Image Processing Reference
In-Depth Information
Fig. B.2
(
a
) Adding two
vectors. (
b
) Subtracting two
vectors
Fig. B.3
The angle between
two vectors
5
5
2
0
5
7
5
+
2
p
3
=
p
1
+
p
2
=
+
=
=
(B.13)
+
5
0
In the same way we can calculate the difference of
p
1
and
p
2
as
5
5
2
0
5
3
5
−
2
p
4
=
p
1
−
p
2
=
−
=
=
(B.14)
5
−
0
These operations can also be interpreted geometrically as illustrated in Fig.
B.2
.
Two vectors cannot be multiplied but we can calculate the
dot product
between
them. Say we define
p
1
=[
and
p
2
=[
T
T
. The dot product between
ab
]
cd
]
them is then defined as
p
1
•
p
2
=
ac
+
bd
(B.15)
The dot product can also be interpreted geometrically as
p
1
•
p
2
=
p
1
·
p
2
·
cos
V
(B.16)
is the length of vector
p
2
, and
V
is the
angle between the vectors, see Fig.
B.3
. Note that it is always the smallest of the two
possible angles that is calculated using Eq.
B.16
, i.e., 0°
p
1
is the length of vector
p
1
,
p
2
where
≤
V
≤
180°. The biggest
angle is found as
V
big
=
360°
−
V
.
B.6
Matrix
When we have multiple vectors we can represent them as one entity denoted a
ma-
trix
. For example,
p
1
=[
and
p
2
=[
T
T
ab
]
cd
]
can be represented as
