On an αth-order fractional Radon transform and a wave type of equation

The Fourier slice theorem holds for the classical Radon transform. In this paper, we consider a fractional Radon transform for which a sort of Fourier slice theorem also holds, and then present an inversion formula. The fractional Radon transform is shown to be characterized by the multi-dimensional case of a wave type of equation in analogy to the classical Radon transform.     

Fast Laplace Transforms for the Exponential Radon Transform

The Fourier slice theorem for the standard Radon transform generalizes to a Laplace counterpart when considering the exponential Radon transform. We show how to use this fact in combination with algorithms for the unequally spaced fast Laplace transform to construct fast and accurate methods for computing both the forward exponential Radon transform and the corresponding back-projection operator.     

Radon变换和衰减Radon变换的分析研究

衰减Radon变换出现在单光子放射型计算机层析成像中。本文首先回顾和研究了Radon变换和衰减Radon变换及其反演的有关结论，进而提出了Tretiak-Metz结果的一种新证明方法，对于一般对象，本文用变换方法非滤子背投影法导出了衰减Radon变换的反演公式。     

A Support Theorem for the Complex Radon Transform of Distributions

The properties of the complex Radon transform of compactly supported distributions are considered. For such distributions, we prove a support theorem allowing us to describe the support of the distribution in terms of the support of its Radon transform.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

Support Theorems for the Radon Transform and Cramér-Wold Theorems

This article presents extensions of the Cramér-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distribution theory and Fourier analysis we show that the presented injectivity results for the Radon transform lead to Cramér-Wold type results for measures. One purpose of this article is to contribute to making known to probabilists interesting results for the Radon transform that have been developed essentially during the 1980s and 1990s.     

A local support theorem for the radiation fields on asymptotically euclidean manifolds

We prove a local support theorem for the radiation fields on asymptotically euclidean manifolds which generalizes the local support theorem for the Radon transform.     

About asymptotic formulas for the inverse Radon transform

We give a proof and generalizations of the Gelfand-Graev asymptotic formula (formulated in 1962 for the odd-dimensional inverse Radon transform).     

A NOTE ON SINGULAR VALUE DECOMPOSITION FOR RADON TRANSFORM IN R~n

The singular value decomposition is derived when the Radon transform is restricted to functions which are square integrable on the unit ball in Rn with respect to the weight Wλ(x). It fulfilles mainly by means of the projection-slice theorem.The range of the Radon transform is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of the those basis functions are given. This immediately leads to an inversion formula by series expansion and range characterizations.     

A concise quaternion geometry of rotations

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two‐way correspondence between geometrical objects in the space of real unit quaternions representing rotations and geometrical objects constituted by directions in the three‐dimensional space subjected to these rotations. In this way a purely geometrical proof of the spherical Ásgeirsson's mean value theorem and a geometrical interpretation of integrals related to the spherical Radon transform of a probability density functions of unit quaternions are accomplished. Copyright © 2004 John Wiley & Sons, Ltd.     

On injectivity of the combinatorial Radon transform of order five

In the present work, we give a proof of the injectivity of the combinatorial Radon transform of order five.     

A new complex approach towards approximate inverse and generalized Radon transform

In this paper, we first consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm by means of the residue theorem of complex analysis, the approximate inverse, Gaussian mollifier and integral equations. And we successfully extend Natterer’s results to the generalized Radon transform with non-uniform attenuation. Finally, we investigate the smoothing properties of the generalized Radon transform.     

The Radon transform on hyperbolic space

The Radon transform that integrates a function in ⁿ , the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.     

A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.     

An elementary proof of the triangle inequality for the Wasserstein metric

We give an elementary proof for the triangle inequality of the -Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.

     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

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 radon transforms of a combinatorial geometry,I

Over a field of characteristic zero, the rank of the point-copoint incidence matrix of a combinatorial geometry of rank ? 2 equals the number of points. The proof uses a finite analogue of the Radon transform.