorithm for sparse-view CT

In computed tomography (CT) there are many situations where reconstruction may need to be performed with sparse-view data. In sparse-view CT imaging, strong streak artifacts may appear in conventionally reconstructed images due to the limited sampling rate, compromising image quality. Compressed sensing (CS) algorithm has shown potential to accurately recover images from highly undersampled data. In the past few years, total variation (TV)-base compressed sensing algorithms have been proposed to suppress the streak artifact in CT image reconstruction. In this paper, we formulate the problem of CT imaging under transform sparsity and sparse-view constraints, and propose a novel compressed sensing-based algorithm for CT image reconstruction from few-view data, in which we simultaneously minimize the ℓ1 norm, total variation and a least square measure. The main feature of our algorithm is the use of two sparsity transforms: discrete wavelet transform and discrete gradient transform, both of which are proven to be powerful sparsity transforms. Experiments with simulated and real projections were performed to evaluate and validate the proposed algorithm. The reconstructions using the proposed approach have less streak artifacts and reconstruction errors than other conventional methods

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

Application of constrained optimisation techniques in electrical impedance tomography

A Constrained Optimisation technique is described for the reconstruction of temporal resistivity images. The approach solves the Inverse problem by optimising a cost function under constraints, in the form of normalised boundary potentials. Mathematical models have been developed for two different data collection methods for the chosen criterion. Both of these models express the reconstructed image in terms of one dimensional (I-D) Lagrange multiplier functions. The reconstruction problem becomes one of estimating these 1-D functions from the normalised boundary potentials. These models are based on a cost criterion of the minimisation of the variance between the reconstructed resistivity distribution and the true resistivity distribution. The methods presented In this research extend the algorithms previously developed for X-ray systems. Computational efficiency is enhanced by exploiting the structure of the associated system matrices. The structure of the system matrices was preserved in the Electrical Impedance Tomography (EIT) implementations by applying a weighting due to non-linear current distribution during the backprojection of the Lagrange multiplier functions. In order to obtain the best possible reconstruction it is important to consider the effects of noise in the boundary data. This is achieved by using a fast algorithm which matches the statistics of the error in the approximate inverse of the associated system matrix with the statistics of the noise error in the boundary data. This yields the optimum solution with the available boundary data. Novel approaches have been developed to produce the Lagrange multiplier functions. Two alternative methods are given for the design of VLSI implementations of hardware accelerators to improve computational efficiencies. These accelerators are designed to implement parallel geometries and are modelled using a verification description language to assess their performance capabilities