Image Processing Reference
In-Depth Information
2
2
Note that this estimator is unbiased because
E
[
〈〉=
m
]
E m
[
]
= +
A
2
2
σ
2
.
If the
noise variance
σ
2
is assumed to be known, an unbiased estimator of
A
2
is given by
ˆ
2
2
A
=〈 〉− σ
2
2
.
(4.82)
m
c
Taking the square root of Equation 4.64 gives the conventional estimator of
A
[39,40,34,35]:
=〈 〉−
2
m
2
σ
2
.
(4.83)
A
c
4.4.3.1.3 Discussion
The parameter to be estimated is the signal
A
. Obviously,
A
is known
a priori
to
be real valued and nonnegative. However, this
a priori
knowledge has not been
incorporated into the conventional estimation procedure. Consequently, the con-
ventional estimator given in Equation 4.83, may reveal estimates that violate
the
a priori
knowledge and are therefore physically meaningless. This is the case
when becomes negative. Therefore, cannot be considered a useful estima-
tor of
A
if the probability that is negative differs from zero significantly. To
determine this probability, the PDF of
A
,
c
A
c
2
A
c
A
c
2
A
c
2
is required. It can be derived from
N
−
1
yNA
(
)
1
y
NA
2
yNA
+
(
)
2
py
()
=
exp
−
I
ε
(),
y
(4.84)
y
N
−
1
2
σ
(
)
2
σ
σ
2
2
2
2
where is given by the sum of
N
real and
N
imaginary, independent, squared,
Gaussian-distributed variables (as discussed in Subsection 4.2.4) i.e.,
y
N
N
(
)
=
∑
∑
2
2
2
y
=
+
.
(4.85)
ww
m
rn
,
in
,
n
n
=
1
n
=
1
The deterministic signal component of
is given by
NA
2
.
From Equation 4.81
y
ˆ
2
and Equation 4.82, we have that
Ay
=/
N
−
2
σ
2
, and with (cf.
Appendix
)
ˆ
ˆ
pA NpNA
A
2
=
2
+ σ
2
2
,
(4.86)
c
y
c
ˆ
2
c
A
c
2
the PDF of
becomes explicitly
N
−
1
ˆ
ˆ
A
+
2
2
σ
+
A
2
2
2
2
N
A
2
+
2
σ
2
ˆ
c
=
pA
c
exp
−
N
2
2
c
ˆ
2
σ
2
A
2
σ
2
A
c
(4.87)
ˆ
NA
A
2
+
2
σ
2
ˆ
c
×
I
ε
A
c
+
2
σ
.
2
2
N
−
1
σ
2
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