Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials

Radon transforms: Unitarization, Inversion and Wavefront sets

The first contribution of this thesis is a new approach based on the theory of group representations in order to solve in a general an unified way the unitarization and inversion problems for generalized Radon transform associated to dual pairs (G/K,G/H) of homogeneous spaces of a locally compact group G, where K and H are closed subgroups of G. Precisely, under some technical assumptions, if the quasi-regular representations of G acting on L^2(G/K) and L^2(G/H) are irreducible, then the Radon transform, up to a composition with a suitable pseudo-differential operator, can be extended to a unitary operator intertwining the two representations. If, in addition, the representations are square integrable, an inversion formula for the Radon transform based on the voice transform associated to these representations is given. Several examples are discussed. The second purpose of the thesis is to investigate the connection between the shearlet transform and the wavelet transform, which has to be found in the Radon transform in affine coordinates. This link yields a formula for the shearlet transform that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives both for finding a new algorithm to compute the shearlet transform of a signal and for the inversion of the Radon transform. Furthermore, we study the role of the Radon transform in microlocal analysis, especially in the resolution of the wavefront set in shearlet analysis. We propose a new approach based on the wavelet transform and on the Radon transform which clarifies how the ability of the shearlet transform to characterize the wavefront set of signals follows directly by the combination of the microlocal properties inhereted by the one-dimensional wavelet transform with a sensitivity for directions inhereted by the Radon transform. Finally, the last chapter of the thesis is devoted to the extension of the shearlet transform to distributions. Our main results are continuity theorems for the shearlet transform and its transpose, called the shearlet synthesis operator, on various test function spaces. Then, we use these continuity results to develop a distributional framework for the shearlet transform via a duality approach. This work arises from the lack in the theory of a complete distributional framework for the shearlet transform and from the link between the shearlet transform with the Radon and the wavelet transforms, whose distribution theory is a deeply investigated and well known subject in applied mathematics

Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography

In this work we introduce a new Radon transform which arises from a new modality of Compton Scattering Tomography (CST). This new system is made of a single detector rotating around a fixed source. Unlike some previous CST, no collimator is used at the detector. Such a system allows us to collect scattered photons coming from two opposite sides of the source-detector segment, hence the manifold of the associated Radon transform is a family of double circular arcs. As first main theoretical result, an analytic inversion formula is established for this new Radon transform. This is achieved through the formulation of the transform in terms of circular harmonic expansion satisfying the consistency conditions in Cormack's sense. Moreover, a fast and efficient numerical implementation via an alternative formulation based on Hilbert transform is carried out. Simulation results illustrate the theoretical feasibility of the new system. From a practical point of view, an uncollimated detector system considerably increases the amount of collected data, which is particularly significant in a scatter imaging system.Comment: 14 pages, 5 figure

On the determination of a function from its conical radon transform with a fixed central axis

Over the past decade, a Radon-type transform called a conical Radon transform, which assigns to a given function its integral over various sets of cones, has arisen in the context of Compton cameras used in single photon emission computed tomography. Here, we study the conical Radon transform for which the central axis of the cones of integration is fixed. We present many of its properties, such as two inversion formulas, a stability estimate, and uniqueness and reconstruction for a local data problem. An existing inversion formula is generalized and a stability estimate is presented for general dimensions. The other properties are completely new results.clos

Higher-rank wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

Let Mn, m be the space of real n × m matrices which can be identified with the Euclidean space Rn m. We introduce continuous wavelet transforms on Mn, m with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on Mn, m and coincide with classical ones in the rank-one case m = 1. We prove an analog of Calderón\u27s reproducing formula for L2-functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on Mn, m. We also introduce continuous ridgelet transforms associated to matrix planes in Mn, m. An inversion formula for these transforms follows from that for the Radon transform. The new approach makes it possible to reconstruct a function on Rn m from data on a set of planes of zero measure. © 2006 Elsevier Inc. All rights reserved