The Euler characteristics of discrete objects and discrete quasi-objects

Assuming planar 4-connectivity and spatial 6-connectivity, we first introdu ce the curvature indices of the boundary of a discrete object, and, using t hese indices of points, we define the vertex angles of discrete surfaces as an extension of the chain codes of digital curves, Second, we prove the re lation between the number of point indices and the numbers of holes, genus, and cavities of an object. This is the angular Euler characteristic of a d iscrete object, Third, we define quasi-objects as the connected simplexes. Geometric relations between discrete quasi-objects and discrete objects per mit us to define the Euler characteristic for the planar 8-connected, and t he spatial 18- and 26-connected objects using these for the planar 4-connec ted and the spatial 6-connected objects. Our results show that the planar 4 -connectivity and the spatial 6-connectivity define the Euler characteristi cs of point sets in a discrete space. Finally, we develop an algorithm for the computation of these characteristics of discrete objects. (C) 1999 Acad emic Press.