Digital Geometry in Image Processing

Distance Functions in Digital Geometry 87

6. Let us consider two additional well-behaved conditions for a n N-

Sequence—one stronger and one weaker.

• An N-Sequence B is strongly well-behaved iff S(B(i, j)) ≥

c

S(B(1, j)), ∀i∀j, 1 ≤ i, j ≤ p.

• An N- Se quence B is weakly well-behaved iff

f(i) + f (j) ≤



f(i + j), i + j ≤ p,

f(p) + f(i + j − p), i + j > p.

Prove the following for hyperoctagonal distance s from [59, 56, 65]:

(a) If a B is strongly well-behaved, it is well-behaved.

(b) If a B is strongly well-behaved, d(B) is a metric.

(c) If a B is well-behaved, it is weakly well-be haved.

(d) If a B is not weakly well-behaved, d(B) ca nnot be a metric.

(e) In 2-D, a B is strongly well-behaved iff it is weakly-well-behaved.

7. Let S(p) = {B : |B| = p} be the ...