Three Problems in Discrete Tomography: Reconstruction,Uniqueness, and Stability

When the available X-ray data is insucient, the accuracy of the tomographic reconstruction is likely to be inadequate. The assumption that the densities of the object materials are restricted to few known grey level values is on the basis of the denition of Discrete Tomography and permits partially to deal with the accuracy problem. The objects studied here are lattice sets and their X-rays in a lattice direction count the number of their points lying on each line parallel to the given direction. In the literature, the reconstruction, and uniqueness issues have been intensively studied from dierent viewpoints. In the general setting, the problems are easy when the X-rays are taken from two directions, and become hard for more than two directions. Analogously negative results arise when the stability of the reconstruction task is studied. A common way to deal with these problems is to incorporate a priori knowledge about the object. Therefore, in the literature special classes of geometric objects are considered such as convex, Q-convex and additive lattice sets. A dierent restriction, in the same spirit, consists in considering bounded sets, i.e., subsets of a given rectangular grid. We review and discuss the three problems especially for Q-convex and bounded additive lattice sets. Finally we discuss some recent results, and perspectives.