Discrete tomography: Determination of finite sets by Xrays
Authors:
R. J. Gardner and Peter Gritzmann
Journal:
Trans. Amer. Math. Soc. 349 (1997), 22712295
MSC (1991):
Primary 52C05, 52C07; Secondary 52A20, 52B20, 68T10, 68U05, 82D25, 92C55
MathSciNet review:
1376547
Fulltext PDF Free Access
Abstract 
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Abstract: We study the determination of finite subsets of the integer lattice , , by Xrays. In this context, an Xray of a set in a direction gives the number of points in the set on each line parallel to . For practical reasons, only Xrays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of (i.e., finite subsets with ) are determined, among all such sets, by their Xrays in these directions. We also show that three Xrays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's Xray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in have the property that convex subsets of are determined, among all such sets, by their Xrays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.
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 2.
 F. Beauvais, Reconstructing a set or measure with finite support from its images, Ph. D. dissertation, University of Rochester, Rochester, New York, 1987.
 3.
 G. Bianchi and M. Longinetti, Reconstructing plane sets from projections, Discrete Comp. Geom. 5 (1990), 223242. MR 91f:68200
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Additional Information
R. J. Gardner
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, Washington 982259063
Email:
gardner@baker.math.wwu.edu
Peter Gritzmann
Affiliation:
Fb IV, Mathematik, Universität Trier, D54286 Trier, Germany
Email:
gritzman@dm1.unitrier.de
DOI:
http://dx.doi.org/10.1090/S0002994797017418
PII:
S 00029947(97)017418
Keywords:
Tomography,
discrete tomography,
Xray,
projection,
lattice,
lattice polygon,
convex body,
$p$adic valuation
Received by editor(s):
October 3, 1995
Additional Notes:
First author supported in part by the Alexander von Humboldt Foundation and by National Science Foundation Grant DMS9501289; second author supported in part by the Deutsche Forschungsgemeinschaft and by a Max Planck Research Award.
Article copyright:
© Copyright 1997
American Mathematical Society
