Image Processing Reference
In-Depth Information
A
quite arbitrary, solution to this problem is to artificially put the estimator
to
rms
A
2
zero whenever
is negative:
!
!
ˆ
ˆ
A
if
A
≥
0
2
2
ˆ
=
(4.128)
A
rms
ˆ
0
if
A
<
0
.
2
A
rms
The PDF of
is then given by
NK
AK
A
−
2
ˆ
ˆ
0
4
px x
N
A
2
+
σ
2
(
ˆ
(
ˆ
∫
pA
)
=
δ
A
)
( )
+
rm
s
r s
ˆ
ˆ
A
rms
rms
A
2
σ
2
2
rms
2
−
K
σ
ˆ
ˆ
AK
2
+
σ
2
A
AK
2
+
σ
2
+
A
2
ˆ
rms
rms
×−
exp
N
I
ε
(
A
rms
),
NK
2
σ
−
1
σ
2
2
2
(4.129)
where denotes the Dirac delta function. Notice that the first term of Equation 4.129
vanishes for high SNR. The bias of the modified estimator Equation 4.128 can be
computed from
δ
()
.
∞
∫
[
ˆ
E
A
]
=
p
( )
xxdx
(4.130)
ˆ
rms
A
rms
0
Figure 4.9 shows the bias for various values of
K
and
N
. In general, the bias
of the modified estimator is significantly smaller compared to the bias of the
10
K = 6; N = 4
K = 4; N = 4
K = 2; N = 4
K = 6; N = 8
K = 4; N = 8
K = 2; N = 8
K = 6; N = 16
K = 4; N = 16
K = 2; N = 16
1
0.1
1.5
2
2.5
3
3.5
4
4.5
5
SNR
FIGURE 4.9
Relative bias of the RMS estimator of
A
for
K
= 2, 4, and 6 and
N
= 4, 8,
and 16. The true value is
A
= 100.
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