  This paper studies the problem of reconstructing binary matrices that are
only accessible through few evaluations of their discrete X-rays. Such question
is prominently motivated by the demand in material science for developing a
tool for the reconstruction of crystalline structures from their images
obtained by high-resolution transmission electron microscopy. Various
approaches have been suggested for solving the general problem of
reconstructing binary matrices that are given by their discrete X-rays in a
number of directions, but more work have to be done to handle the ill-posedness
of the problem. We can tackle this ill-posedness by limiting the set of
possible solutions, by using appropriate a priori information, to only those
which are reasonably typical of the class of matrices which contains the
unknown matrix that we wish to reconstruct. Mathematically, this information is
modelled in terms of a class of binary matrices to which the solution must
belong. Several papers study the problem on classes of binary matrices on which
some connectivity and convexity constraints are imposed. We study the
reconstruction problem on some new classes consisting of binary matrices with
periodicity properties, and we propose a polynomial-time algorithm for
reconstructing these binary matrices from their orthogonal discrete X-rays.
