Discretization in 2D and 3D orders Michel Couprie, Gilles Bertrand, * and Yukiko Kenmochi Laboratoire A 2 SI, ESIEE Cit e Descartes B.P. 99, 93162 Noisy-Le-Grand Cedex, France Received 15 May 2002; received in revised form 4 September 2002; accepted 17 October 2002 Abstract Among the different discretization schemes that have been proposed and studied in the literature, the supercover is a very natural one, and furthermore presents some interesting properties. On the other hand, an important structural property does not hold for the super- cover in the classical framework: the supercover of a straight line (resp. a plane) is not a dis- crete curve (resp. surface) in general. We follow another approach based on a different, heterogenous discrete space which is an order, or a discrete topological space in the sense of Paul S. Alexandroff. Generalizing the supercover discretization scheme to such a space, we prove that the discretization of a plane in R 3 is a discrete surface, and we prove that the discretization of the boundary of any closed convex set X is equal to the boundary of the discretization of X . Ó 2003 Elsevier Science (USA). All rights reserved. Keywords: Discretization; Topology; Orders; Supercover; Discrete surfaces 1. Introduction An abundant literature is devoted to the study of discretization schemes. Let E be an ‘‘Euclidean’’ space, and let D be a ‘‘discrete’’ space related to E . Typically, one can take E¼ R n and D¼ Z n (n ¼ 2; 3), but we do not limit ourselves to this case. A dis- cretization scheme associates, to each subset X of E; a subset DðX Þ of D which is called the discretization of X . Different discretization schemes have been proposed and compared with respect to some fundamental geometric, topological and struc- Graphical Models 65 (2003) 77–91 www.elsevier.com/locate/gmod * Corresponding author. Fax: +33-1-45-92-66-99. E-mail addresses: coupriem@esiee.fr (M. Couprie), bertrand@esiee.fr (G. Bertrand), kenmochy @esiee.fr (Y. Kenmochi). URL: www.esiee.fr/~coupriem/sdi_eng. 1524-0703/03/$ - see front matter Ó 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S1524-0703(03)00003-1