Image Processing Reference
In-Depth Information
Π
1
(u, v; 1)
=
{(2, 3), (3, 3), (4, 3), (5, 4), (6, 5), (6, 6),
(7, 7), (8, 8), (7, 8), (6, 8), (5, 8)}
Π
2
(u, v; 1)
=
{(2, 3), (3, 3), (4, 3), (5, 4), (6, 5), (6, 6),
=
(7, 7), (6, 8), (5, 8)}
Π
3
(u, v; 1)
=
{(2, 3), (3, 3), (3, 4), (3, 5), (4, 5), (5, 5),
=
(5, 6), (5, 7), (5, 8)}
Note:
|Π
1
| = 10
&
|Π
2
| = |Π
3
| = 8
Π
4
(u, v; 2)
=
{(2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 5),
=
(6, 6), (5, 7), (5, 8)}
Π
5
(u, v; 2)
=
{(2, 3), (3, 4), (3, 5), (4, 6), (5, 7), (5, 8)}
Π
6
(u, v; 2)
=
{(2, 3), (2, 4), (2, 5), (3, 6), (4, 7), (5, 8)}
Note:
|Π
4
| = 8
and
|Π
5
| = |Π
6
| = 5
3-D
: Consider
u = (3, 5, 6)
and
v = (2, 7, 4)
.
Π
1
(u, v; 1)
=
{(3, 5, 6), (4, 5, 6), (4, 6, 6), (4, 6, 7), (4, 5, 7), (3, 5, 7),
(3, 6, 7), (3, 6, 6), (2, 6, 6), (2, 6, 5), (2, 7, 5), (2, 7, 4)}
Π
2
(u, v; 1)
=
{(3, 5, 6), (3, 5, 6), (3, 6, 6), (3, 7, 6), (3, 7, 5), (2, 7, 4)}
Π
3
(u, v; 1)
=
{(3, 5, 6), (2, 5, 6), (2, 6, 6), (2, 6, 5), (2, 6, 4), (2, 7, 4)}
Note:
|Π
1
| = 11
and
|Π
2
| = |Π
3
| = 5
Π
4
(u, v; 2)
=
{(3, 5, 6), (3, 4, 6), (2, 4, 5), (2, 5, 6), (2, 6, 5), (2, 7, 4)}
Π
5
(u, v; 2)
=
{(3, 5, 6), (2, 6, 6), (2, 7, 5), (2, 7, 4)}
Π
6
(u, v; 2)
=
{(3, 5, 6), (3, 6, 5), (3, 7, 5), (2, 7, 4)}
Note:
|Π
4
| = 5
and
|Π
5
| = |Π
6
| = 3
Π
7
(u, v; 3)
=
{(3, 5, 6), (4, 5, 7), (4, 6, 6), (3, 7, 5), (2, 7, 4)}
Π
8
(u, v; 3)
=
{(3, 5, 6), (2, 6, 5), (2, 7, 4)}
Π
9
(u, v; 3)
=
{(3, 5, 6), (3, 6, 5), (2, 7, 4)}
Note:
|Π
7
| = 4
and
|Π
8
| = |Π
9
| = 2
€
Different paths in 2-D and 3-D are illustrated in Figs. 2.3 and 2.4, respec-
tively.
As we observe in the above example, there are several (often, infinitely
many) O(m)-paths between two points and finitely many of them may be
shortest.
For example, in Example 2.4, in 2-D Π
∗
2
and Π
∗
3
are O(1) shortest paths
∗
∗
while Π
6
are O(2) shortest paths. Π
1
and Π
4
are other paths. Similar
observations are made in 3-D.
5
and Π
