Digital Signal Processing Reference
In-Depth Information
The real-domain fuzzy rank and spatial vectors
r
and
s
are now defined in
an analogous manner to their crisp counterparts:
R
[1 :
N
]
R
T
[1 :
N
]
=
=
.
r
and
s
(2.25)
Similarly, the crisp and fuzzy spatial and rank order vectors are given by
x
=
Rx
L
and
x
L
=
R
T
x
.
A careful inspection of the fuzzy terms indicates that the ranges of val-
ues have been increased beyond that of their crisp counterpart. Specifically,
x
i
,
x
[1
,
l
=
1
l
]. To yield more intuitive values,
these terms can be restricted to the same range as their crisp counterpart by
normalizing the rows or columns of the fuzzy SR matrix. Because the rows
and columns correspond to space and rank, respectively, we designate
R
,
l
=
1
x
l
], and
r
i
,
s
i
∈
)
∈
[
x
(
j
(
1
)
and
R
L
to be the row (spatial) and column (rank) normalized fuzzy SR matrices.
The normalized fuzzy rank and spatial index vectors are now given by
R
[1 :
N
] and
s
R
L
[1 :
N
]. Similarly,
x
=
R
x
L
and
x
L
R
L
x
.
r
=
=
=
Given this normalization, the appropriate bounds hold
r
i
, s
(
j
)
∈
[1
,N
] and
[
x
(
1
)
,x
(
N
)
]. Carrying out the matrix expressions for a single term
yields the following expressions for
r
i
and
x
(
j
)
:
x
i
, x
(
j
)
∈
j
=
1
j R
i,
(
j
)
r
i
=
,
(2.26)
j
=
1
R
i,
(
j
)
and
i
=
1
x
i
R
i,
(
j
)
x
)
=
.
(2.27)
i
=
1
(
j
R
i,
(
j
)
Thus
r
i
is a normalized weighted sum of the integers 1
,
2
,
...
,N
and
x
(
j
)
is a
normalized weighted sum of the samples
x
1
,x
2
,
...
,x
N
. The weights in each
case are the affinity relations between samples.
To illustrate the value in utilizing fuzzy relations, consider again the exam-
ple
x
=
[10
,
1
,
2]
T
and Gaussian membership function (
σ
=
3). The fuzzy SR
matrix in this case is
0
.
0111
0
.
0286
1
.
0000
R
=
1
.
0000
0
.
9460
0
.
0111
,
(2.28)
0
.
9460
1
.
0000
0
.
0286
and the normalized fuzzy SR matrices are
0
.
0107
0
.
0275
0
.
9618
R
=
0
.
5110
0
.
4834
0
.
0057
(2.29)
0
.
4791
0
.
5065
0
.
0145
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