Reconstruction of Discrete Sets from Two or More Projections in Any Direction

During the workshop entitled “Discrete Tomography”, held in Volkrange on March 22, 1999, A. Kuba presented the open problem of reconstructing discrete sets satisfying the properties of connectivity and convexity by projections taken along many directions. In this paper, we study this problem, considering a similar property of discrete sets: the Q-convexity. In fact this property contains a certain kind of connectivity and convexity. The main result of this paper is a polynomial-time algorithm which is able to reconstruct Q-convex sets from their projections, when the directions of the projections and the ones of the Q-convexity are the same. Moreover, the algorithm works for any finite number of directions. keywords: algorithms, combinatorial problems, convexity, discrete tomography, discrete sets.