Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, Reconstructing convex polyominoes from their horizontal and vertical projections, Theoretical Comput. Sci. 155 (1996), 321-347. CMP 96:09
  • 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 Comput. Geom. 5 (1990), no. 3, 223–242. MR 1036872,
  • 4. Shi-kuo Chang, The reconstruction of binary patterns from their projectionw, Comm. ACM 14 (1971), 21–25. MR 0285156,
  • 5. H. E. Chrestenson, Solution to Problem 5014, Amer. Math. Monthly 70 (1963), 447-448.
  • 6. M. G. Darboux, Sur un problème de géométrie élémentaire, Bull. Sci. Math. 2 (1878), 298-304.
  • 7. Herbert Edelsbrunner and Steven S. Skiena, Probing convex polygons with X-rays, SIAM J. Comput. 17 (1988), no. 5, 870–882. MR 961045,
  • 8. P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. II. Discrete case, Discrete Math. 91 (1991), no. 2, 149–159. MR 1124762,
  • 9. R. J. Gardner, Geometric Tomography, Cambridge University Press, New York, 1995. CMP 96:02
  • 10. R. J. Gardner and P. Gritzmann, Successive determination and verification of polytopes by their X-rays, J. London Math. Soc. (2) 50 (1994), 375-391.
  • 11. R. J. Gardner, P. Gritzmann, and D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays, (1996), in preparation.
  • 12. R. J. Gardner and P. McMullen, On Hammer’s X-ray problem, J. London Math. Soc. (2) 21 (1980), no. 1, 171–175. MR 576194,
  • 13. R. Gordon and G. T. Herman, Reconstruction of pictures from their projections, Comm. Assoc. Comput. Machinery 14 (1971), 759-768.
  • 14. Fernando Q. Gouvêa, 𝑝-adic numbers, Universitext, Springer-Verlag, Berlin, 1993. An introduction. MR 1251959
  • 15. A. Heppes, On the determination of probability distributions of more dimensions by their projections, Acta Math. Acad. Sci. Hungar. 7 (1956), 403-410. MR 19:70f
  • 16. C. Kisielowski, P. Schwander, F. H. Baumann, M. Seibt, Y. Kim, and A. Ourmazd, An approach to quantitative high-resolution transmission electron microscopy of crystalline materials, Ultramicroscopy 58 (1995), 131-155.
  • 17. Neal Koblitz, 𝑝-adic numbers, 𝑝-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003
  • 18. D. Kölzow, A. Kuba, and A. Volčič, An algorithm for reconstructing convex bodies from their projections, Discrete Comput. Geom. 4 (1989), no. 3, 205–237. MR 988743,
  • 19. G. G. Lorentz, A problem of plane measure, Amer. J. Math. 71 (1949), 417-426. MR 10:519c
  • 20. A. Rényi, On projections of probability distributions, Acta Math. Acad. Sci. Hungar. 3 (1952), 131-142. MR 14:771e
  • 21. Herbert John Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. MR 0150048
  • 22. P. Schwander, C. Kisielowski, M. Seibt, F. H. Baumann, Y. Kim, and A. Ourmazd, Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy, Physical Review Letters 71 (1993), 4150-4153.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52C05, 52C07, 52A20, 52B20, 68T10, 68U05, 82D25, 92C55

Retrieve articles in all journals with MSC (1991): 52C05, 52C07, 52A20, 52B20, 68T10, 68U05, 82D25, 92C55

Additional Information

R. J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063

Peter Gritzmann
Affiliation: Fb IV, Mathematik, Universität Trier, D-54286 Trier, Germany

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