Digital Signal Processing Reference
In-Depth Information
1
N
(
R
where
C
w
=
is the covariance of the noise on the covariance data. Since
usually the astronomical sources are much weaker than the noise (often at least by a
factor 100), we can approximate
R
⊗
R
)
n
I
,
≈
˙
n
. If the noise is spatially white,
˙
n
=
σ
we obtain for the covariance of
i
D
n
i
D
)
≈
σ
J
4
N
M
s
M
s
.
cov
(
The variance in the image is given by the diagonal of this expression. From this and
the structure of
M
s
A
and the structure of
A
, we can see that the variance on
each pixel in the dirty image is constant,
=(
◦
A
)
n
J
2
N
, but that the noise on the image
is correlated, possibly leading to visible structures in the image. This is a general
phenomenon.
Similar equations can be derived for the MVDR image and the AAR image.
σ
/
(
)
5.3
Weighted Least Squares Imaging
At this point, the deconvolution problem can be formulated as a maximum
likelihood (ML) estimation problem, and solving this problem should provide a
statistically efficient estimate of the parameters. Since all signals are assumed to be
i.i.d. Gaussian signals, the derivation is standard and the ML estimates are obtained
ln
tr
R
−
1
R
ˆ
{
,
ˆ
}
=
arg min
,
|
R
(
,
)
|
+
(
,
)
.
(26)
where
|·|
denotes the determinant.
R
(
,
)
is the model, i.e., vec
(
R
(
,
)) =
r
=
M
.
In this subsection, we will consider the overparametrized case, where each pixel
in the image corresponds to a source. In this case,
M
is a priori known, the model is
linear, and the ML problem reduces to a Weighted Least Squares (WLS) problem to
match
r
to the model
r
:
()
C
−
1
/
2
2
2
H
C
−
1
w
=
ˆ
arg min
(
r
−
r
)
=
arg min
(
r
−
M
)
(
r
−
M
)
(27)
w
where we fit the “data”
r
to the model
r
=
M
. The correct weighting is the inverse
of the covariance of the residual,
w
=
r
−
r
, i.e., the noise covariance matrix
C
w
=
. For this, we may also use the estimate
C
w
obtained by using
R
instead
of
R
. Using the assumption that the astronomical sources are much weaker than the
noise we could contemplate to use
R
R
1
N
(
⊗
R
)
≈
˙
n
for the weighting. If the noise is spatially
n
I
, the weighting can then even be omitted.
white,
˙
n
=
σ
