Biorthogonal decompositions of the Radon transform by multivariate interpolation

The concept of biorthogonal and singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. By application of the theory of multivariate interpolation, e. g. the set of Lagrange polynomials with respect to the space of homogeneous spherical polynomials, we determine new biorthogonal decompositions of the Radon transform. We consider the case of functions with support in the unit ball and the case of functions with support ?^r. In both cases we assume that the functions are square integrable with respect to some weight functions. In the important special case of square integrable functions with respect to the unit ball the structure of the biorthogonal decompositions is easier in comparison with the known singular and biorthogonal decompositions. Especially the calculation of the unknown expansion coefficients can be done by using arbitrary fundamental systems (μ-resolving data set in terms of tomography with a minimum number of nodes) and simplifies essentially. The decompositions are based on a system of zonal (ridge) Gegenbauer (ultraspherical) polynomials which are used in the theory of the Radon transform and in the field of numerical algorithms for the inversion of the transform.     

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.     

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.     

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.     

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Integration over the Space of Functions and Poincaré Series Revisited

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.     

Biorthogonal decompositions of the Radon transform

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     

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.     

The Radon–Kipriyanov transform of the generalized spherical mean of a function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform K_γ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.     

Fourier Reconstruction of Functions from their Nonstandard Sampled Radon Transform

In this article, we suggest a new Fourier transform based algorithm for the reconstruction of functions from their nonstandard sampled Radon transform. The algorithm incorporates recently developed fast Fourier transforms for nonequispaced data. We estimate the corresponding aliasing error in dependence on the sampling geometry of the Radon transform and confirm our theoretical results by numerical examples.     

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion of G^∞-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.     

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.     

$BMO, H^1$, and Calderón-Zygmund operators for non doubling measures

Given a Radon measure on , which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space when is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming doubling. For instance, Calderón-Zygmund operators which are bounded on are also bounded from into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space . Using a sharp maximal operator it is shown that operators which are bounded from into the new BMO space and from its predual into must be bounded on , . From this result one can obtain a new proof of the T(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on with functions of the new BMO are bounded on . Received February 18, 2000 / Published online October 11, 2000     

On topological Radon transformations

We construct a functor, which we call the topological Radon transform, from a category of complex algebraic varieties with morphisms given by divergent diagrams, to constructible functions. The topological Radon transform is thus the composition of a pull-back and a push-forward of constructible functions. We show that the Chern-Schwartz-MacPherson transformation makes the topological Radon transform of constructible functions compatible with a certain homological Verdier-Radon transform. We use this set-up to prove, given a projective variety X, a formula for the Chern-Mather class of the dual variety in terms of that of X.     

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 new proof of the support theorem and the range characterization for the Radon transform

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.     

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.     

The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials I: Mellin Transform Techniques

The Fourier transform of orthogonal polynomials with respect to their own orthogonality measure defines the family of Fourier–Bessel functions. We study the asymptotic behaviour of these functions and of their products, for large real values of the argument. By employing a Mellin analysis we construct a general framework to exhibit the relation of the asymptotic decay laws to certain dimensions of the orthogonality measure, that are defined via the divergence abscissas of suitable integrals. The unifying r?le of Mellin transform techniques in deriving classical and new results is underlined. Submitted: November 5, 2004. Accepted: January 6, 2006.     

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.