The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete Tomography since many types of integral matrices can be written as linear combinations of permutation matrices. This paper is an invitation to the study of X-rays of permutations from a combinatorial point of view. We p...

Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. In [1], an algorithm which performs four of these filling operations has a time complexity of O(N logN), where N is the size of projections, and leads to a reconstruction algorithm for convex polyominoes running in O(N logN)-time. In this paper we fir...

An extension of the basic image reconstruction problem in discrete tomography is considered: given a graph G = (V, E) and a family P of chains Pi together with vectors h(Pi) = (h 1 i , ..., h k i ), one wants to find a partition V , ..., V k of V such that for each Pi and each color j, |V ∩Pi| = hji . An interpretation in terms of scheduling is presented. We consider special cases of graphs and...

Reconstructing binary matrices from their row, column, diagonal, and antidiagonal sums (also called projections) plays a central role in discrete tomography. One of the main difficulties in this task is that in certain cases the projections do not uniquely determine the binary matrix. This can yield an extremely large number of (sometimes very different) solutions. This ambiguity can be reduced...

A graph-theoretical model is used to show that a special case of image reconstruction problem (with 3 colors) can be solved in polynomial time. For the general case with 3 colors, the complexity status is open. Here we consider that among the three colors there is one for which the total number of multiple occurrences in a same line (row or column) is bounded by a fixed parameter. There is no a...