Image Processing Reference
In-Depth Information
TABLE 2.12: Optimal neighborhood value for least relative error.
n m
opt
0
m
opt
1
m
opt
2
m
opt
3
m
opt
4
1
1
1
1
1
1
2
2
1
1
1
1
3
2
2
2
2
2
4
2
2
2
2
2
5
2
2
2
2
2
6
2
2
2
2
2
7
3
2
2
2
2
8
3
2
2
2
2
9
3
2
2
2
2
10
3
2
2
2
2
Reprinted from
Information Sciences
59(1992), P. P. Das, J. Mukherjee and B. N. Chatterji, The t-Cost Distance
in Digital Geometry, 1-20, Copyright (1992), with permission from Elsevier.
The maxima of relative error between δ
m
and E
n
is defined as: R(m,n) =
|E
n
(x)−δ
m
(x)|/E
n
(x).
max
x∈R
n
€
The following theorem is proved in [66].
Theorem 2.36.
max(n/m−
√
A(m,n)
=
n,
⌊m⌋+ (m−⌊m⌋)
2
−1)
√
=
max(
n(r/r
I
−1),r
C
/r−1)
√
R(m,n)
=
max(
n/m−1,1−1/
(⌊m⌋+ (m−⌊m⌋)
2
)
=
max(r/r
I
−1),1−r/r
C
)
where r
I
and r
C
are the radii of the inscribed and circumscribed hyperspheres,
respectively (Section 2.5.3.3).
€
2.6.4 Error of t-Cost Distance
In this section we explore the suitability of a t-cost distance in n-D for the
approximation of E
n
. A measurement of error between D
t
and E
n
has been
presented from [58].
We quote the theorems below. The proofs are from [58].
Theorem 2.37. Absolute error a(u) is unbounded, that is, ∀M, M ∈ R
+
,
∃u ∈ Z
n
such that a(u) = |E
n
(u)−D
t
(u)| > M.
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