Conical Uniqueness Sets for the Spherical Radon Transform

Let K be a cone in Rⁿ. Then K is a uniqueness set for the sphericalRadon transform if and only if it is not contained in the zeroset of any (nontrivial) homogeneous harmonic polynomial. A localversion of this result is also proved. 1991 Mathematics SubjectClassification 44A12.     

傅立叶变换与小波变换的比较教学

本文针对小波变换教学中小流变换概念理解困难的问题，提出了一种比较教学方法，通过分析小波变换与傅立叶变换之间的联系，并从四个方面进行对比，清楚地描述了小波变换的本质，从而对加深对小波变换的理解。     

Radon Transform on Sobolev Spaces

Siberian Mathematical Journal - The Radon transform  $ R $ maps a function  $ f $ on  $ {??}^{n} $ to the family of the integrals of  $ f $ over...     

Radon Transform on the Torus

We consider the Radon transform on the (flat) torus \mathbbTⁿ = \mathbbRⁿ/\mathbbZⁿ{\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^n} defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give a characterization of the image of the space of smooth functions on \mathbbTⁿ{\mathbb{T}^{n}} .     

The Lagrangian Radon Transform and the Weil Representation

We consider the operator $\mathcal {R}$ , which sends a function on ${\mathbb {R}}^{2n}$ to its integrals over all affine Lagrangian subspaces in ${\mathbb {R}}^{2n}$ . We discuss properties of the operator $\mathcal {R}$ and of the representation of the affine symplectic group in several function spaces on ${\mathbb {R}}^{2n}$ .     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

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.     

Spherical Singular Integrals, Monogenic Kernels and Wavelets on the Three-Dimensional Sphere

The construction of wavelets relies on translations and dilations which are perfectly given in . On the sphere translations can be considered as rotations but it is difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches to obtain wavelets which are worth to be considered. The first concept goes back to W. Freeden and collaborators who define wavelets by means of kernels of spherical singular integrals. The other concept developed by J.P. Antoine and P. Vandergheynst is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define singular integrals and kernels of singular integrals on the three dimensional sphere which are also approximate identities. In particular the Cauchy kernel in Clifford analysis is a kernel of a singular integral, the singular Cauchy integral, and an approximate identity. Furthermore, we will define wavelets on the 3-dimensional sphere by means of kernels of singular integrals. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš Received: October, 2007. Accepted: February, 2008.     

Radon变换在断层成像中的应用

介绍了著名的 Radon变换和断层成像的图像重建 .并通过图像实例直观地展示了重建效果 .     

On the Inversion of the xfct Radon Transform

It has been recently shown by Fokas [ 1 - 3 ] and Novikov [ 4 ] that the spectral analysis of a particular partial differential equation yields the inversion formula for the problem of computerized emission tomography. In this paper, we show that a similar analysis can be made for the case of X‐ray fluorescence tomography.     

Radon变换反演问题的数值方法

利用Tchebycheff多项式和古典Radon变换反演公式,本文得到了Tchebycheff变换对,从而导出了数值反演结果．     

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.     

Frames for the Continuous Spherical Wavelet Transform

In this paper we will construct frames for the continuous spherical wavelet transform. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling Regular Textures in Images Using the Radon Transform

Journal of Applied and Industrial Mathematics - The Radon transform is a major integral transform in computed tomography and a widely applied technique in computer vision and image analysis which...     

The Generalized Radon Transform on the Plane, the Inverse Transform, and the Cavalieri Conditions

In the two-dimensional case, the generalized Radon transform takes each function supported in a disk to the values of the integrals of that function over a family of curves. We assume that the curves differ only slightly from straight lines and the network formed by these curves has the same topological structure as the network of straight lines. Thus, the generalized Radon transform specifies a function on the set of straight lines. Under these conditions, we obtain a solution of the inversion problem for the generalized Radon transform and indicate a Cavalieri condition describing the range of this transform in the space of functions on the set of straight lines.     

Geometric Analysis and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets

Due to its unequalled advantages, the magnetic resonance imaging (MRI) modality has truly revolutionized the diagnosis and evaluation of pathology. Because many morphological anatomic details that may not be visualized by other high tech imaging methods can now be readily shown by diagnostic MRI, it has already become the standard modality by which all other clinical imaging techniques are measured. The unique quantum physical basis of the MRI modality combined with the imaging capabilities of current computer technology has made this imaging modality a target of interdisciplinary interest for clinicians, physicists, biologists, engineers, and mathematicians. Due to the fact that MRI scanners perform corticomorphic processing, this modality is by far more complex than all the other high tech clinical imaging techniques. The purpose of this paper is to outline a phase coherent wavelet approach to Fourier transform MRI. It is based on distributional harmonic analysis on the Heisenberg nilpotent Lie group G and the associated symplectically invariant symbol calculus of pseudodifferential operators. The contour and contrast resolution of MRI scans which is controlled by symplectic filter bank processing gives the noninvasive MRI modality superiority over X-ray computed tomography (CT) in soft tissue differentiation.     

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 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.