Total curvature and intersection tomography
type: Item
instance of: scholarly article
title: Total curvature and intersection tomography
zbMATH Open document ID: 0887.53050
author: Bernhard Rupp
published in: Advances in Mathematics
publication date: 1998-01-14T00:00:00Z
review text: A basic inverse problem of integral geometry is the following one: When is a ``one-dimensional'' set in the plane determined by the number of intersections with a variable line, i.e., by its Crofton transform [\textit{H. Steinhaus}, Colloq. Math. 3, 1-13 (1954; Zbl 0057.14602)]? In this article the author takes the class of \(K\)-sets, i.e., generalized sets, represented by finite collections of closed rectifiable curves of finite total absolute curvature. He shows how to reconstruct \(K\)-sets in the Euclidean plane from their projections onto variable lines (Theorem 1.1) [cf. also \textit{H. Fast}, Real. Anal. Exch. 19, 59-80 (1994; Zbl 0797.28002); \textit{T. J. Richardson}, Discrete Comput. Geom. 16, 21-31 (1996; Zbl 0859.53052)]. Furthermore, he proves a projection formula, relating the total absolute curvature of \(K\)-sets in the Euclidean plane to the average total variation of their projections (Theorem 1.2).
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 92C55
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 1066076
zbMATH Keywords: intersection tomography
zbMATH Keywords: stereology
zbMATH Keywords: total curvature
zbMATH Keywords: inverse problem
zbMATH Keywords: Crofton transform
zbMATH Keywords: \(K\)-sets
zbMATH Keywords: rectifiable curves
reviewed by: Eberhard Teufel
MaRDI profile type: Publication
full work available at URL: aima.1996.1640
OpenAlex ID: W2009357361
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DOI: 10.1006/AIMA.1996.1640
description: scientific article; zbMATH DE number 1066076