On inversion and Sobolev estimate results about the weighted Radon transform

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space.     

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.     

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.     

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.     

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.     

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.     

DERIVATION AND ANALYSIS OF A NEW RECONSTRUCTION FORMULA FROM CONE-BEAM PROJECTIONS

A new method to compute the first derivative of 3-D Radon transform is given for cone-beam data taken from any orbit. Smith [Ⅰ1] and Grangeat [5] even derived cone-beam inversion formulas which are the basic work in fully 3-D image reconstruction algorithm and are used extensively now. In this paper we will give a new inversion formula and a simple necessary and sufficient condition which guarantees the complete reconstruction algorithm.     

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.     

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.     

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 the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

Semyanistyi’s Integrals and Radon Transforms on Matrix Spaces

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝ ⁿ . We obtain an extended version of Fuglede’s formula, connecting generalized Semyanistyi’s integrals, Radon transforms and Riesz potentials on the space of real rectangular matrices. This result combined with the matrix analog of the Hilbert transform leads to variety of new explicit inversion formulas for the Radon transform of functions of matrix argument. The authors were supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). The first author was also supported by Abraham and Sarah Gelbart Research Institute for Mathematical Sciences. The second author was also supported by the NSF grants EPS-0346411 (Louisiana Board of Regents) and DMS-0556157).     

A singular value decomposition for the radon transform in n-dimensional euclidean space

We construct the singular value decomposition of the Radon transform when the Radon transform is restricted to functions which are either square integrable on the unit disc in IR ⁿ with respect to one of the weights (1-r ²)^n/2-λ: or square integrable on IR ⁿ with respect to exp(r ²). An application to calculating mollifiers for approximate inversion of the sampled Radon transform is discussed.     

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.     

Convolution–backprojection method for the k-plane transform, and Calderón''s identity for ridgelet transforms

We develop a convolution–backprojection method for the k-plane Radon transform , . A slight modification of this method gives an explicit inversion formula for in terms of the corresponding wavelet-like transforms (or the k-plane ridgelet transforms), and a generalization of Calderón's reproducing formula.     

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…     

Extensions of some fixed point theorems of Rhoades

The classical Radon transform, R, maps an integrable function in Rⁿ to its integrals over all n ? 1 dimensional hyperplanes, and the exterior Radon transform is the transform R restricted to hyperplanes that do not intersect a given disc. A singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc. This decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space. A modification of this method is given that will exactly invert functions of compact support. These results generalize theorems of R. M. Perry and the author. A singular value decomposition for the Radon transform that integrates over spheres in Rⁿ containing the origin is also given. This follows from the singular value decomposition for R and yields the null space and a new inversion method for this transform.     

The generalized totally geodesic Radon transform and its application to texture analysis

The generalized totally geodesic Radon transform associates the mean values over spherical tori to a function f defined on ??³??, where the elements of ??³ are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the probability density function corresponding to the angle distribution function W. Eventually, this communication suggests a new approach to recover an approximation of f from data sampling W. At the same time it provides additional clarification of a recently suggested method applying reproducing kernels and radial basis functions by instructive insight into its involved geometry. The focus is on the correspondence of geometrical and group features rather than on the mapping of functions and their spaces. Copyright © 2008 John Wiley & Sons, Ltd.     

Remarks on the degenerate Radon transform in

The aim of this note is to prove endpoint boundedness of the generalized Radon transform which was introduced by Phong and Stein. M. Christ's combinatorial method is used to obtain restricted weak type at the endpoints. Also we show that the results of this note are essentially optimal.