Fast Filling Operations Used in the Reconstruction of Convex Lattice Sets

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 first improve the implementation of these four filling operations to a time complexity of O(N), and additionally we provide an implementation of a fifth filling operation (introduced in [2]) in O(N logN) that permits to decrease the overall time-complexity of the reconstruction algorithm to O(N logN). More generally, the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with Z) can be done in O(N logN)-time.