Approximate X-rays reconstruction of special lattice sets

Sometimes the inaccuracy of the measurements of the X-rays can give rise to an inconsistent reconstruction problem. In this paper we address the problem of reconstructing special lattice sets in Z2 from their approximate X-rays in a finite number of prescribed lattice directions. The class of “strongly Q-convex sets” is taken into consideration and a polynomial time algorithm for reconstructing members of that class with line sums having possibly some bounded differences with the given X-ray values is provided. In particular, when these differences are zero, the algorithm exactly reconstructs any set. As a result, this algorithm can also be used to reconstruct convex subsets of Z2 from their exact X-rays in a finite set of suitable prescribed lattice directions.