Image Processing Reference
In-Depth Information
If this system has no solutions, then
py
(
,...,
y
)
=
0
. If it has a single solu-
y
1
n
tion
x
=
(
x n
,...,
)
T
, then
1
px
(
,...,
x
)
x
1
n
py
(
,...,
y
)
=
,
(4.176)
y
1
n
|
Jx
(
,...,
x
)
|
1
n
where
g
x
g
x
1
1
...
1
n
Jx
(
,...,
x
)
=
...
...
...
(4.177)
1
n
g
x
g
x
n
....
n
1
n
is the Jacobian of the transformation in Equation 4.175. If it has several solutions,
then we add the corresponding terms as in Equation 4.169.
4.7.3
A PPROXIMATION OF THE M EAN OF A R ANDOM V ARIABLE
4.7.3.1
Theorem
The mean of a function
ygx
=
()
of a random variable
x
is given by
+∞
[( ]
gx
=
gxp xdx
x
() ()
.
(4.178)
E
−∞
If
x
is concentrated near its mean
µ
, the mean of
g ()
can be approximated
by a Taylor expansion about
g ()
µ
:
1
gx
()
g
()
µ
+ ′
g
( (
µ
x
−++ !
µ
)
g
()
n
( (
µ
x
µ
).
n
(4.179)
n
Taking the expectation value of both sides yields
1
2
1
[( ]
gx
g
()
µ
+
g
′′
()
µ
Var
()
x
++ !
g n
()
()
µ µ
.
(4.180)
E
n
n
4.7.3.2
Example
As an example, consider the estimator of the signal amplitude given in Equation 4.65
as . To find the mean value of , an expansion about the mean
of the argument of the square root is employed. If we write
= 〈〉−
2
A
A
m
2
σ
2
c
c
2
y
=〈 〉−
m
2
σ
2
,
 
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