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