Image Processing Reference
In-Depth Information
From the presented above DEDR strategie, one can deduce that the solution to the
optimization problem found in the previous study (Shkvarko, 2006) results in
n
1
F =
KS R
,
(9)
n
1
where
K
= (
SR S
+
A
-1
)
-1
(10)
represents the so-called regularized reconstruction operator;
R
is the
noise whitening
filter, and the adjoint (i.e. Hermitian transpose) SFO
S
+
defines the matched spatial filter in
the conventional signal processing terminology.
2.3.2 DEDR strategy for uncertain operational scenario
To optimize the search for the desired SO
F
in the
uncertain
operational scenario with the
randomly perturbed SFO (4), the
extended
DEDR
strategy was proposed in (Shkvarko, 2006)
F =
arg min
F
max
p
{
ext
(
F
)}
(11)
2
|| ||
()
subject to <||
Δ
||
2
>
p
(
Δ
)
(12)
where the conditioning term (12) represents the worst-case statistical performance (WCSP)
regularizing constraint imposed on the unknown second-order statistics <||
Δ
||
2
>
p
(
Δ
)
of
the random distortion component
Δ
of the SFO matrix (4), and the DEDR “extended risk”
is defined by
~
~
ext
(
F
)
=
tr{<(
F - I
)
A
(
F - I
)
+
>
p
(
Δ
)
} + tr{
FR
n
F
+
}
(13)
where the regularization parameter
and the metrics inducing weight matrix
A
compose
the processing level “degrees of freedom” of the DEDR method.
To proceed with the derivation of the robust SFO (11), the risk function (13) was next
decomposed and evaluated for its the maximum value applying the Cauchy-Schwarz
inequality and Loewner ordering (Greco & F. Gini, 2007) of the weight matrix
A
I
with
the scaled Loewner ordering factor = min{
:
A
} = 1. With these robustifications, the
extended DEDR strategy (11) is transformed into the following optimization problem
F =
min
F
{
(
F
)
}
(14)
with the
aggregated
DEDR risk function
(F)} = tr{(FS - I)A(FS - I)
+
} + tr{F
R
F
+
},
(15)
Where
RR
= (R
n
+ I); = / 0.
(β)
(16)
The optimization solution of (14) follows a structural extension of (9) for the augmented
(diagonal loaded)
R
that yields
1
KSR
,
F =
(17)
