Methods of integral geometry and recovering a function with compact support from its projections in unknown directions
type: Item
instance of: scholarly article
title: Methods of integral geometry and recovering a function with compact support from its projections in unknown directions
zbMATH Open document ID: 0654.44004
DOI: 10.1007/BF00140119
published in: Acta Applicandae Mathematicae
publication date: 1988-01-01T00:00:00Z
review text: The author discusses a method for recovering a function \(d(x_ 1,x_ 2)\) with compact support describing the distribution density of the matter of a planar particle from given projections in unknown directions: A planar asymmetric particle is rotated by unknown angles \(\phi_ 1,...,\phi_ n\) around the origin as its center of mass and projected onto the line \(x_ 2=0\). The problem is to find the relative angles \(\phi_ k-\phi_ 1\), \(k=2(1)n\). Let be \(d_{\phi}(\vec x):=d(X_{\phi}\vec x)\) with the rotation matrix \(D_{\phi}\) and \(M_ k(\phi):\int d_{\phi}(x_ 1)x_ 1^ kdx_ 1\) the kth moment. All arguments of recovering the angles (n\(\geq 7)\) are based on the fact that \(M_ k(\phi)\) is a homogeneous trigonometric polynomial of degree k determinated by its values at \((k+1)\) points. So one can recover the (s,t)th moments of \(d(x_ 1,x_ 2)\) with \(s+t\leq k\) and approximate \(d(x_ 1,x_ 2)\) by a polynomial. The result may be applied in tomography.
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 4067569
zbMATH Keywords: reconstruction from projections
zbMATH Keywords: integral geometry
zbMATH Keywords: Radon transform
zbMATH Keywords: electron microcopy
zbMATH Keywords: tomography
author: A. B. Goncharov
reviewed by: Wolfram Luther
MaRDI profile type: Publication
cites work: Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos
cites work: The X-ray transform
full work available at URL: https://doi.org/10.1007/bf00140119
OpenAlex ID: W2079477296
description: scientific article; zbMATH DE number 4067569