A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [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]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

Mixed norm estimate for Radon transform on weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We will discuss about the mapping property of Radon transform on L ^p spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy-Littlewood-Sobolev lemma imply weighted inequality for the Radon transform.     

高维加权Bergman空间上的Toeplitz算子

本文研究了高维加权Bergman空间Ap(Bn,dVpφ)(1＜p＜∞).上的Toeplitz算子.利用Toeplitz算子的Berezin变换,获得了Ap(Bn,dVp)(1＜p＜∞)上具有L∞(Bn)符号的Toeplitz算子的有限乘积的有限和是紧算子的一些等价刻画,推广了加权Bergman空间Ap(D,dmpφ)上的Toeplitz算子的有限乘积的有限和是紧的当且仅当它的Berezin变换在单位圆盘的边界消失为0的结论     

Singularities of the radon transform of a class of piecewise smooth functions in <Emphasis Type="Italic">R</Emphasis><Superscript>2</Superscript>

A class of piecewise smooth functions in R2 is considered.The propagation law of the Radon transform of the function is derived.The singularities inversion formula of the Radon transform is derived from the propagation law.The examples of singularities and singularities inversion of the Radon transform are given.     

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。     

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.     

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.     

Diophantine Integral Geometry

In his works [1], [2] and [3], the author succeeded in establishing several inversion formulas for Radon transform on Euclidean space, Damek-Ricci space and also on a finite set. The present paper deals with Radon transform R on discrete hyperplanes in the lattice defined by linear diophantine equations. More precisely, we study carefully various natural questions in this context: specific properties of the discrete Radon transform R and its dual R*, inversion formula for R (see Theorem 4.1) and also an appropriate support theorem in the discrete case (see Theorem 5.3).      

On the Mehler-Fock Transform in Lp-Space

The present paper is devoted to study the transform by the index of the Legendre function which is known as the Mehler-Fock transform. Mapping properties of the Mehler-Fodr transform in the weighted space L_p(ω(t); IR₊) are given as the inversion formula. The image space is also characterized.     

On the approximate inverse method in SPECT image reconstruction

ABSTRACT

In this paper, we first study the smoothing properties of the attenuated Radon transform with complex-valued coefficients. Then on the basis of the framework of Sobolev spaces and the method of the approximate inverse, we derive a reconstruction algorithm. Finally, we extend the results of Rigaud and Lakhal to the attenuated Radon transform with complex-valued coefficients.     

指数型Radon变换的一些结果

该论文主要研究在平面情形下指数型 Radon 变换的连续性,得到了它的近似反演公式，并对近似反演的数值解法加以改进.借助一些技巧,该文从理论上还建立了精确反演公式，从而推广了古典 Radon 变换的相应结果.     

Injectivity of rotation invariant windowed Radon transforms

We consider rotation invariant windowed Radon transforms that integrate a function over hyperplanes by using a radial weight (called window). T. Quinto proved their injectivity for square integrable functions of compact support. This cannot be extended in general. Actually, when the Laplace transform of the window has a zero with positive real part δ, the windowed Radon transform is not injective on functions with a Gaussian decay at infinity, depending on δ. Nevertheless, we give conditions on the window that imply injectivity of the windowed Radon transform on functions with a more rapid decay than any Gaussian function.     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

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.     

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.     

Inverse radon transform with one-dimensional wavelet transform

1. IntroductionSince Radon obtained the inverse formula of Radon transform in 1917, different inversemethods such as Fourier inversion, convolution back-projection inversion etc. have beeninvestigated['l']. Wavelet as a useful tool is interested in the inversion of Radon transformin recent years['--']. The application of wavelet analysis to Radon transform was proposedin I4] and [5]. An inversion formula based on continuous wavelet transform was derivedin [6] and [7]. This formula was based…     

The Radon–Kipriyanov transform of the generalized spherical mean of a function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform K_γ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.     

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

A one‐dimensional Radon transform on SO(3) and its application to texture goniometry

We consider a one‐dimensional Radon transform on the group SO (3), which is motivated by texture goniometry. In particular, we will derive several inversion formulae and compare them with the inversion of the one‐dimensional spherical Radon transform on ??³ for even functions. Copyright © 2005 John Wiley & Sons, Ltd.