Image Processing Reference
In-Depth Information
Fig. 10.3
Image scaling using forward mapping. Notice the black pattern, which is a result of the
inherent problem related to forward mapping
10.2.1 Backward Mapping
The solution is to avoid forward mapping and instead use
backward mapping
. Back-
ward mapping maps from
g(x
,y
)
to
f(x,y)
. That is, it goes through the
output
image
,
g(x
,y
)
, one pixel at a time (using two for-loops) and for each position
(x
,y
)
it uses the
inverse transformation
to calculate
(x, y)
. That is, it finds out
where in the input image a pixel must come from in order to be mapped to
(x
,y
)
.
The principle is illustrated in Fig.
10.2
(b). The inverse transformation is found by
matrix inversion of the transformation matrix as
⎡
⎤
⎡
⎤
⎡
⎤
−
1
x
y
1
x
y
1
a
1
a
2
a
3
b
1
b
2
b
3
001
⎣
⎦
=
⎣
⎦
⎣
⎦
·
(10.10)
For scaling, rotation and shearing the inverse matrices look like the following:
⎡
⎣
⎤
⎦
,
⎡
⎣
⎤
⎦
1
/S
x
00
0
/S
y
cos
θ
sin
θ
0
−
Scaling:
0
Rotation:
sin
θ
cos
θ
0
0
0
1
0
0
1
⎡
⎤
1
−
B
x
0
1
⎣
⎦
Shearing:
−
B
y
1
0
1
−
B
x
B
y
0
0
1
−
B
x
B
y
So, if we want to implement a program that as input takes an image f(x,y) and
as output gives a scaled image g(x',y'), then it could look something like this in
C-code:
Image_Width_Output
=
Image_Width_Input
∗
Sx ;
∗
Image_Height_Output
=
Image_Height_Input
Sy ;
