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
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