1. Pixel Relations

2. Neighboring, Adjacency
N4(p): 4-neighbors
ND(p): 4-diagonal neighbors
N8(p) = N4(p) ND(p) : 8-neighbors
Adjacency:
V: a set of gray levels s.t. if a pixel’s gray level in V, it will be used to establish adjacency.
4-adjacency
8-adjacency
m-adjacency p p q s p r q s p r

3. Path, Connectivity, Region
a path from p = (s,t) to q = (x,y) is a set adjacent points
P(p,q) = {(s,t), …, (xi, yi), …, (x, y)}
If (s, t) = (x, y), P is a closed path.
p and q are connected in S if P(p,q)  S. The set {q; P(p,q)  S} is a Connected Component of p  S.
S is a connected set if it has only one connected component. S is also called a Region.
The boundary of a region R = {p; p  R, exists q  R, q  N(p)}
If R is entire image, its boundary is the first and last rows and columns of pixels.

4. Distance Metrics Normed distance Euclidean distance
Given pixels p, q, r, a function D(p,q) is a distance function if
Euclidean distance
L2 norm
City-block distance
L1 norm
Chessboard distance
L norm