The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk under the additional condition that the curves that do not meet coincide with the corresponding straight lines.     

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.     

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}} .     

Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

The Radon transform $\mathcal{R}f$ of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f∈L ₂(SO(3)) from $\mathcal{R}f\in L_{2}(S^{2}\times S^{2})$ which is known only on a discrete set of points. Since one has only partial information about $\mathcal{R}f$ the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform $\mathcal{R}f$ of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.     

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...     

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.     

Application of the Regularization Method to the Numberical Inversion of a Class of Generalized Radon Transform

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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.     

Spherical Radon Transform and Wavelets

In this paper we shall approach the problem of localization and inversion of spherical Radon transform. This motivates the construction of particular wavelets, based on a suitable representation of the unitary group via Gel'fand's projective model Pℝ³     

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.     

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 ${\mathbb {R}}^{2n}$ .     

BV Capacity on Generalized Grushin Plane

In this paper we introduce and investigate the so-called BV capacity on the generalized Grushin plane \(\mathbb {G}^2_\alpha \), thereby discovering some sharp trace and BV isocapacity inequalities on \(\mathbb {G}^2_\alpha \).     

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.     

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.     

Fourier Transform on the Lobachevsky Plane and Operational Calculus

Functional Analysis and Its Applications - The classical Fourier transform on the line sends the operator of multiplication by $$x$$ to $$ifrac{d}{dxi}$$ and the operator $$frac{d}{d x}$$ of...     

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...     

离散余弦变换的一种推广

离散余弦变换(DCT)在数字信号、图像处理、频谱分析、数据压缩和信息隐藏等领域有着广泛的应用.推广离散余弦变换,给出一个包含三个参数的统一表达式,并证明在许多情形新变换是正交变换.最后给出一种新型离散余弦变换,并证明它是正交变换.     

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 .