We present a new theoretical model and reconstruction results for a new class of fast x-ray CT machine -- the Real Time Tomography (RTT) system, which uses switched sources and an offset detector array. We begin by reviewing elementary properties of the Radon and X-ray transforms, and limited angle tomography. Through the introduction of a new continuum model, that of sources covering the surface of a cylinder in R³, we show that the problem of three-dimensional reconstruction from RTT data reduces to inversion of the three-dimensional Radon transform with limited angle data. Using the Paley-Wiener theorem, we then prove the existence of a unique solution and give comments on stability and singularity detection. We show, first in the two-dimensional case, that the conjugate gradient least squares algorithm is suitable for CT reconstruction. By exploiting symmetries in the system, we then derive a method of applying CGLS to the three-dimensional inversion problem using stored matrix coefficients. The new concept of source firing order is introduced and formalised, and some novel visualisations are used to show how this affects aspects of the geometry of the system. We then perform a detailed numerical analysis using the condition number and SVD of the forward projection matrix $A$, to show that the choice of firing order affects the conditioning of the problem. Finally, we give reconstruction results from both simulated phantoms and real experimental data that support the numerical analysis.

Computed Tomography (CT) is the standard in medical imaging field. In this study, we look at the curvelet transform in an attempt to use it as a basis for representing a function. In doing so, we seek a way to reconstruct a function from the Radon data that may produce clearer results. Using curvelet decomposition, any known function can be represented as a sum of curvelets with corresponding coefficients. It can be shown that these corresponding coefficients can be found using the Radon data, even if the function is unknown. The use of curvelets has the potential to solve partial or truncated Radon data problems. As a result, using a curvelet representation to invert radon data allows the chance of higher quality images to be produced. This paper examines this method of reconstruction for computed tomography (CT). A brief history of CT, an introduction to the theory behind the method, and implementation details will be provided.

The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform.

In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization.

Thermoacoustic tomography (TAT) is a newly emerging modality in biomedical imaging. It combines the good contrast of electromagnetic and good resolution of ultrasound imaging. The mathematical model of TAT is the observability problem for the wave equation: one observes the data on a hyper-surface and reconstructs the initial perturbation. In this dissertation, we consider several mathematical problems of TAT. The first problem is the inversion formulas. We provide a family of closed form inversion formulas to reconstruct the initial perturbation from the observed data. The second problem is the range description. We present the range description of the spherical mean Radon transform, which is an important transform in TAT. The next problem is the stability analysis for TAT. We prove that the reconstruction of the initial perturbation from observed data is not H¨older stable if some observability condition is violated. The last problem is the speed determination. The question is whether the observed data uniquely determines the ultrasound speed and initial perturbation. We provide some initial results on this issue. They include the unique determination of the unknown constant speed, a weak local uniqueness, a characterization of the non-uniqueness, and a characterization of the kernel of the linearized operator.

Multiples in seismic data pose serious problems to seismic interpreters for both AVO studies and interpretation of stacked sections. Several methods have been practiced with varying degrees of success to suppress multiples in seismic data. One family of velocity filters for demultiple operations using Radon transforms traditionally face challenges when the water column is shallow. Additionally, the hyperbolic Radon Transform can be computationally expensive. In this thesis, I introduce a novel multiple suppression technique in the t-x-p domain, where p is the local slope of seismic events that aims at tackling some of the aforementioned limitations, and discuss the advantages and scope of this approach. The technique involves essentially two steps: the decomposition part and the suppression part. Common Mid-Point (CMP) gathers are taken and transformed from the original t-x space to the extended t-x-p space and eventually to the t0-x-p space, where t0 is the zero offset traveltime. Multiplication of the gather in the extended space with Gaussian tapering filters, formed using the difference of the powers of the intrinsically calculated velocities in terms of t0 , x and p using analytical relations and the picked primary velocities and stacking along the p axis produces gathers with multiples suppressed. / text

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Orientador: Euclides de Mesquita Neto / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-09T07:13:04Z (GMT). No. of bitstreams: 1 Adolph_Marco_D.pdf: 9999341 bytes, checksum: 0336316213626e1dcf851637c49e82be (MD5) Previous issue date: 2006 / Resumo: O objetivo deste trabalho foi desenvolver uma formulação para obtenção de Funções de Green e estados auxiliares para os problemas viscoelastodinâmicos tridimensionais. Os problemas são descritos com auxílio de equações diferenciais, as quais foram solucionadas utilizando as transformadas de Radon e Fourier, bem como condições de contorno específicas. A transformada inversa é realizada numericamente. Essa formulação resulta em soluções com apenas uma integral imprópria e uma limitada, porém, quando se adota transformada de Fourier, são obtidas duas integrais ilimitadas. Para os problemas isotrópicos, essa formulação permite o desacoplamento do problema tridimensional em dois problemas auxiliares bidimensionais. As respostas de tensão e deslocamento foram deduzidas para os problemas de semi-espaço com cargas concentrada e distribuída, bem como de espaços completos submetidos a carregamento distribuído. A implementação numérica foi validada com auxílio das soluções estáticas de Cerruti, Boussinesq e Kelvin, bem como com implementações numéricas dinâmicas baseadas na transformada dupla de Fourier. Foram obtidos bons resultados. É apresentada a formulação para semi-espaços anisotrópicos / Abstract: The purpose of this study was to develop a formulation to obtain Green's functions and auxiliary states in order to solve three-dimensional viscoelastodynamic problems. The problems are described with the aid of differential equations, which were solved by using the Radon and Fourier transforms, as well as specific boundary conditions. The inverse transform was numerically accomplished. This formulation resulted in solutions with a single (one) improper integral and a finite one, however, in case of using the Fourier transform, two infinite integrals are obtained. As for the isotropic-related problems, this formulation allows to break up the threedimensional problem into two auxiliary two-dimensional problems. The stress and displacement solutions were obtained for problems related to half-space under concentrated and distributed loads, as well as to whole-spaces under a distributed load. The numerical implementation was validated by using the Cerruti, Boussinesq and Kelvin¿s static solutions, and numerical dynamic responses based on the double Fourier transform. Thus, effective results were obtained. In addition, it was included the formulation for anisotropy half-spaces / Doutorado / Mecanica dos Solidos / Doutor em Engenharia Mecânica

La transformée de Radon généralisée est une extension de la transformée de Radon qui généralise ses courbes de projection. Ce mémoire présente de nouveaux formalismes théoriques à la transformée de Radon Généralisée discrète. Les approches proposées dans ce mémoire ont différentes propriétés. Nous citons principalement : l'aspect modèle où chaque point dans l'espace de Radon correspond à un modèle dans l'espace spatial. Il est le résultat de la somme des pixels appartenant au modèle, la projection multidirectionnelle dans le sens que le domaine transformé de Radon se constituera au fur et à mesure que les courbes effectuent une rotation, selon le même principe utilisé dans la transformée de Radon classique et l'inversion exacte qui signifie la reconstruction exacte de l'image initiale à partir de l'espace de Radon de telle sorte que l'image reconstruite à partir de l'espace de Radon est égale en tout point à l'image initiale. La première approche proposée, appelée la transformée de Radon Généralisée Discrète multidirectionnelle est basée sur un formalisme algébrique défini par une multiplication matricielle entre des matrices de projection et l'image. Cette transformée permet une projection multidirectionnelle vu que les matrices de projection sont définies pour sélectionner des courbes épousant différentes directions. Cette transformée a l'avantage de ne poser aucune contrainte sur la nature des courbes projetées tout en permettant une inversion exacte. Nous avons appliqué la nouvelle transformée dans le domaine de la reconnaissance de formes, plus précisément dans la reconnaissance des bâtiments de forme rectangulaire dans des images satellitaire de haute résolution. En partant du principe qu'une courbe est transformée en un point de forte intensité dans l'espace de Radon, notre méthode de reconnaissance adoptée est basée sur l'étude de l'espace de Radon dans le but d'en extraire les pics. Ces derniers portent les informations cherchées sur la forme à identifier, à savoir ses paramètres, sa localisation et son orientation. Une deuxième approche appelée transformée de radon discrète polynomiale a été également proposée. Cette transformée projette une image discrète suivant des courbes polynômiales de différents degrés et orientations. Cette approche, fondée sur des propriétés arithmétiques, est également exactement inversible et multi directionnelle. Nous avons appliqué cette approche dans la reconnaissance des empreintes digitales. Les résultats montrent la précision de la méthode pour la détection de la position et de la direction des courbes polynomiales. Des propriétés intéressantes comme l'invariance aux transformations comme la rotation, la translation et le bruit caractérisent cette approche / The Generalized Radon transform is an extension of the Radon transform which generalizes its projection curves. This paper presents new theoretical formalism to the generalized discrete radon transform. The approaches proposed in this paper have different properties. We mainly cite: the model aspect where each point in Radon space corresponds to a model in spatial space. It is the result of the sum of the pixels belonging to the model, the multi-projection which means that the Radon transform domain will be constructed as the curves are rotated according to the same principle used in the classical Radon transform and exact inversion which means the exact reconstruction of the original image from Radon space so that the reconstructed image is equal in all pixels to the original image. The first proposed approach, called the Generalized Discrete Radon transform is based on an algebraic formalism defined by a matrix multiplication between the projection matrices and the image. This transform allows multidirectional projection since the projection matrices are defined to select curves following different directions. This transform has the advantage of not posing any constraints on the nature of the projected curves while allowing an exact inversion. We applied the new transformed in the field of pattern recognition, specifically in recognition of rectangular buildings in satellite images of high resolution. Assuming that a curve is transformed into an intensive point in the Radon space, our adopted recognition method is based on the study of Radon space in order to extract the peaks. These point out the needed information to identify the pattern, i.e., its parameters, its location and orientation. A second approach called polynomial discrete Radon transform was also proposed. This transform projects a discrete image following polynomial curves of different degrees and directions. This approach, based on arithmetic properties, is exactly reversible and multi-directional. We applied this approach to fingerprint recognition. The results show the precision of the method on detecting the position and direction of polynomial curves but also interesting properties such as invariance transformations such as rotation, translation and noise

The goal of this study is the recovery of functions and finite parametric distributions from their spherical means over spheres and designing a general formula or algorithm for the reconstruction of a function f via its spherical mean transform. The theoretical study is and supported with a numerical implementation based on radar data. In this study, we approach the reconstruction problem in two different way. The first one is to show how the reconstruction problem could be converted to a Prony-type system of equations. After solving this Prony-type system of equations, one can extract the parameters that describe the corresponding functions or distributions efficiently. The second way is to solve this problem via a backprojection procedure.