Reconstruction of convex lattice sets from tomographic projections in quartic time

Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. Many algorithms have been published giving fast implementations of these operations, and the best running time [S. Brunetti, A. Daurat, A. Kuba, Fast filling operations used in the reconstruction of convex lattice sets, in: Proc. of DGCI 2006, in: Lecture Notes in Comp. Sci., vol. 4245, 2006, pp. 98–109] is O(N2logN) time, where N is the size of projections. In this paper we improve this result by providing an implementation of the filling operations in O(N2). As a consequence, we reduce the time-complexity of the reconstruction algorithms for many classes of lattice sets having some convexity properties. In particular, the reconstruction of convex lattice sets satisfying the conditions of Gardner–Gritzmann [R.J. Gardner, P. Gritzmann, Discrete tomography: Determination of finite sets by X-rays, Trans. Amer. Math. Soc. 349 (1997) 2271–2295] can be performed in O(N4)-time.