Image Processing Reference
In-Depth Information
The only essential variable is the signal parameter
A
. The stationary point that may
become degenerate is the point
A
0 (degeneracy occurs whenever Equation 4.104
becomes equal to zero). If the ln
L
function is Taylor expanded about the stationary
point
A
=
=
0, we yield
La
b
c
ln
=+
!
A
2
+
AOA
4
+
(
6
),
(4.107)
2
4
!
with
N
−
∑
1
m
2
1
2
σ
2
N
n
a
=
1
−
,
(4.108)
σ
m
σ
4
2
n
n
=
N
∑
1
mN
2
n
b
=
−
,
(4.109)
2
σσ
4
2
n
=
N
3
8
∑
m
2
n
c
=−
,
(4.110)
σ
8
n
=
1
and the order symbol of Landau. Notice that because the ln
L
function is
symmetric about
A
O
()
.
0, the odd terms are absent in Equation 4.107. In order
to investigate if the expansion up to the quartic term in Equation 4.107 is
sufficient, it has to be determined whether the coefficients may change sign
under influence of the observations. It is clear from Equation 4.109 that the
coefficient
b
may change sign. The coefficient , however, will always be
negative, independent of the particular set of observations. This means that the
expansion (Equation 4.107) is sufficient to describe the possible structures of
the ln
L
function. Consequently, the study of the ln
L
as a function of the
observations can be replaced by a study of the following quartic Taylor poly-
nomial in the essential variable
A
:
=
c
b
c
A
2
+
A
4
,
(4.111)
2
!
4
!
where the term has been omitted because it does not influence the structure.
The polynomial in Equation 4.111 is always stationary at
A
a
0. This will be a
minimum, a degenerate maximum, or a maximum when is positive, equal to
zero, or negative, respectively. It follows directly from Equation 4.111 that ln
L
has two additional stationary points (being maxima) if is positive, that is, if
Equation 4.106 is met. Note that the condition in (Equation 4.106) is always met
=
b
b
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