New range theorems for the dual Radon transform

Three new range theorems are established for the dual Radon transform : on functions that do not decay fast at infinity (and admit an asymptotic expansion), on , and on . Here , and acts on even functions .

     

Invariant differential operators and the range of the matrix Radon transform

In this paper we characterize the range of the matrix Radon transform by invariant differential operators. This generalizes analogous results for the d-plane transform in Rn.     

An inversion formula for the dual horocyclic Radon transform on the hyperbolic plane

Consider the Poincare unit disk model for the hyperbolic plane H ². Let Ξ be the set of all horocycles in H ² parametrized by (θ, p), where e^iθ is the point where a horocycle ξ is tangent to the boundary |z| = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R* : μ(θ, p) → (z) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that P_m(d/dp)(μ_m(p)e^p) be even for all m ∈ ?. Here P_m(d/dp) is a family of differential operators introduced by Helgason, and μ_m(p) are the coefficients of the Fourier series expansion of μ(θ, p). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new proof of the support theorem and the range characterization for the Radon transform

The aim of this note is to give a new and elementary proof of the support theorem for the Radon transform, which is based only on the projection theorem and the Paley-Wiener theorem for the Fourier transform. The idea is to solve a certain system of linear equations in order to determine the coefficients of a homogeneous polynomial (interpolation problem). By the same method, we get a short proof of the range characterization for Radon transforms of functions supported in a ball.     

Biorthogonal decompositions of the Radon transform

Summary. The concept of singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. A singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators , . In this paper we consider a new approach which generalizes this concept. By application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator. With respect to the Radon transform for functions with support in the unit ball we apply the system of Appell polynomials which is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions. Vice versa by application of our decompositions we are able to prove new properties of the Appell polynomials. Received October 19, 1993     

Numerical inversion of the Radon transform

Summary The problem of inverting the Radon transform, i.e. the reconstruction of a function inR ² from its line integrals arises e.g. in computerized tomography and in nondestructive testing. In the present paper the least squares method with piecewise constant trial functions is investigated. An error estimate is derived. An implementation using the fast Fourier transform is described and numerical results are reported.     

On the inversion of the attenuated Radon transform

We given an inversion formula for the attenuated Radon transform $$\bar \omega _L $$ for constant attenuation μ which is accurate up toO(µ ⁴) We also derive an integral equation of the second kind for the solution ofR _μf=g. The implementation of both methods for the inversion ofR _μ consists basically of a generalized filtered backprojection algorithm. Numerical examples are presented.     

Range conditions for the exponential Radon transform

A new approach to the range conditions for the exponential Radon transform is introduced. Partially supported by NSF EPSCOR Grant and Wesley Foundation Grant no. 9012019. Partially supported by NSF Grant 33-1807-231.     

On the Radon transform over isotropic d-planes

研究了速降函数在R2n中迷向d-平面上的Radon变换,d＜n,并分别给出d=1,d=2的情况下,此Radon变换的像满足的二阶偏微分方程组.     

Inversion of the Radon transform with incomplete data

The Radon transform R(p, θ), θ∈S^n?1, p∈?¹, of a compactly supported function f(x) with support in a ball B_a of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on S^n?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.     

Some notes on the Radon transform and integral geometry

This is an expanded version of a lecture given in Vienna on January 17, 1990 in which the significance of a famous paper by Radon, and the developments originating from it, were discussed.     

Sobolev estimates for the Radon transform of Melrose and Taylor

We derive the regularity properties of the Radon transform of Melrose and Taylor for the scattering on a compact, convex obstacle with a smooth boundary. The result is formulated in terms of the highest order of contact of tangent lines with the boundary of an obstacle. The main ingredients of the proof are the estimates for degenerate oscillatory integral operators and almost orthogonal decompositions. © 1998 John Wiley & Sons, Inc.     

The identification problem for the exponential Radon transform

The exponential Radon transform, which arises in single photon emission computed tomography, is defined by ? ?(μ:ω,s) = ∫_R?(sω + tomega;^?) e^μt dt?. Here ? is a compactly supported distribution in the plane which represents the location and intensity of a radio-pharmaceutical in a body of constant, but unknown, attenuation μ, and ω is a direction. The identification problem is to determine the attenuation μ from the data ?? with ? unknown. We will show that μ can be determined from the data if and only if ? is not a radial distribution and give formulae for computing μ when ? is not radial.     

Radon transform on spaces of constant curvature

A correspondence among the totally geodesic Radon transforms-as well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.