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
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