Image Processing Reference
In-Depth Information
Note that the rotation is done counterclockwise since the y-axis is pointing down-
wards. If we wish to do a clockwise rotation we can either use
θ or change the
transformation to
x
y
cos θ
x
y
x = x ·
cos θ + y ·
sin θ
sin θ
=
·
(10.6)
sin θ
cos θ
y =− x ·
sin θ + y ·
cos θ
10.1.4 Shearing
To shear an image means to shift pixels either horizontally, B x , or vertically, B y .
The difference from translation is that the shifting is not done by the same amount,
but depends on where in the image a pixel is. In Fig. 10.1 (e) B x =−
0 . 5 and B y =
0.
The transformation is defined as
x
y
1
x
y
x =
+
·
x
y
B x
B x
=
·
(10.7)
B y
1
y =
x
·
B y +
y
10.1.5 Combining the Transformations
The four transformations can be combined in all kinds of different ways by multiply-
ing the matrices in different orders, yielding a number of different transformations.
One is shown in Fig. 10.1 (f). Instead of defining the scale factors, the shearing fac-
tors and the rotation angle, it is common to merge these three transformation into
one matrix. The combination of the four transformations is therefore defined as
x
y
a 1
x
y
a 3
b 3
x =
a 1 ·
x
+
a 2 ·
y
+
a 3
a 2
=
·
+
(10.8)
b 1
b 2
y =
b 1 ·
x
+
b 2 ·
y
+
b 3
and this is the affine transformation. Below the relationships between Eq. 10.8 and
the four above mentioned transformations are listed.
a 1
a 2
a 3
b 1
b 2
b 3
Translation
1
0
x
0
1
y
Scaling
S x
0
0
0
S y
0
Rotation
cos θ
sin θ
0
sin θ
cos θ
0
Shearing
1
B x
0
B y
1
0
Often homogeneous coordinates are used when implementing the transformation
since they make further calculations faster. In homogeneous coordinates, the affine
transformation becomes
x
y
1
a 1 a 2 a 3
b 1 b 2 b 3
001
x
y
1
=
·
(10.9)
where a 3 =
x and b 3 =
y .
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