Digital Signal Processing Reference
In-Depth Information
, a vector-based member-
ship function can be defined, for example, based on the Gaussian function,
Following selection of the distance metric
D
(
·
,
·
)
2
2
e
−
D
(
a
,
b
)
/
2
σ
µ
G
(
a
,
b
)
=
,
(2.52)
where
, as before, is the spread parameter. The multivariate median affine
and center affine filters are now simply defined as
σ
i
=
1
w
i
R
i,
(δ)
x
i
MAFF[
x
]
=
(2.53)
i
=
1
|
w
R
i,
(δ)
|
i
and
i
=
1
w
(
i
)
R
δ
,
(
i
)
x
(
i
)
i
=
1
|
w
(
i
)
|
=
.
CAFF[
x
]
(2.54)
R
δ
,
(
i
)
The fuzzy vector median and fuzzy vector weighted median filters are formed
as similar multivariate extensions:
i
=
1
x
i
R
i,
(δ)
FMED[
x
]
=
x
(δ)
=
(2.55)
i
=
1
R
i,
(δ)
and
FWMED[
x
]
=
MED[
w
1
x
1
,
w
2
x
2
,
...
,
w
N
x
N
]
(2.56)
=
FMED[
w
1
x
1
,
w
2
x
2
,
...
,
w
N
x
N
]
.
(2.57)
To illustrate the advantages of using fuzzy techniques in a multivariate
application, we reconsider the edge filtering example in Figure 2.9 and extend
it to vector data. Figure 2.11a shows a two-dimensional directional signal
with an abrupt edge transition. Gaussian noise-corrupted observations of this
(a)
(b)
(c)
FIGURE 2.11
Filtering of directional signal corrupted by additive Gaussian noise: (a) original signal, (b) vector
median, and (c) fuzzy vector median ensemble outputs.
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