Gradually Varied Surfaces and Gradually Varied Functions

The gradually varied surface was introduced and studied in digital and discrete spaces by Chen. The basic idea of introducing gradually varied surfaces is to employ a purely discrete interpolation algorithm to fit a discrete surface when the desired surface is not required to be “smooth.” In this paper, we generalize the concept of gradually varied surfaces. A digital surface is defined as a mapping f from an n-dimensional digital manifold D into an m-dimensional grid space Σm. A discrete surface is said to be gradually varied if two points in D, p, and q are adjacent, implying f(p) and f(q) are adjacent in Σm. We have proved the following constructive theorem: Let i Σm be an indirectly adjacent grid space. Given a subset J of D and a mapping fJ : J → i Σm, if the distance of any two points p and q in J is not less than the distance of fJ (p) and fJ (q) in i Σm, then there exists an extension mapping f of fJ , such that the distance of any two points p and q in D is not less than the distance of f(p) and f(q) in i Σm. That is to say, the guiding point set (J, f(J)) has a gradually varied surface fitting. In other words, any digital manifold (graph) can normally immerse an arbitrary i Σm. We also show that any digital manifold (graph) can normally immerse an arbitrary tree T . Furthermore, we will discuss the gradually varied function, which is a gradually varied surface in the case of m = 1 in i Σm or integer set Z. An envelope theorem, a uniqueness theorem, and an extension theorem which concerns preserving the same norm, are obtained. Finally, we will show an optimal uniform approximation theorem of gradually varied functionals and develop an efficient algorithm for the approximation.