Digital Signal Processing Reference
In-Depth Information
•
Partial ordering
(P-ordering), where the input data are partitioned
into smaller groups, which are then ordered
•
Conditional ordering
(C-ordering), where multivariate samples are
ordered conditionally on one of their marginal sets of observations
12.4.1.1
R-ordering Filters
Let
F
1
(filter length). The noisy image vectors inside the filtering window
W
will be
denoted as
F
j
,
(
x
)
be a multichannel image and let
W
be a window of finite size
n
+
j
=
0
,
1
,
...
,n
.Ifthe distance between two vectors
F
i
,
F
j
is
denoted as
ρ(
F
i
,
F
j
)
, then the scalar quantity
n
R
i
=
0
ρ(
F
i
,
F
j
)
(12.31)
j
=
is the aggregated distance associated with the noisy vector
F
i
inside the pro-
cessing window. Assuming a reduced ordering of the
R
i
—
R
)
≤
R
)
≤
...
≤
(
0
(
1
R
(τ )
≤
...
,
≤
R
—implies the same ordering of the corresponding vectors
(
n
)
F
i
:
F
. Nonlinear ranked-type multichannel filters de-
fine the vector
F
(
0
)
as the output of the filtering operation. This selection is
due to the fact that vectors that diverge greatly from the data population
usually appear in higher-indexed locations in the ordered sequence.
90
,
91
;
F
;
...
;
F
;
...
;
F
(
0
)
(
1
)
(τ )
(
n
)
12.4.1.2 Vector Median Filter
The best-known member of the family of the ranked-type multichannel fil-
ters is the
vector median filter
(VMF).
9
,
10
,
92
-
99
The definition of the multichannel
median is a direct extension of the ordinary scalar median definition with the
L
1
or
L
2
norm utilized to order vectors according to their relative magni-
tude differences.
92
The output of the VMF is the pixel
F
∗
∈
W
for which the
following condition is satisfied:
n
n
F
∗
,
F
j
)
≤
0
ρ(
0
ρ(
F
i
,
F
j
)
,i
=
0
,
...
,n
.
(12.32)
j
=
j
=
It has been observed through experimentation that the VMF discards impulses
and preserves edges and details in the image.
92
However, its performance in
the suppression of additive white Gaussian noise, which is frequently en-
countered in image processing, is inferior to that of the arithmetic mean filter
(AMF). If a color image is corrupted by both additive Gaussian noise and
impulsive noise, an effective filtering scheme should make an appropriate
compromise between the AMF and the VMF.
12.4.1.3 Extended Vector Median Filter
The VMF concept may be combined with linear filtering when the vector
median is inadequate for filtering out noise (such as in the case of additive
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