Reconstructing digital sets from X-rays

Abstract

In this paper, we study the problem of determining digital sets by means of their X-rays. An X-ray of a digital set F in a direction u counts the number of points in F on each line parallel to u. A class Φ's of digital sets is characterized by the set U of directions if among all Ф's elements, each element in Ф is determined by its X-rays in U's directions. This discrete tomography's problem is of primary importance in reconstructing three-dimensional crystals from two-dimensional images taken by an electron microscope by measuring the number of atoms lying on each line in some directions (see [16]). There are some classes of digital sets that satisfy some connection and convexity conditions and that cannot be characterized by any set of directions [2]. Gardner and Gritzmann [10] show that there are some sets of four prescribed directions that characterize the class of totally convex sets (sets which are convex with respect to all the directions). We make the conjecture that there is a certain set U of four directions that characterizes the class of convex polyominoes (sets which are convex with respect to only two directions: horizontal and vertical). In order to give experimental evidence of this conjecture, we present a polynomial algorithm that reconstructs convex polyominoes from their X-rays.