Digital Signal Processing Reference
In-Depth Information
This approach is in some way similar to the technique we propose in Section
12.5, in which the filters based on digital path are introduced. In the new
approach, instead of looking for subwindows with similar pixels, we investi-
gate digital paths linking the central pixel with pixels belonging to the filter
window.
Another class of adaptive algorithms is based on the rank transformations,
defined using an ordering operator whose goal is the transformation of the set
of pixels lying in a given filtering window
W
into a monotonically increasing
sequence
{
F
0
,F
1
,
...
,F
n
}→{
F
,F
,
...
,F
)
}
, with the property:
F
)
≤
(
)
(
)
(
(
0
1
n
k
F
,k
=
0
,
...
,n
−
1. In this way the rank operator is defined on the ordered
(
k
+
1
)
values from the set
{
F
,
...
,F
)
}
and has the form
(
0
)
(
n
n
n
1
Z
F
0
=
0
(
k
)
F
(
k
)
,
Z
=
0
(
k
)
,
(12.10)
=
=
k
k
where
k
are nonzero weighting (ranking) coefficients. Taking appropriate
ranking coefficients allows the definition of a variety of useful operators. The
sequence
•
{
...
}
1
,
1
,
,
1
corresponds to the moving average operator.
•
{
0
,
...
,
0
,
=
1
,
0
,
...
,
0
}
,
m
=
n
/
2, generates the median (for even
m
number of neighbors
n
).
•
{
0
,
...
,
0
,
m
−
α
=
1
= ··· =
m
=···=
m
+
α
=
1
,
0
,
...
,
0
}
,0
≤
α
≤
m
defines the
α
-trimmed mean, which is a compromise between the
median (
α
=
0) and the moving average (
α
=
m
).
•
{
1
,
0
,
...
,
0
,
1
}
determines the so-called mid-range filter.
The standard median exploits the rank-order information (order statistics)
to eliminate impulsive noise. This filter substitutes the corrupted pixel with
the middle-position element (median) of the ordered input samples. Since its
introduction, it has been extensively studied and extended to the weighted
median and its special case center-weighted median filter.
The median filter is one of the most commonly used nonlinear filters. It has
the ability to attenuate strong impulse noise while preserving image edges.
Its major drawback, however, is that it wipes out structures that are of the
size of the filter window, and this effect causes the texture of a filtered image
to be strongly distorted. Another drawback of the standard median is that it
inevitably alters the details of the image not distorted by the noise process;
because the standard median cannot distinguish between the corrupted and
original pixels, and whether a pixel is corrupted or not, it is replaced by the
local median within a filtering window. Therefore, a trade-off between the
suppression of noise and the preservation of fine image details and edges has
to be found. This can be accomplished in different ways; the goal, however,
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