Digital Signal Processing Reference
In-Depth Information
Now, let
r
=
max
j
r
(
X
j
)
. Then, for
X
j
∈ M
, expanding the product in
Equation 5.14 gives
+
O
K
2
N
−
r
−
1
2
(
X
i
−
X
j
)
1
P
K
(
X
j
)
=
.
(5.15)
K
2
i, X
i
=
X
j
K
2
N
−
r
, the second term is
Because the first term in Equation 5.15 is
O
(
1
/
)
negligible for small values of
K
, and
ˆ
β
0
can be calculated as
ˆ
β
0
=
arg min
X
j
∈M
P
K
(
X
j
)
2
(
X
i
−
X
j
)
=
arg min
X
j
∈M
K
2
i, X
i
=
X
j
=
arg min
X
j
∈M
X
j
|
X
i
−
X
j
|
.
i, X
i
=
An immediate consequence of the mode property is the fact that running-
window smoothers based on the mode-myriad are
selection-type
,inthe sense
that their output is always, by definition, one of the samples in the input
window. This “selection” property, shared also by the median, makes mode-
myriad smoother a suitable framework for image processing, where the ap-
plication of selection-type smoothers has been shown to be convenient.
23
,
24
5.3.1
Geometrical Interpretation
Myriad estimation, defined in Equation 5.5, can be interpreted in a more
intuitive manner. As depicted in Figure 5.3a, it can be shown that the sample
myriad,
ˆ
β
K
,isthe value that minimizes the product of distances from point
A
to the sample points
X
1
,X
2
,
=
β
,
...
,X
6
. Any other value, such as
X
A
A
K
K
x
x
^
β
^
β
X
5
X
1
X
2
X
3
X
6
X
4
X
5
X
1
X
2
'
X
3
X
6
X
4
β
(b)
(a)
FIGURE 5.3
(a) The sample myriad,
ˆ
β
,minimizes the product of distances from point
A
to all samples. Any
=
β
,produces a higher product of distances; (b) the myriad as
K
is
other value, such as
x
reduced.
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