Digital Signal Processing Reference
In-Depth Information
images require dedicated hardware solutions and sometimes even supercomputers.
This section will describe some of the algorithms for deconvolution. Additional
4.1
The CLEAN Algorithm
is obtained via
a sequential Least Squares fitting method. The algorithm is based on the assumption
that the sky is mostly empty, and consists of a set of discrete point sources. The
brightest source is estimated first, its contribution is subtracted from the dirty image,
then the next brightest source is subtracted, etc.
The algorithm further assumes that
B
p
)
and the known dirty beam
B
(
p
)
, the desired image
I
(
p
)
has its peak at the origin. Inside the
loop, a candidate location
p
q
is selected as the location of the largest peak in
I
D
(
(
p
)
p
)
,
the corresponding power
ˆ
2
σ
q
is estimated, and subsequently a small multiple of
q
B
σ
ˆ
(
p
−
p
q
)
is subtracted from
I
D
(
p
)
. The objective is to minimize the residual,
until it converges to the noise level:
q
0
while
I
D
(
=
p
)
is not noise-like:
⎡
⎣
q
=
q
+
1
p
q
=
arg max
p
I
D
(
p
)
q
ˆ
σ
=
I
D
(
p
q
)
/
B
(
0
)
ˆ
q
B
I
D
(
p
)
:
=
I
D
(
p
)
−
γ
σ
(
p
−
p
q
)
, ∀
p
2
q
B
synth
(
ˆ
I
clean
(
p
)=
I
D
(
p
)+
∑
q
γ
σ
p
−
p
q
)
, ∀
p
.
γ
≤
The scaling parameter
1 is called the loop gain; for accurate convergence
it should be small because the estimated location of the peak is at a grid point,
whereas the true location of the peak may be in between grid points.
B
synth
(
)
is a
“synthetic beam”, usually a Gaussian bell-shape with about the same beam width as
the main lobe of the dirty beam; it is introduced to mask the otherwise high artificial
resolution of the image.
In current imaging systems, instead of the subtractions on the dirty image, it is
considered more accurate to do the subtractions on the covariance data
R
m
instead,
p
R
m
:
R
m
−
γ
q
a
q
(
H
=
σ
ˆ
m
)
a
q
(
m
)
and then to recompute the dirty image. For efficiency, usually a number of peaks are
estimated from the dirty image together, the covariance is updated for this ensemble,
and then the residual image is recomputed.
