Digital Spaces

Our aim is to provide a mathematical framework for a theory of objects and their surfaces which is applicable to multidimensional discrete spaces. Our motivation comes from practical applications in which boundaries need to be identified in multidimensional data sets with the further aim of displaying them on a computer screen (see Figure 1.1.3). Our definitions are biased towards such applications. One of our aims is to characterize surfaces with a well-determined inside and outside and to define boundaries of objects so that they are indeed surfaces of this type. (In particular, this means that our surfaces must have an orientation, so that we can tell which side is the inside as opposed to the outside.) Furthermore, we want to make our presentation general enough to incorporate many of the reasonable but ad hoc ways that notions, such as “connectedness” and “boundary,” may be defined in digital geometry. We need a framework appropriate for a mathematical treatment of the intuitive notion of a “surface with a connected inside and a connected outside” (a “Jordan surface”) in the discrete multidimensional environment.