Reconstruction of convex bodies from brightness functions
type: Item
instance of: scholarly article
title: Reconstruction of convex bodies from brightness functions
zbMATH Open document ID: 1035.52001
author: Peyman Milanfar
author: Richard J. Gardner
published in: Discrete \& Computational Geometry
publication date: 2003-08-21T00:00:00Z
review text: The brightness function of a convex body in \(\mathbb{R}^{n}\) gives the \((n-1)\)-dimensional volume of its orthogonal projections onto hyperplanes. By Aleksandrov's projection theorem any two origin-symmetric convex bodies which have same brightness function must be equal. The authors provide algorithms for reconstructing an approximation to an unknown origin-symmetric convex body from finitely many values of its brightness function. Different variants, including an oracle-polynomial time algorithm, are discussed, and all of these algorithms proceed in two phases. First the values of the brightness function are used to find the surface area measure of an approximating polytope, then the polytope itself is reconstructed from its area measure. The convergence of the polytope to the body is proved under certain assumptions on the body, and the proof also provides estimates on the rate of convergence. Implementations of the presented algorithms suggest that they can work even with noisy data.
reviewed by: C. Peri
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B55
zbMATH DE Number: 1969088
zbMATH Keywords: computerized tomography
zbMATH Keywords: polynomial time algorithm
zbMATH Keywords: brightness function
zbMATH Keywords: converx body in \(\mathbb{R}^n\)
zbMATH Keywords: volume
zbMATH Keywords: orthogonal projections onto hyperplanes
zbMATH Keywords: approximating polytope
describes a project that uses: Qhull
MaRDI profile type: Publication
full work available at URL: s00454-002-0759-2
OpenAlex ID: W2000380654
DOI: 10.1007/S00454-002-0759-2
Recommended article: Determination of convex bodies by their brightness functions
Recommended article: Reconstruction problems of convex bodies from surface area measures and lightness functions
Recommended article: Reconstruction of convex bodies from moments
Recommended article: Reconstruction of convex bodies of revolution from the areas of their shadows
Recommended article: An algorithm for reconstructing convex bodies from their projections
Recommended article: Reconstruction of convex bodies from surface tensors
Recommended article: Q4038559
Recommended article: On the illumination of smooth convex bodies
Recommended article: Recovery of a convex body from the set of its projections
description: scientific article; zbMATH DE number 1969088