Image Processing Reference
In-Depth Information
Figure 2.2
A simple example with the original image
I
o
on the left and the noise corrupted ver-
sion
I
n
on the right. The images differ by 26 pixels.
ground and also subtracted pixels from the foreground. (It is assumed here that the
black pixels are the foreground and white are the background). The total number of
pixels differing between the two images is 26.
Starting with the noise-corrupted image
I
n
, the objective is to find a filter to
recover the original image
I
o
. In practice, this may not be possible. The design task
therefore reduces to finding the optimum filter
ψ
opt
out of all possible filters
ψ
that
minimizes the difference between the filtered noisy image
(
I
n
) and the original
I
o
.
In the language of statistics, an optimal estimator is being sought. Its task is to esti-
mate the true value of the image pixels from a noise-corrupted version.
ψ
2.1 Error Criterion
Thus far, the words
best
and
optimum
have been used loosely and have not been
given any specific mathematical definition. In quantitative terms, they require a
measure of similarity. The measure usually used in this context is the mean-abso-
lute error (MAE). Another measure is the mean-square error (MSE), and the rela-
tive merits can be debated for grayscale images. For binary images, MAE and MSE
are identical.
Given two images
I
1
(
r
,
c
) and
I
2
(
r
,
c
) with the same number of
R
rows and
C
columns, their MAE is defined as
C
−
1
R
−
1
1
)
∑
∑
(
)
)
(
(
MAE I
,
I
=
Irc Irc
,
−
,
.
(2.1)
12
1
2
RC
c
=
0
r
=
0
The optimum filter is therefore defined as the one that minimizes the difference
between the ideal image
I
o
and the filtered version of the noisy image
ψ
(
I
n
),
(
)
)
(
MAE
ψ
I
,
I
.
(2.2)
n
o
For binary images, the MAE consists of just two types of errors:
