Image Processing Reference
In-Depth Information
)
)
(
(
∆
01
,
+
∆
10
,
(
)
(
)
MAE
ψ
I
,
I
=
,
(2.3)
n
o
RC
where
∆
(0, 1) equals the number of pixels for which
ψ
(
I
n
)
=
0and
I
o
=1,and
∆
(1, 0)
equals the number of pixels for which
(
I
n
)
=
1 and
I
o
=0.
An error occurs only at those locations where the filter output and the ideal im-
age differ. For each location where this occurs, the contribution to the total MAE is
precisely one pixel.
Note: The MAE gives equal weighting to
ψ
(1, 0) errors (i.e.,
pixels that should have been set to black but have been missed, and those that have
been set to black but should not have been). There may be cases where different
weightings for these two types of errors may be appropriate.
∆
(0, 1) errors and
∆
2.2 Filter Constraints
Any practical filter that can be designed to operate on an image must be constrained
in some way. There are an infinite number of possible filters that may take many
different forms. An unconstrained filter for image restoration would be absurdly
large. Consider the image shown in Fig. 2.3. An unconstrained filter would require
that every output pixel had a different filter, and require that every one of those fil-
ters was a function of every pixel in the input image. This would be a true optimal
filter. In fact, it would be a number of different filters, since each pixel would re-
quire its own estimator. Such a totally unconstrained filter would clearly be imprac-
tical. However, it is possible to constrain the problem to make it practical and at the
same time produce acceptable results even though the filter used would be
suboptimal.
Two commonly used constraints are the
window constraint
and
translation
invariance
.
Figure 2.3
Unconstrained Filter. An unconstrained filter requires that every output pixel has a
different filter and every one of these filters is a function of every pixel in the input image. This
type of filter is unrealistic and practical results are usually achieved by using filters that are
both windowed and translation invariant.
