Image Processing Reference
In-Depth Information
2.6.1 Notions of Error
We first formally define the notions of errors.
Definition 2.38. Errors are computed between pairs of points in n-D as fol-
lows: ∀u,v ∈ Z
n
a(u,v)= absolute error between d(u,v;N(·)) and Euclidean distance
E
n
(u,v): a(u,v) = |E
n
(u,v) −d(u,v;N(·))|
r(u,v)= relative error between d(u,v;N(·)) and Euclidean distance
E
n
(u,v): r(u,v) =
a(u,v)
E
n
(u,v)
Using translation invariance, we often deal with a(x) and r(x).
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Definition 2.39. REL
i
(N(·),n) represents the maximum of relative er-
ror of type i, 0 ≤i ≤ 4 for a given N(·) in n-D:
REL
0
(N(·),n)
=
max
u∈Z
n
r(u)
a(u)
d(N(·);u)
REL
1
(N(·),n)
=
max
u∈Z
n
a(u)
max{d(N(·);u),E
n
(u)}
REL
2
(N(·),n)
=
max
u∈Z
n
a(u)
|d(N(·);u)+E
n
(u)|
REL
3
(N(·),n)
=
max
u∈Z
n
a(u)
√
REL
4
(N(·),n)
=
max
u∈Z
n
d(N(·);u)
2
+E
n
(u)
Note that unlike REL
0,1
(N(·),n), the last three relative errors REL
2−4
(N(·),n)
are normalized:
REL
i
≤ 1,
2 ≤ i ≤ 4.
If the neighborhood set N(·) is characterized by a parameter λ, the opti-
mal choice of this parameter minimizing some maximum error (as above) is
denoted by λ
opt
i
. That is,
REL(λ
opt
i
,n) = min
λ
REL
i
(λ,n).
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For example, for m-Neighbor distance λ = m, or for t-cost distance λ = t.
The range of λ is decided by the type of the parameter and lies between 1 and
n for m and t. Wider ranges are also possible for parameters of other N(·)'s.
We often write REL
0
simply as REL if the context is clear.
2.6.2 Error of m-Neighbor Distance
It has been shown in [69] that the maximum relative error REL(m,n) is
bounded by a real constant, whereas, unfortunately, the absolute error has no
