Digital Signal Processing Reference
In-Depth Information
It can be seen that each local extremum of
Q
(β)
, including the weighted
myriad
ˆ
, can be written as a
weighted mean
of the input samples
X
i
. Because
the weights
h
i
β
(β)
are always positive, the right-hand side of Equation 5.45 is
(
X
(
1
)
,X
(
N
)
)
in
, confirming that all the local extrema lie within the range of the
input samples. By defining the mapping
i
=
1
h
i
X
i
i
=
1
h
i
(β)
(β)
·
=
T
(β)
(5.46)
,orthe roots of
Q
(β)
the local extrema of
Q
(β)
,are seen to be the
fixed points
of
T
(
·
)
:
β
∗
=
(β
∗
).
(5.47)
The following
fixed point iteration
results in an efficient algorithm to compute
these fixed points:
T
i
=
1
h
i
(β
)
·
X
i
1
m
β
=
T
(β
)
=
i
=
1
h
i
.
(5.48)
m
+
m
(β
)
m
In the classical literature, this is also called the
method of successive approxima-
tion
for the solution of the equation
.
28
It has been proved that the
iterative method of Equation 5.48 converges to a fixed point of
T
β
=
T
(β)
(
·
)
; thus,
=
β
∗
=
(β
∗
).
lim
m
→∞
β
T
(5.49)
m
The speed of convergence of the iterative algorithm (Equation 5.48) depends
on the initial value
β
0
.Asimple approach to selecting
ˆ
β
0
is to assign it the
value equal to that of the input sample
X
i
, which leads to the smallest cost
P
(
X
i
)
.
5.6.2 Fixed Point Weighted Myriad Search
Step 1:
Select the initial point
ˆ
β
0
among the values of the input samples:
ˆ
β
=
arg min
X
i
P
(
X
i
).
0
Step 2:
Using
ˆ
β
0
as the initial value, perform
L
iterations of the fixed point
recursion
of Equation 5.48. The final value of these
iterations is then chosen as the weighted myriad:
β
m
+
1
=
T
(β
m
)
ˆ
ˆ
T
(
L
)
(
β
FP
=
β
0
).
This algorithm can be compactly written as
T
(
L
)
arg min
X
i
ˆ
β
FP
=
P
(
X
i
)
.
(5.50)
=
0 (meaning that no fixed point iterations
are performed), the above algorithm computes the selection weighted myriad.
Note that for the special case
L
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