Image Processing Reference
In-Depth Information
or as a function of only
A
N
∑
Am
−
NA
2
n
ln
L
∼
ln
I
.
(4.97)
0
σ
2
σ
2
2
n
=
1
The ML estimate is then found from the global maximum of ln
L
:
{
}
(
)
=
arg max ln
L
.
(4.98)
A
ML
A
Note that Equation 4.98 cannot be solved analytically. Finding the maximum
of the (log-)likelihood is therefore a numerical optimization problem.
4.4.3.1.5 Discussion
It is not possible to find the maximum of the ln
L
function directly because the
parameter
A
enters that function in a nonlinear way. Therefore, finding the max-
imum of the ln
L
function will, in general, be an iterative numerical process.
In order to get some insight into the properties of the ML estimator, the structure
of the ln
L
function is now studied. This structure is established by the number and
nature of the stationary points of the function. Stationary points are defined as points
where the gradient vanishes, i.e., where
∂
∂
ln
L
=
0
(4.99)
A
Substituting Equation 4.97 into Equation 4.99, we obtain
N
∑
∂
∂
Am
−=
NA
n
ln
I
0
.
(4.100)
0
A
σ
σ
2
2
n
=
1
Given that
dI
()
z
ν
ν
=
I
()
z
z
+
Iz
(),
(4.101)
ν
+1
ν
dz
or explicitly that the derivative of
I
0
(
z
), with respect to
z
, equals
I
1
(
z
), it follows that
( )
( )
N
Am
I
n
∑
m
NA
1
σ
2
n
−=
0
,
(4.102)
σ
Am
σ
2
2
I
n
0
2
n
=
1
σ
Search WWH ::
Custom Search