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 , an algorithm which performs four of these filling operations has a time complexity of O(N 2logN), where N is the size of projections, and leads to a reconstruction algorithm for convex polyominoes running in O(N 6 logN)-time. In this paper we first improve the implementation of these four filling operations to a time complexity of O(N 2), and additionally we provide an implementation of a fifth filling operation (introduced in ) in O(N 2logN) that permits to decrease the overall time-complexity of the reconstruction algorithm to O(N 4logN). More generally, the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with ℤ2) can be done in O(N 4logN)-time.
KeywordsDiscrete Tomography Convexity Filling Operations
- 8.Brunetti, S., Daurat, A.: Reconstruction of discrete sets from two or more X-rays in any direction. In: Proc. of IWCIA 2000, Université de Caen, pp. 241–258 (2000)Google Scholar
- 9.Knuth, D.E.: Balanced Trees (section 6.2.3). In: Sorting and Searching. The Art of Computer Programming, vol. 3, pp. 458–475. Addison-Wesley, Reading (1998)Google Scholar
- 10.Brunetti, S., Daurat, A.: Reconstruction of Q-convex sets. In: Herman, G.T., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications. Appl. Numer. Harmon. Anal., Birkhäuser (to appear)Google Scholar
- 14.Daurat, A.: Convexité dans le plan discret. Application à la tomographie. Ph.D thesis, LLAIC1, and LIAFA Université Paris 7 (2000)Google Scholar