Discrete multidimensional Jordan surfaces

We introduce a mathematical framework suitable for a general theory of surfaces, objects, and their borders and boundaries in multidimensional discrete spaces. Our motivation comes from practical applications, where objects and their boundaries need to be identified in multidimensional data sets with the further aim of displaying them on a computer screen. Our definitions are biased toward such applications. In particular, we desire to characterize surfaces with a well-determined inside and outside and to define object borders and boundaries so that they will indeed be surfaces of this type. Furthermore, we make our presentation general enough to incorporate many of the reasonable but ad hoc ways that notions of “object,” “border,” and “boundary” may be defined in digital geometry. Some basic theorems are proven, showing that the framework is appropriate for the 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.