Image Processing Reference
In-Depth Information
We present a few results that are used in the measurement Theorem 2.19.
These follow from binomial expansion and enumerations for solutions of simple
integer equations.
Lemma 2.10. ∀r,t ∈ N the following holds: h
n
(r,s) =
2
3
n
⌊s/(r+1)⌋
n
i
i
j
s + i−1−(r + 1)j
i−1
(−1)
n
(−2)
i
×
4
(−1)
j
5
i=⌈s/r⌉
j=0
where h
n
(r,s) = the number of distinct ways to select x from Z
n
to satisfy the
equation
n
|x
i
| = s,0 ≤ x
i
≤r,1 ≤ i≤ n.
i=1
h
n
(r,s) is a polynomial of degree n− 1 in s with rational coe
cients.
€
Corollary 2.8. Special cases of h
n
(r,s):
8
<
:
0
s < 0,
1
s = 0,
h
n
(r,s) =
n
t
t−1 + s
s
(−1)
n
n
t=1
(−2)
t
r ≥ s.
€
Now we present the surface area theorem from [70].
Theorem 2.19. ∀m,n ∈ N, m ≤ n, and r ∈ N, surf(m,n;r)=
2
3
⌊m(r−1)/r⌋
(m−k)r−m
mr
n
k
4
5
.2
k
+
h
n−k
(r−1,s)
h
n
(r,s)
k=1
s=0
s=m(r−1)+1
where h
n
(r,s) is as defined in Lemma 2.10.
Proof. surf(m,n;r) is the number of distinct solutions in x ∈ Z for d
m
(x) =
r. The counting is done with the help of the above lemmas.
Corollary 2.9. surf(m,n;r) and vol(m,n;r) are polynomials in r with ra-
tional coe
cients of degree n−1 and n, respectively.
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Example 2.15. Let us take an example to illustrate the theorem. Consider
n = 3, m = 2, r = 2 to list the points on the surfaces:
S(2,3; 2)
= {(±2,0,0),(0,±2,0),(0,0,±2),
(±2,±1,0),(±2,0,±1),(±1,±2,0),
(±1,0,±2),(0,±1,±2),(0,±2,±1),
(±2,±2,0),(±2,0,±2),(0,±2,±2),
(±2,±1,±1),(±1,±2,±1),(±1,±1,±2),
(±1,±1,±1)}
S(2,3; 1)
= {(±1,0,0),(0,±1,0),(0,0,±1)
(±1,±1,0),(±1,0,±1),(0,±1,±1)}
S(2,3; 0)
= {(0,0,0)}
