Digital Signal Processing Reference
In-Depth Information
1
0.95
0.9
0.85
Identity filter
Median filter
0.8
0.75
0.7
0
20
40
60
80
100
120
SNR(dB)
FIGURE 2.1
Correlation coefficient between an MA process and the identity and (window size 5) median
filter outputs as a function of SNR in the Laplacian noise-corrupted observation.
2.2.2
Spatial-Rank Ordering
To formally relate the spatial ordering and rank ordering of samples in a signal
processing application, consider again the typical case in which an observa-
tion window passes over an observation sequence in a predefined scanning
pattern. At each location
n
the observation window covers
N
samples, which
can be indexed according to their spatial location and written in vector form:
,x
N
[
n
]]
T
x
[
n
]
=
[
x
1
[
n
]
,x
2
[
n
]
,
...
.
(2.13)
The subscript
has now been added to explicitly indicate that the samples
are indexed according to their natural spatial order within the observation
signal or image. A second natural ordering of the observed samples is rank
order, which yields the order statistics of the observation samples,
≤
≤···≤
.
x
(
1
)
[
n
]
x
(
2
)
[
n
]
x
(
N
)
[
n
]
(2.14)
Writing the order statistics in vector form yields the rank order observation
vector:
[
n
]]
T
x
L
[
n
]
=
[
x
[
n
]
,x
[
n
]
,
...
,x
.
(2.15)
(
1
)
(
2
)
(
N
)
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