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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Discrete tomography:
Determination of finite sets by X-rays


Authors: R. J. Gardner and Peter Gritzmann
Journal: Trans. Amer. Math. Soc. 349 (1997), 2271-2295
MSC (1991): Primary 52C05, 52C07; Secondary 52A20, 52B20, 68T10, 68U05, 82D25, 92C55
MathSciNet review: 1376547
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Abstract: We study the determination of finite subsets of the integer lattice ${\Bbb Z}^n$, $n\ge 2$, by X-rays. In this context, an X-ray of a set in a direction $u$ gives the number of points in the set on each line parallel to $u$. For practical reasons, only X-rays 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 ${\Bbb Z}^n$ (i.e., finite subsets $F$ with $F={\Bbb Z}^n\cap {\mathrm {conv}}\,F$) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in ${\Bbb Z}^2$ have the property that convex subsets of ${\Bbb Z}^2$ are determined, among all such sets, by their X-rays 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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Additional Information

R. J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
Email: gardner@baker.math.wwu.edu

Peter Gritzmann
Affiliation: Fb IV, Mathematik, Universität Trier, D-54286 Trier, Germany
Email: gritzman@dm1.uni-trier.de

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01741-8
PII: S 0002-9947(97)01741-8
Keywords: Tomography, discrete tomography, X-ray, 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 DMS-9501289; second author supported in part by the Deutsche Forschungsgemeinschaft and by a Max Planck Research Award.
Article copyright: © Copyright 1997 American Mathematical Society