Regularizing the method of reconstructing a function on the basis of its spherical Radon transform
type: Item
instance of: scholarly article
title: Regularizing the method of reconstructing a function on the basis of its spherical Radon transform
zbMATH Open document ID: 1259.65208
DOI: 10.3103/S0278641912010098
author: O. V. Shestakov
published in: Moscow University Computational Mathematics and Cybernetics
publication date: 2012-11-20T00:00:00Z
review text: Let \(f\in C_0^{\infty}(U)\), where \(U\) is the unit disc in \({\mathbb R}^2\). The spherical Radon transform is determined as \(Rf(p,r)=(1/2\pi) \int_{S^1} f(p+r\theta) d\theta\), \(S^1=(\cos\varphi,\sin\varphi)\), \(\varphi\in [0,2\pi)\), \(r>0\). Recently, D. Finch and M. Haltmeier have obtained the following inversion formula: \[ f(x)=(1/2\pi) \Delta_x \int_{S^1} \int_0^2 r Rf(p,r) \log | r^2-| x-p|^2| dr dp. \] The author suggests a regularized version of the formula applicable to the case where spherical projections are registered with a noise for a finite number of points \(p\). Error estimates for the reconstruction procedure are given.
reviewed by: Mikhail Yu. Kokurin
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 65R32
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 6108659
zbMATH Keywords: spherical Radon transform
zbMATH Keywords: tomographic images
zbMATH Keywords: regularization
zbMATH Keywords: incomplete data
zbMATH Keywords: error estimates
MaRDI profile type: Publication
cites work: Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact solution
cites work: Inversion of Spherical Means and the Wave Equation in Even Dimensions
cites work: A solution of the computer tomography paradox and estimating the distances between the densities of measures with the same marginals
cites work: Estimation of the distance between the densities of probabilistic measures with close projections
cites work: The Radon transform
cites work: Inversion of the attenuated Radon transform
description: scientific article; zbMATH DE number 6108659