Image Processing Reference
In-Depth Information
4. As an exception to symmetric and isotropic (Section 2.2.1.1) neighbor-
hood sets, many anisotropic yet symmetric neighborhoods in 2-D also
define metrics [62, 141].
N
1
=
{(±1, 0), (0, ±1)}
d
N
1
(x)
=
|x
1
| + x
2
|
N
2
A
=
{(±1, 0), ±(1, 1)}
d
N
2
A
(x)
=
max(|2x
1
− x
2
|,|x
2
|)
N
2
B
=
{(±1, 0), ±(1, −1)}
d
N
2
B
(x)
=
max(|2x
1
+ x
2
|, |x
2
|)
N
2
C
=
{(0, ±1), ±(1, 1)}
d
N
2
C
(x)
=
max(|2x
2
+ x
1
|, |x
1
|)
N
2
D
=
{(0, ±1), ±(1, −1)}
d
N
2
D
(x)
=
max(|2x
2
− x
1
|,|x
1
|)
N
3
A
=
{(±1, 0), (0, ±1), ±(1, 1)}
d
N
3
A
(x)
=
max(|x
1
|, |x
2
|, |x
1
− x
2
|)
N
3
B
=
{(±1, 0), (0, ±1), ±(1, −1)}
d
N
3
B
(x)
=
max(|x
1
|, |x
2
|, |x
1
+ x
2
|)
N
4
A
=
{(±1, 0), (±1, ±1)}
d
N
4
A
(x)
=
max(2⌈(|x
1
| − |x
2
|)/2⌉, 0) + |x
2
|
N
4
B
=
{(0, ±1), (±1, ±1)}
d
N
4
B
(x)
=
max(2⌈(|x
2
| − |x
1
|)/2⌉, 0) + |x
1
|
N
5
=
{(±1, 0), (0, ±1), (±1, ±1)}
d
N
5
(x)
=
max(|x
1
|, |x
2
|)
′
(a) Prove that the above d
s are metrics and give the lengths of the
shortest paths for the corresponding neighborhood sets.
(b) Study the disks of these distances.
(c) Two metrics d
1
and d
2
over Z
2
are isomorphic if there exists a bijec-
tion λ : Z
2
→ Z
2
such that ∀x,y ∈ Z
2
, d
1
(λ(x),λ(y)) = d
2
(x,y)
and d
2
(λ
−1
(y)) = d
1
(x,y). Prove that the following sets
of metrics are isomorphic to each other: {d
N
2
A
,d
N
2
B
,d
N
2
C
,d
N
2
D
−1
(x),λ
},
{d
N
3
A
,d
N
3
B
} and {d
N
4
A
,d
N
4
B
}.
5. The t−cost-m−Neighbor (or tCmN) [167] neighborhood set N(t,m,n),
is defined ∀n,m,t ∈ P, 1 ≤ m ≤ n, 1 ≤ t ≤ n as {w :
w ∈ {0,±1}
n
,h
n
(w) ≤ m} with an associated cost function δ(w) =
min(t,h
n
(w)) where h
n
(·) is the component sum function (Definition
2.5). It induces tCmN paths π(u,v;t,m : n) and the tCmN norm de-
fined as ∀x ∈ Z
n
,
d(x; t,m : n)
max
i=0
S
i
(x),
=
P
P
i
j=1
f
j
(x) + [minimum(1, (t − i)/(m − i))
n
j=i+1
f
j
(x)],
S
i
(x)
=
0 ≤ i ≤ n,
where
(t − i)/(m − i), 0 ≤ (t − i)/(m − i) < 1,
1,
minimum(1, (t − i)/(m − i))
=
(t − i) ≥ (m − i).
and f
i
(·) is the component function (Definition 2.4).
Note that, for i = m, minimum(1, (t − i)/0) = 1 and 0 ≤ minimum(1, (t − i)/(m −
i)) ≤ 1.
Prove the following for the tCmN neighborhood [167] ∀n,1 ≤ m ≤ n,
∀t,1 ≤ t ≤n:
(a) d(t,m : n) is a metric over Z
n
.
(b) ∀x ∈ Z
n
, d(x;t,m : n) = |π
∗
(x;t,m : n)|.
(c) d(t,m : n) is a generalization of m-Neighbor [60] and t-cost [58]
distances. That is, d
m
= d(1,m : n) and D
t
= d(t,n : n).
