Discrete tomography determination of bounded lattice sets from four X-rays

Discrete Tomography: Determination of Finite Sets by X-Rays

We study the determination of finite subsets of the integer lattice Zn, 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 a...

Reconstruction of lattice sets from infinite X-rays

For any infinite non-null vectors there is always a subset of Z2 whose X-rays along fixed directions are the given vectors. If there are only two directions and the vectors are periodic, then the set can be chosen periodic, it’s not true with more than two directions.

Reconstruction of Lattice Sets From Infinite Rays

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 Z 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 ...

Unique reconstruction of bounded sets in discrete tomography

We consider the problem of recognizing arbitrary finite subsets of Z2 by X-rays corresponding to a small set of directions S. We show that if we fix any rectangle A in Z2 then there exists a so called valid set S of four directions (at least when A is not too ”small”) depending only on the size of A such that any two subsets of A can be distinguished, using these directions only. By our approac...