Graph-based Approach

Abstract

In this approach, it is possible to show that some particular properties of a digital concept are the same as the corresponding properties of its continuous original. The considerations in Chapter 1 demonstrate that it is necessary to introduce a special graph structure into a discrete representation in order to ensure the required properties of a digital analog of a continuous concept. Thus, while defining a digital analog, one must specify at the same time in whichgraph structure this analog is defined. For example, if one defines a digital analog of a simple closed curve as a subgraph of the digital plane Z ² in which each point is 4-adacent to exactly two other points, then the Jordan curve theorem holds for this curve in a graph with changeable 4/8-adjacency or in a well-composed graph. However, as demonstrated in Chapter 1, the Jordan curve theorem does not hold for the simple closed 4-curve in a graph with only one 4-adjacency relation. If we use the changeable ad- acenc 4/ 8 , we have 4-connectedness for the foreground and 8-connectedness for the j Y background. Consequently, connected components are not intrinsic features of the digital representation and the neighborhoods of points are not homogeneous as it is the case for ℝ² with the usual topology.