Discrete Tomography: Determination of Finite Sets by X-Rays

We study the determination of finite subsets of the integer lattice En, n > 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 En (i.e., finite subsets F with F = En n 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 22 have the property that convex subsets of 22 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.

[1]  P. Fishburn,et al.  Sets Uniquely Determined by Projections on Axes I , 1990 .

[2]  G. Darboux,et al.  Sur un problème de géométrie élémentaire , 1878 .

[3]  Shi-Kuo Chang,et al.  The reconstruction of binary patterns from their projections , 1971, Commun. ACM.

[4]  R. Gardner Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[5]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[6]  Jeffrey C. Lagarias,et al.  Sets uniquely determined by projections on axes II Discrete case , 1991, Discret. Math..

[7]  Steven Skiena,et al.  Probing Convex Polygons with X-Rays , 1988, SIAM J. Comput..

[8]  Gabriele Bianchi,et al.  Reconstructing plane sets from projections , 1990, Discret. Comput. Geom..

[9]  George G. Lorentz A Problem of Plane Measure , 1949 .

[10]  Peter Gritzmann,et al.  Successive Determination and Verification of Polytopes by their X-Rays , 1992, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[11]  Peter Schwander,et al.  An approach to quantitative high-resolution transmission electron microscopy of crystalline materials , 1995 .

[12]  Richard Gordon,et al.  Reconstruction of pictures from their projections , 1971, CACM.

[13]  Kim,et al.  Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy. , 1993, Physical review letters.

[14]  Maurice Nivat,et al.  Reconstructing Convex Polyominoes from Horizontal and Vertical Projections , 1996, Theor. Comput. Sci..

[15]  Attila Kuba,et al.  An algorithm for reconstructing convex bodies from their projections , 1989, Discret. Comput. Geom..

[16]  P. McMullen,et al.  On Hammer's X-Ray Problem , 1980 .

[17]  A. Rényi,et al.  On projections of probability distributions , 1952 .

[18]  N. Koblitz p-adic Numbers, p-adic Analysis, and Zeta-Functions , 1977 .

[19]  A. Heppes On the determination of probability distributions of more dimensions by their projections , 1956 .