Image Processing Reference
In-Depth Information
simple case of certain operational scenario (Henderson & Lewis, 1998), (Shkvarko, 2008), the
discrete-form (i.e. matrix-form) SFO
S
is assumed to be deterministic, i.e. the random
perturbation term in (4) is irrelevant,
Δ
=
0
.
The digital enhanced RS imaging problem is formally stated as follows (Shkvarko, 2008): to
map the scene pixel frame image
B
via lexicographical reordering
B
=
L
{
b
} of the SSP
vector estimate
b
reconstructed from whatever available measurements of independent
realizations of the recorded data vector
u.
The reconstructed SSP vector
b
is an estimate of
the second-order statistics of the scattering vector
e
observed through the perturbed SFO
(4) and contaminated with noise
n
; hence, the RS imaging problem at hand must be
qualified and treated as a statistical nonlinear inverse problem with the uncertain operator.
The high-resolution imaging implies solution of such an inverse problem in some optimal
way. Recall that in this paper we intend to follow the unified descriptive experiment design
regularized (DEDR) method proposed originally in (Shkvarko, 2008).
2.3 DEDR method
2.3.1 DEDR strategy for certain operational scenario
In the descriptive statistical formalism, the desired SSP vector
b
is recognized to be the
vector of a principal diagonal of the estimate of the correlation matrix
R
e
(
b
), i.e.
b
= {
ˆ
e
R
}
diag
.
b
= {
ˆ
e
Thus one can seek to estimate
R
}
diag
given the data correlation matrix
R
u
pre-
estimated empirically via averaging
J
recorded data vector snapshots {
u
(
j
)
}
1
ˆ
u
J
j
uu
} =
uu
,
Y
=
R
= aver
j
{
(5)
() ()
j
j
1
J
()
j
()
j
J
by determining the solution operator (SO)
F
such that
b
= {
ˆ
e
R
}
diag
= {
FYF
+
}
diag
(6)
where {·}
diag
defines the vector composed of the principal diagonal of the embraced
matrix.
To optimize the search for
F
in the
certain
operational scenario the DEDR strategy was
proposed in (Shkvarko, 2006)
F
min
F
{
(
F
)},
(7)
(
F
)
=
trace{(
FS - I
)
A
(
FS - I
)
+
} +
trace{
FR
n
F
+
}
(8)
that implies the minimization of the weighted sum of the systematic and fluctuation errors
in the desired estimate
b
where the selection (adjustment) of the regularization parameter
and the weight matrix
A
provide the additional experiment design degrees of freedom
incorporating any descriptive properties of a solution if those are known a priori (Shkvarko,
2006). It is easy to recognize that the strategy (7) is a structural extension of the statistical
minimum risk estimation strategy for the nonlinear spectral estimation problem at hand
because in both cases the balance between the gained spatial resolution and the noise energy
in the resulting estimate is to be optimized.
