Exact inversion of the conical Radon transform with a fixed opening angle (original) (raw)

On a Class of Generalized Radon Transforms and Its Application in Imaging Science

perso-etis.ensea.fr

Integral transforms based on geometrical objects, i.e. the so-called generalized Radon transforms, play a key role in integral geometry in the sense of I M Gelfand. In this work, we discuss the properties of a newly established class of Conical Radon Transforms (CRT), which are defined on sets of circular cones having fixed axis direction and variable opening angle. In particular, we describe its inversion process, i.e. the recovery of an unknown function from the set of its integrals on cone surfaces, or its conical projections. This transform is the basis for a new gamma-ray emission imaging principle, which works with Compton scattered radiation and offers the remarkable advantage of functioning with a fixed detector instead of a rotating one, as in conventional emission imaging modalities.

Inversion of a Class of Circular and Elliptical Radon Transforms

Contemporary mathematics, 2015

The paper considers a class of elliptical and circular Radon transforms appearing in problems of ultrasound imaging. These transforms put into correspondence to an unknown image function f in 2D its integrals Rf along a family of ellipses (or circles). From the imaging point of view, of particular interest is the circular geometry of data acquisition. Here the generalized Radon transform R integrates f along ellipses (circles) with their foci (centers) located on a fixed circle C. We prove that such transforms can be uniquely inverted from radially incomplete data to recover the image function in annular regions. Our results hold for cases when f is supported inside and/or outside of the data acquisition circle C.

Inversion of the circular Radon transform on an annulus

Inverse Problems, 2010

The representation of a function by its circular Radon transform (CRT) and various related problems arise in many areas of mathematics, physics and imaging science. There has been a substantial spike of interest towards these problems in the last decade mainly due to the connection between the CRT and mathematical models of several emerging medical imaging modalities. This paper contains some new results about the existence and uniqueness of the representation of a function by its circular Radon transform with partial data. A new inversion formula is presented in the case of the circular acquisition geometry for both interior and exterior problems when the Radon transform is known for only a part of all possible radii. The results are not only interesting as original mathematical discoveries, but can also be useful for applications, e.g. in medical imaging.

Numerical inversion of circular arc Radon transform

IEEE Transactions on Computational Imaging, 2016

Circular arc Radon (CAR) transforms associate to a function, its integrals along arcs of circles. The inversion of such transforms is of natural interest in several imaging modalities such as thermoacoustic and photoacoustic tomography, ultrasound, and intravascular imaging. Unlike the full circle counterpart-the circular Radon transform-which has attracted significant attention in recent years, the CAR transforms are scarcely studied objects. In this paper, we present an efficient algorithm for the numerical inversion of the CAR transform with fixed angular span, for the cases in which the support of the function lies entirely inside or outside the acquisition circle. The numerical algorithm is noniterative and is very efficient as the entire scheme, once processed, can be stored and used repeatedly for reconstruction of images. A modified numerical inversion algorithm is also presented to reduce the artifacts in the reconstructed image which are induced due to the limited angular span. Index Terms-Circular arc Radon transform, circular Radon transform, streak artifacts, trapezoidal product integration method, truncated singular value decomposition, volterra integral equations.

Radon transforms on a class of cones with fixed axis direction

Journal of Physics A: Mathematical and General, 2005

Integral transforms which map functions on R 3 onto their integrals on circular cones having fixed axis direction and variable opening angle are introduced and studied as generalizations of the known Radon transform. Besides their intrinsic mathematical interest, they serve as backbone support to emission imaging based on Compton scattered radiation, the way the standard Radon transform does for emission imaging based on non-scattered radiation. In this work, we establish its basic properties and prove analytically its invertibility. Formulae to express it in terms of the standard Radon transform (or vice versa) are given. We also discuss some extensions as applications.

Generalized transforms of Radon type and their applications

Proceedings of Symposia in Applied Mathematics, 2006

These notes represent an extended version of the contents of the third lecture delivered at the AMS Short Course "Radon Transform and Applications to Inverse Problems" in Atlanta in January 2005. They contain a brief description of properties of some generalized Radon transforms arising in inverse problems. Here by generalized Radon transforms we mean transforms that involve integrations over curved surfaces and/or weighted integrations. Such transformations arise in many areas, e.g. in Single Photon Emission Tomography (SPECT), Electrical Impedance Tomography (EIT) thermoacoustic Tomography (TAT), and other areas.

Radon Transform Inversion using the Shearlet Representation

2009

The inversion of the Radon transform is a classical ill-posed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level , the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error O(log( −1 ) 4/5 ), as → 0. Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets.

An inversion method for the cone-beam transform

Medical Imaging 2006: Physics of Medical Imaging, 2006

This paper presents an alternative formulation for the cone-beam projections given an arbitrary source trajectory and detector orientation. This formulation leads to a new inversion formula. As a special case, the inversion formula for the spiral source trajectory is derived.

Inversion of a New V-line Radon Transform and its Numerical Analysis

AIP Conference Proceedings, 2010

A new Radon transform defined on a discontinuous curve formed by a pair of half-lines forming a letter V is defined and studied. We establish its analytic inverse formula, its related filtered back-projection reconstruction procedure and its numerical analysis. These theoretical results allow the reconstruction of two-dimensional images of a radiating object from its Compton scattered rays measured on a one-dimensional collimated camera. Numerical simulations results illustrate the performance of the new imaging process.

Partial inversion of the 2D attenuated Radon transform with data on an arc

2017

In two dimensions, we consider the problem of inversion of the attenuated X-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the subdomain encompassed by this arc. The attenuation is assumed known in this subdomain. The method of proof uses the range characterization in terms of a Hilbert transform associated with A-analytic functions in the sense of Bukhgeim.