Discretization, Digitization, and Embedding

Abstract

This chapter deals with the relationships among objects in discrete, digital, and continuous spaces. If Chap. 7 can be viewed as covering the theoretical aspect of discrete spaces and its objects, then this chapter can be viewed as covering the practical methods to make actual moves from one space to another. Likewise, Chap. 9 then connects classical mathematics to discrete and digital topology. Two of the most important operations are discretization (obtaining a discrete object from continuous space) and embedding (putting a discrete or digital object into a continuous space). To obtain a discrete object, we need to retain its geometric and tropologic identity, such as distance measurements and genus (holes). In general, a discrete object differs from a continuous object in that it has two basic measures: graph distance and Euclidean distance. The graph distance measures the number of edges between two notes as discrete sampling points. Euclidean distance can be viewed as the weight on the edges. Embedding is putting a discrete object back into a continuous space. For instance, putting a weighted graph into 3D Euclidean space, we must not allow two edges to cross each other. In practice, discretization and embedding are not separated. They are two parts of a unified process called mesh generation in computer graphics. Other embedding methods, such as the piecewise linear reconstruction method and more sophisticated polynomial fitting and B-spline methods are discussed in Chap. 11.