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
.
