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.     

Local algorithms in exterior tomography

Exterior tomographic data are taken over lines outside a central region, and such data occur in the industrial nondestructive evaluation of large objects such as rockets. We explain, using microlocal analysis, which singularities are well reconstructed from exterior data, and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform [E.T. Quinto, Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform, J. Math. Anal. Appl. 95 (1983) 437–448]. We extend Lambda Tomography to exterior data and to limited angle exterior data. The algorithm is tested on industrial data from Perceptics, Inc.     

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.     

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).      

Integration of positive constructible functions against Euler characteristic and dimension

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411-416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula.     

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.     

一阶非线性周期方程的奇异点方法

本文应用奇异点理论,在g(x)为凹(凸)型函数时,给出周期系统(?)+a(t)g(x)=h(t)整体等价于Whitney意义下的尖点映射的结果.精确地说,算子Fx(t)=(?)+a(t)g(x(t))的奇异值集F(∑)为单连通超曲面并且将C[0,1]分成两个连通分支A1和A3,使得:(1)对周期为1的连续函数p(t)∈A1有唯一解.(2)对周期为1的连续函数p(t)∈A3恰有三个周期解.进一步,尖点集C的像集F(C)是C[0,1]中的,余维数等于2的子流形.对p∈F(C)有唯一解,而对p(t)∈F(∑)\F(C)恰有两个周期解.     

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.     

The inversion problem and applications of the generalized radon transform

We prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation. We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.     

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.     

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.     

The raised-cosine wavelets in computerized tomography

In Computerized Tomography (CT), an image must be recovered from its sampled projections in the form of values of the Radon transform. In this work a method of recovering the image is based on the properties of the raised-cosine wavelet. This wavelet has a closed form which allows for certain precomputations and avoids convolution. The rate of convergence of the resulting algorithm to the image density function is found under suitable hypotheses. This algorithm is then tested on the standard Shepp–Logan     

Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis

Texture analysis is used here as short term for analysis of crystallographic preferred orientation. Its major mathematical objective is the determination of a reasonable orientation probability density function and corresponding crystallographic axes probability density functions from experimentally accessible diffracted radiation intensity data. Since the spherical axes probability density function is modelled by the one-dimensional Radon transform for SO(3), the problem is its numerical inversion. To this end, the Radon transform is characterized as an isometry between appropriate Sobolev spaces. The mathematical foundations as well as first numerical results with zonal basis functions are presented.     

Boundary singularities with a simple decomposition

We define the decomposition of a boundary singularity as a pair (a singularity in the ambient space together with a singularity of the restriction to the boundary). We prove that the Lagrange transform is an involution on the set of boundary singularities that interchanges the singularities that occur in the decomposition of a boundary singularity. We classify the boundary singularities for which both of these singularities are simple. Bibliography: 8 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 55–69, 1991.     

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 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.     

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.     

Singularities of the solution of the laplacian in domains with circular edges

In this paper we generalize the results from [4] to special domains with curved edges. For general elliptic boundary value problems the behavior of the solutions near arbitrary, smooth edges is analyzed by Maz'ja and Rossmann [3]. First following Dauge [1] we derive a regularity theorem for the solution of the Dirichlet problem of the Laplacian with a decomposition into edge singularities of nontensor product form. In this case the regularity of the remainder term in the decomposition corresponds to the one in the two-dimensional case [2]. Following [4] we obtain a refined decomposition where all singularity terms are of tensor product form. We illustrate our results with several examples.