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...     

On reconstruction of density from a finite set of values of the radon transformation

In this paper, we give estimates of approximate (with respect to various metrics) reconstruction of smoothed density from a finite set of values of the Radon transformations for various smoothing kernels. Bibligraphy: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 181–185. Translated by S. Yu. Pilyugin.     

Pseudopolynomial algorithms for the solution of backpack problems

Classical backpack problems are considered and computational procedures for exact solutions are suggested. The method implemented in the algorithms combines dynamic programming with sequential construction and elimination of inefficient alternatives; it produces a gain in computer operation speed.Translated from Dinamicheskie Sistemy, No. 5, pp. 98–103, 1985.     

Mutation Polynomials and Oriented Matroids

Several polynomials are of use in various enumeration problems concerning objects in oriented matroids. Chief among these is the Radon catalog. We continue to study these, as well as the total polynomials of uniform oriented matroids, by considering the effect on them of mutations of the uniform oriented matroid. The notion of a ``mutation polynomial' is introduced to facilitate the study. The affine spans of the Radon catalogs and the total polynomials in the appropriate rational vector spaces of polynomials are determined, and bases for the Z -modules generated by the mutation polynomials are found. The Radon polynomials associated with alternating oriented matroids are described; it is conjectured that a certain extremal property, like that held by cyclic polytopes among simplicial polytopes, is possessed by them. Received November 20, 1998, and in revised form August 21, 1999. Online publication May 19, 2000.     

Numerical inversion of the Radon transform

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.     

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。     

Convergence analysis of the loping OS-EM iterative version of the circular radon transform

The loping OS-EM iteration is a numerically efficient regularization method for solving ill-posed problems. In this article we investigate the loping OS-EM iterative method in connection with the circular Radon transform. We show that the proposed method converges weakly for the noisy data. Numerical tests are presented for a linear problem related to photoacoustic tomography.     

CONVERGENCE ANALYSIS OF THE LOPING OS-EM ITERATIVE VERSION OF THE CIRCULAR RADON TRANSFORM

The loping OS-EM iteration is a numerically efficient regularization method for solving ill-posed problems. In this article we investigate the loping OS-EM iterative method in connection with the circular Radon transform. We show that the proposed method converges weakly for the noisy data. Numerical tests are presented for a linear problem related to photoacoustic tomography.     

Local tomography with nonsmooth attenuation

Local tomography for the Radon transform with nonsmooth attenuation is proposed and justified. The main theoretical tool is analysis of singularities of pseudodifferential operators with nonsmooth symbols. Results of numerical testing of local tomography are presented.

     

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.     

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.     

On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inL ^p and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.Dedicated to Constantine Dafermos on his 60^th birthday     

Radon transforms and lamplighter random walks

We compute the spectrum of the lamplighter random walk in the case where the underlying graph is a path, representing the state space as a product of two rooted q-ary trees and using suitable Radon transforms. We analyze with the same techniques two additional examples in which the action of the automorphism group on the state space has orbits and the restriction on each orbit is not multiplicity free. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

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.     

Computation of the differentiation matrix for the Hermite interpolating polynomials

We obtain formulas for computing the elements of the differentiation matrix for special cases of the Hermite interpolating polynomials. They are expressed in terms of the elements of the differentiation matrices of the Lagrange interpolating polynomials in various systems of interpolation nodes, which can easily be calculated on a computer. These formulas find application in numerical realization of collocation finiteelement methods for solving differential problems.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 43–49.     

Numerical solution of some three-dimensional problems of elasticity theory by Vekua's method

We consider the solution of the problem of elastic equilibrium of a three-dimensional orthotropic plate in the absence of displacements on the end surfaces under the action of forces applied to the lateral surfaces. The solution of the original problem by Vekua's method is reduced to the solution of a recursive sequence of two-dimensional problems. A numerical solution of these problems is obtained by computer using the finite-difference method. The effect of the number of Legendre polynomials on the accuracy with which the boundary conditions are satisfied is investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 77–84, 1986.     

Radon transformations and zonoids

The class of compact sets known as zonoids or Steiner's (compact) sets, i.e., compact sets that are positive linear combinations (possibly, continuous ones) of segments, are described in terms of the Radon transformation.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 254–260, February, 1996.This research was partially supported by the Russian Foundation for Basic Research.     

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.