Image Processing Reference
In-Depth Information
6. Let us consider two additional well-behaved conditions for an 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-Sequence B is weakly well-behaved iff
f(i + j), i + j ≤p,
f(p) + f(i + j −p), i + j > p.
f(i) + f(j) ≤
Prove the following for hyperoctagonal distances 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-behaved.
(d) If a B is not weakly well-behaved, d(B) cannot 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 set of all N-Sequences of length p
in 2-D. Define a relation R over S(p) as: ∀B
1
,B
2
∈ S(p), B
1
R B
2
iff f
1
(i) ≤ f
2
(i), ∀1 ≤ i ≤ p where f(i) is the sum sequence. Prove
(Theorem 3 in [63]) that ∀p ≥ 1, < S(p),R > is a distributive lattice
with the least element S
0
= {{1}} and the greatest element S
1
= {{2}}.
n−1
i=0
s
i
r
i
, the surface
area of the hypersphere H(m,n;r) of d
m
is a polynomial of degree n−1
in r with rational coe
cients (s
i
). Devise a direct method to compute
these coe
cients for any given m and n without using the expression
presented in Theorem 2.19.
8. From Corollary 2.9 we know that surf(m,n;r) =
Hint: Numerically evaluate surf(m,n;r) for r = 0,1,··· ,n − 1 and
equate with the polynomial above to get n simultaneous equations
in n unknown s
′
i
s. Solve these equations by matrix method using
rational algebra. For details see [70].
Repeat the above for vol(m,n;r) and verify Table 2.9.
9. Using the approach from Exercise 8, compute a table like Table 2.9 for
surface and volume expressions for hyperspheres of t-cost distance D
t
for n = 1,2,··· ,5. Assume that surface and volume expressions are
polynomials with rational coe
cients.
10. Compute the corners of hyperspheres for the following N-Sequences:
(a) Show that for n = 2 and B = {1}, vertices of H(B;r) are φ((r,0)).
(b) Show that for n = 3 and B = {1,3,3}, vertices of H(B;r) are
φ((r,⌊2r/3⌋,⌊2r/3⌋)).
