Gradually varied surface and its optimal uniform approximation
Li Chen · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
A new digital surface called the gradually varied surface is introduced and studied in digital spaces, especially in digital manifolds. In this paper, we have proved a constructive theorem: Let i_(Sigma) m be an indirectly adjacent grid space. Given a subset J of D and a mapping fJ : J yields i_(Sigma) m, 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_(Sigma) m, 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_(Sigma) m, we call such f a gradually varied surface. We also show that any digital manifold (graph) can normally immerse an arbitrary tree T. Furthermore, we discuss the gradually varied function. An envelop theorem, a uniqueness theorem, and an extension theorem concerned with having the same norm are obtained. Finally, we show an optimal uniform approximation theorem of gradually varied functionals and develop an efficient algorithm for the approximation.