Image Processing Reference

In-Depth Information

10.1 Affine Transformations

The class of
affine transformations
covers four different transformations, which are

illustrated in Fig.
10.1
. These are: translation, rotation, scaling and shearing.

10.1.1 Translation

Let us now look at the transformations in Fig.
10.1
and define their concrete map-

ping equations. Translation is simply a matter of shifting the image horizontally

and vertically with a given off-set (measured in pixels) denoted
x
and
y
.For

translation the mapping is thus defined as

x

y

x

y

x

y

x
=
x
+
x

y
=

⇒

=

+

(10.3)

y

+

y

So if
x

=

100 and
y

=

100 then each pixel is shifted 100 pixels in both the

x- and y-direction.

10.1.2 Scaling

When scaling an image, it is made smaller or bigger in the x- and/or y-direction. Say

we have an image of size 300

×

200 and we wish to transform it into a 600

×

100

image. The x-direction is then scaled by: 600
/
300

=

2. We denote this the x-scale

factor and write it as
S
x
=

1
/
2. Together this means

that the pixel in the image
f(x,y)
at position
(x, y)
=
(
100
,
100
)
is mapped to a new

position in the image
g(x
,y
)
, namely
(x
,y
)

2. Similarly
S
y
=

100
/
200

=

=

·

·

=

(
100

2
,
100

1
/
2
)

(
200
,
50
)
.In

general, scaling is expressed as

x

y

S
x

x

y

x
=

·

x

S
x

0

⇒

=

·

(10.4)

0

S
y

y
=

y

·

S
y

10.1.3 Rotation

When rotating an image, as illustrated in Fig.
10.1
(d), we need to define the amount

of rotation in terms of an angle. We denote this angle
θ
meaning that each pixel in

f(x,y)
is rotated
θ
degrees. The transformation is defined as

x

y

cos
θ

x

y

x
=

x

·

cos
θ

−

y

·

sin
θ

−

sin
θ

⇒

=

·

(10.5)

sin
θ

cos
θ

y
=

·

+

·

x

sin
θ

y

cos
θ