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Alexander Katsevich 《Transactions of the American Mathematical Society》2001,353(3):1089-1102
Three new range theorems are established for the dual Radon transform : on functions that do not decay fast at infinity (and admit an asymptotic expansion), on , and on . Here , and acts on even functions .
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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. 相似文献
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Alexander Katsevich 《Mathematische Nachrichten》2005,278(4):437-450
Consider the Poincare unit disk model for the hyperbolic plane H 2. Let Ξ be the set of all horocycles in H 2 parametrized by (θ, p), where eiθ is the point where a horocycle ξ is tangent to the boundary |z| = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R* : μ(θ, p) → (z) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that Pm(d/dp)(μm(p)ep) be even for all m ∈ ?. Here Pm(d/dp) is a family of differential operators introduced by Helgason, and μm(p) are the coefficients of the Fourier series expansion of μ(θ, p). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Bernd Droste 《manuscripta mathematica》1983,42(2-3):289-296
The aim of this note is to give a new and elementary proof of the support theorem for the Radon transform, which is based only on the projection theorem and the Paley-Wiener theorem for the Fourier transform. The idea is to solve a certain system of linear equations in order to determine the coefficients of a homogeneous polynomial (interpolation problem). By the same method, we get a short proof of the range characterization for Radon transforms of functions supported in a ball. 相似文献
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Michael Rosier 《Numerische Mathematik》1995,72(2):263-283
Summary.
The concept of singular value decompositions is a valuable tool
in the examination of ill-posed inverse problems
such as the inversion of the Radon transform. A singular value
decomposition depends on the determination of suitable orthogonal systems
of eigenfunctions of the operators
, .
In this paper we consider a new approach which generalizes this concept.
By application of biorthogonal instead of orthogonal functions we
are able to apply a larger class of function sets in order to
account for the structure of the eigenfunction spaces. Although it is
preferable to use eigenfunctions it is still possible to consider
biorthogonal function systems which are not eigenfunctions of the operator.
With respect to the Radon transform for functions with support in the
unit ball we apply the system of Appell polynomials which is a natural
generalization of the univariate system of Gegenbauer (ultraspherical)
polynomials to the multivariate case. The corresponding biorthogonal
decompositions show some advantages in comparison with the known
singular value decompositions. Vice versa by application of our
decompositions we are able to prove new properties of the Appell
polynomials.
Received October 19, 1993 相似文献
9.
Frank Natterer 《Numerische Mathematik》1978,30(1):81-91
Summary The problem of inverting the Radon transform, i.e. the reconstruction of a function inR
2 from its line integrals arises e.g. in computerized tomography and in nondestructive testing. In the present paper the least squares method with piecewise constant trial functions is investigated. An error estimate is derived. An implementation using the fast Fourier transform is described and numerical results are reported. 相似文献
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Frank Natterer 《Numerische Mathematik》1979,32(4):431-438
We given an inversion formula for the attenuated Radon transform $$\bar \omega _L $$ for constant attenuation μ which is accurate up toO(µ 4) We also derive an integral equation of the second kind for the solution ofR μf=g. The implementation of both methods for the inversion ofR μ consists basically of a generalized filtered backprojection algorithm. Numerical examples are presented. 相似文献
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A new approach to the range conditions for the exponential Radon transform is introduced.
Partially supported by NSF EPSCOR Grant and Wesley Foundation Grant no. 9012019.
Partially supported by NSF Grant 33-1807-231. 相似文献
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KANG Qian-qian 《高校应用数学学报(A辑)》2012,27(2)
研究了速降函数在R2n中迷向d-平面上的Radon变换,d<n,并分别给出d=1,d=2的情况下,此Radon变换的像满足的二阶偏微分方程组. 相似文献
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A. G. Ramm 《Mathematical Methods in the Applied Sciences》1992,15(3):159-166
The Radon transform R(p, θ), θ∈Sn?1, p∈?1, of a compactly supported function f(x) with support in a ball Ba of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on Sn?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data. 相似文献
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S. G. Gindikin 《Monatshefte für Mathematik》1992,113(1):23-32
This is an expanded version of a lecture given in Vienna on January 17, 1990 in which the significance of a famous paper by Radon, and the developments originating from it, were discussed. 相似文献
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Andrew Comech 《纯数学与应用数学通讯》1998,51(5):537-550
We derive the regularity properties of the Radon transform of Melrose and Taylor for the scattering on a compact, convex obstacle with a smooth boundary. The result is formulated in terms of the highest order of contact of tangent lines with the boundary of an obstacle. The main ingredients of the proof are the estimates for degenerate oscillatory integral operators and almost orthogonal decompositions. © 1998 John Wiley & Sons, Inc. 相似文献
19.
Donald C. Solmon 《Mathematical Methods in the Applied Sciences》1995,18(9):687-695
The exponential Radon transform, which arises in single photon emission computed tomography, is defined by ? ?(μ:ω,s) = ∫R?(sω + tomega;?) eμt dt?. Here ? is a compactly supported distribution in the plane which represents the location and intensity of a radio-pharmaceutical in a body of constant, but unknown, attenuation μ, and ω is a direction. The identification problem is to determine the attenuation μ from the data ?? with ? unknown. We will show that μ can be determined from the data if and only if ? is not a radial distribution and give formulae for computing μ when ? is not radial. 相似文献
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Carlos A. Berenstein Enrico Casadio Tarabusi Á rpá d Kurusa 《Proceedings of the American Mathematical Society》1997,125(2):455-461
A correspondence among the totally geodesic Radon transforms-as well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.