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 f_J : J yields i_(Sigma) _m, if the distance of any two points p and q in J is not less than the distance of f_J(p) and f_J(q) in i_(Sigma) _m, then there exists an extension mapping f of f_J, 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.

Date Published: 23 March 1994
PDF: 8 pages
Proc. SPIE 2182, Image and Video Processing II, (23 March 1994); doi: 10.1117/12.171078

Published in SPIE Proceedings Vol. 2182:
Image and Video Processing II
Sarah A. Rajala; Robert L. Stevenson, Editor(s)