Image Processing Reference
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Hence, the condition for the stationary points becomes
( )
( )
N
Am
I
n
1
∑
1
σ
2
A
=
m
.
(4.103)
n
N
Am
I
n
0
2
n
=
1
σ
0 is a stationary point of ln
L
, independent
of the particular data set. The nature of a stationary point is determined by the sign
of the second-order derivative of the function in that point. From this derivative, it
follows whether a stationary point is a minimum or a maximum and whether or not
it is degenerate. From Equation 4.97, the second-order derivative of the ln
L
function
can be computed to yield
It follows from Equation 4.103 that
A
=
( )
(
( )
( )
N
Am
Am
I
I
n
1
2
n
∂
2
ln
L
A
∑
m
2
σ
2
N
1
σ
2
σ
2
n
=
1
−
−
−
.
(4.104)
))
∂
σ
Am
Am
Am
σ
2
4
2
I
I
2
n
n
n
0
2
0
2
n
=
1
σ
σ
From the knowledge that [20]
ν
()
z
Iz
∼
2
Γ
(
ν
+
1
)
when
z
→
0
,
(4.105)
ν
it is easy to verify that
A
=
0 is a minimum of ln
L
whenever
N
1
∑
m
2
> σ
2
2
.
(4.106)
n
N
n
=
1
If this condition is met, the ln
L
function will have two further stationary points
being maxima. This can be seen by studying the possible structures of the ln
L
function using catastrophe theory.
Catastrophe theory
is concerned with the struc-
tural change of a parametric function under influence of its parameters [41]. It
tells us that a structural change of the function is always preceded by a degeneracy
of one of its stationary points. In order to analyze such a structural change, the
parametric function can be replaced by a Taylor expansion of the essential variables
about the latter stationary point. The essential variables correspond to the directions
in which degeneracy may occur. According to the catastrophe theory, the global
structure of a parametric function, with only one essential variable, is completely
set by its Taylor expansion with terms up to the degree to which the coefficient
cannot vanish under the influence of its parameters. The function studied here is
ln
L
as a function of
A
. Its parameters are the observations. Thus, the structural
change of the ln
L
function under the influence of the observations has to be studied.
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