Digital Geometry in Image Processing

32 Digital Geometry in Image Processing

2. Separating Dimension: The dimension m of the separating hyperplane is

bounded by a constant r s uch that 0 ≤ r ≤ m < n. For example, in 2-D,

4-neighbors have r = 1, m = 1 and consequently only line separation is

allowed. 8-neighbors, on the o ther hand, have r = 0, m = 0, 1, and both

point and line separations are allowed. That is, n − m =

P

n

i=1

|w

i

| ≤

n − r.

3. Separating Cost: The cost between neighbors is integ ral. That is, δ(w) ∈

P. Often the cost is taken to be unity.

4. Isotropy and Symmetry: The neighborhoo d is isotropic in all (discr ete)

directions. That is, all permutations and/or reflections of w, φ(w) ∈

N(·).

5. Un iformity: The neighborhood relation is identical at all points along a

path and at