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 convex lattice sets. In \cite{g98}, an algorithm which performs four of these filling operations has a time complexity of $O(N^2\log(N))$ where $N$ is the size of X-rays. In this paper we improve this algorithm by including a fifth filling operation that is needed for the reconstruction with the same time-complexity. As a consequence the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with $\ZZ^2$) can be done in $O(N^4\log(N))$-time.