Inversion formulas for the broken-ray Radon transform

We consider the inverse problem of the broken ray transform (sometimes also referred to as the V-line transform). Explicit image reconstruction formulas are derived and tested numerically. The obtained formulas are generalizations of the filtered backprojection formula of the conventional Radon transform. The advantages of the broken ray transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium simultaneously and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is typically discarded.Comment: To be submitted to Inverse Problem

On the V-Line Radon Transform and Its Imaging Applications

Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results

Scattered Radiation Emission Imaging: Principles and Applications

Imaging processes built on the Compton scattering effect have been under continuing investigation since it was first suggested in the 50s. However, despite many innovative contributions, there are still formidable theoretical and technical challenges to overcome. In this paper, we review the state-of-the-art principles of the so-called scattered radiation emission imaging. Basically, it consists of using the cleverly collected scattered radiation from a radiating object to reconstruct its inner structure. Image formation is based on the mathematical concept of compounded conical projection. It entails a Radon transform defined on circular cone surfaces in order to express the scattered radiation flux density on a detecting pixel. We discuss in particular invertible cases of such conical Radon transforms which form a mathematical basis for image reconstruction methods. Numerical simulations performed in two and three space dimensions speak in favor of the viability of this imaging principle and its potential applications in various fields

Discrete Wavelet Frames on the Sphere

In this paper the Continuous Wavelet Transform on the sphere is exploited to build the asociated Discrete Wavelet Frames. First, the half-continuous frames, i.e. frames where the position remains a continuous variable, is presented and then the fully discrete theory is presented. The notion of controlled frames is introduced, which reflects the particular nature of the underlying theory, particularly the apperant conflict between dilation and the compacity of the sphere. The paper is concluded with some numerical illustrations and future work

Best chirplet chain: near-optimal detection of gravitational wave chirps

The list of putative sources of gravitational waves possibly detected by the ongoing worldwide network of large scale interferometers has been continuously growing in the last years. For some of them, the detection is made difficult by the lack of a complete information about the expected signal. We concentrate on the case where the expected GW is a quasi-periodic frequency modulated signal i.e., a chirp. In this article, we address the question of detecting an a priori unknown GW chirp. We introduce a general chirp model and claim that it includes all physically realistic GW chirps. We produce a finite grid of template waveforms which samples the resulting set of possible chirps. If we follow the classical approach (used for the detection of inspiralling binary chirps, for instance), we would build a bank of quadrature matched filters comparing the data to each of the templates of this grid. The detection would then be achieved by thresholding the output, the maximum giving the individual which best fits the data. In the present case, this exhaustive search is not tractable because of the very large number of templates in the grid. We show that the exhaustive search can be reformulated (using approximations) as a pattern search in the time-frequency plane. This motivates an approximate but feasible alternative solution which is clearly linked to the optimal one. [abridged version of the abstract]Comment: 23 pages, 9 figures. Accepted for publication in Phys. Rev D Some typos corrected and changes made according to referee's comment

Stereographic Frames of Wavelets on the Sphere

In this technical report the Discrete Wavelet Frames are build on the already existing Spherical Continuous Wavelet Transform. The spherical half-continuous frames are explored, i.e when the position on the sphere is kept continuous variable. Then, the controlled frames are introduced, which comes from the particular nature of the underlying theory, namely the conflict between dilation and the compacity of the spherical manifold. The perspectives for the future work are given

A new Compton scattering tomography and its applications in medical imaging

International audienceCompton Scatter Imaging stands out among the novel approaches for exploring the inner parts of bodies because it takes advantage of scattered radiation. Instead of being rejected, as it is in current systems, scattered radiation is used as an imaging agent improving sensitivity and potentially reducing the amount of radiation required. Compton Scatter modalities have complex theoretical foundations involving Radon transforms in different manifolds. Several Compton based systems have been proposed and continue being studied including transmission and emission techniques and combinations of them like the new bimodal emission/transmission systems. Recently, a numerical study of a new modality of Compton tomography has been performed revealing attractive features in the field of non destructive testing of large objects like reinforced concrete or metal rafters. In this text, we complete this analysis studying its application in biomedical imaging. We analyse quality, with testing images intended for biomedical imaging, aswell as applications. Our results may help to develop new reconstruction techniques for bimodal images

The adjoint operator of the Radon transform on rotating V-lines and its role in image reconstruction

International audienceIntegral transformations over conical sets appear as the natural mathematical models in scattered radiation imaging. Compton cameras are emblematic examples of these imaging techniques. The concept of Compton camera originates from the need to improve sensitivity in Single Photon Emission Imaging, which uses hole collimators. It advocates electronic collimation which registers radiation emitted by the radiating object and scattered by a scattering detector placed before an absorption detector. The data consists of three dimensional conical projections of the activity density of a radio-tracer. We consider here a particular two-dimensional Compton camera in which the collected data consists of the set of integrals of the density on rotating V-lines. We present approximate reconstructions obtained both from an adequate back-projection procedure, and from the action of the adjoint operator of this transform. The relation between these operators is also established

The Radon transform on V-lines: artifact analysis and image enhancement

International audienceCompton Scatter Imaging is a promising approach in imaging science. Since it employs scattered radiation that is otherwise rejected, it is claimed to improve the sensitivity of current radiation-based imaging systems. The Radon transform on V-lines models a Compton Camera and its inversion through filtered back-projection leads to an efficient way to reconstruct the density of a radiotracer. Recently, some new properties of the Radon transform on V-lines have been discovered enabling reconstruction of electronic densities in a new modality of Compton Scatter Tomography. The interest on the Radon transform on V-lines is thus renewed and novel strategies must be developed in order to fulfil the quality requirements of its applications. In this paper we study the most representative degradation effects of the Radon transform on V-lines. Particularly, we use micro local analysis to explain its artifacts and apply some strategies to compensate them

Back-projection inversion of a conical Radon transform

In an effort to deal with many ionizing radiation imaging mechanisms involving the Compton effect, we study a Radon transform on circular cone surfaces having a fixed axis direction, which is called here conical Radon transform (CRT). Concretely, we seek to recover a density function (Formula presented.) in (Formula presented.) from its integrals over such circular cone surfaces or its conical projections. Although the existence of the inverse CRT has been established, it is the aim of this work to use this result to extent the concept of back-projection, well known in Computed Tomography (CT) to this type of cone surfaces. We discuss in some details the features of back-projection in relation to the corresponding CRT adjoint operator as well as the filters that arise naturally from the exact solution of the inversion problem. This intuitive approach is attractive, lends itself to efficient computational algorithms and may provide hints and guide for more general back-projection methods on other classes of cone surfaces, for example, occurring in Compton camera imaging. Comprehensive numerical simulations results are presented and discussed to illustrate and validate this approach based on the concept of back-projection.Fil: Cebeiro, Javier. Universidad Nacional de San Martin. Escuela de Ciencia y Tecnología. Centro de Matemática Aplicada; Argentina. University of Cergy-Pontoise; Francia. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Morvidone, Marcela Alejandra. Universidad Nacional de San Martin. Escuela de Ciencia y Tecnología. Centro de Matemática Aplicada; Argentina. Universidad Tecnológica Nacional. Facultad Regional Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Nguyen, M. K.. University of Cergy-Pontoise; Franci