The Euler Characteristics of Discrete Objects and Discrete Quasi-Objects

摘要

Assuming planar 4-connectivity and spatial 6-connectivity, we first introduce the curvature indices of the boundary of a discrete object, and, using these 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 relation 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 discrete object. Third, we define quasi-objects as the connected simplexes. Geometric relations between discrete quasi-objects and discrete objects permit us to define the Euler characteristic for the planar 8-connected, and the spatial 18- and 26-connected objects using these for the planar 4-connected and the spatial 6-connected objects. Our results show that the planar 4-connectivity and the spatial 6-connectivity define the Euler characteristics of point sets in a discrete space. Finally, we develop an algorithm for the computation of these characteristics of discrete objects.