Editing Digital geometry (section)

==Digital space==
A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space.  A 2D image is a function on a 2D digital space (See [[image processing]]).

In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see [[pixel connectivity]]. A digital space and its (digital-)connectivity determine a [[digital topology]].

In digital space, the digitally continuous function (A. Rosenfeld, 1986) and the [[Gradually varied surface|gradually varied function]] (L. Chen, 1989) were proposed, independently.

A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if ''x''  and ''y'' are two adjacent points in a digital space, |''f''(''x'')&nbsp;&minus;&nbsp;''f''(''y'')|&nbsp;≤&nbsp;1.

A gradually varied function is a function from a digital space  <math>\Sigma</math> to <math>\{ A_1, \dots,A_m \}</math> where <math>  A_1< \cdots <A_m </math> and <math> A_i</math> are real numbers. This function possesses the following property:  If ''x'' and ''y'' are two adjacent points in <math>\Sigma</math>, assume <math>f(x)=A_i</math>, then <math>f(y)=A_{i}</math>,  <math>f(x)=A_{i+1}</math>, or <math>A_{i-1}</math>.  So we can see that the gradually varied function is defined to be more general than the digitally continuous function.

An extension theorem related to above functions was mentioned by A. Rosenfeld (1986) and completed by L. Chen (1989). This theorem states: Let <math>D \subset \Sigma</math> and <math>f: D\rightarrow  \{ A_1, \dots,A_m \}</math>. The necessary and sufficient condition for the existence of the gradually varied extension <math>F</math> of <math>f</math> is : for each pair of points <math>x</math> and <math>y</math> in <math>D</math>, assume <math>f(x)=A_i</math> and <math>f(y)=A_j</math>, we have <math>|i-j|\le d(x,y)</math>, where <math>d(x,y)</math> is the (digital) distance between <math>x</math> and <math>y</math>.