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 ([7]) is O(N2 log N) 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. Especially, the reconstruction of convex lattice sets satisfying the conditions of GardnerGritzmann [12] can be performed in O(N4)-time. Key words: Discrete Tomography, Convexity, Filling Operations. A large part of this paper is extracted from the conference article [7] which was a joint work with Attila Kuba. With our deep sorrow, Attila did not see the end of the story. This paper is dedicated to his memory.

• Type: Journal
• Year: 2008
• Where: TCS
• Authors: Sara Brunetti, Alain Daurat
• Keywords: Convexity Constraints, Discrete Tomography, Lattice Sets, TCS 2008, Theoretical Computer Science