Simulations were carried out on single sources as well as on multiple sources. The technique has been tested on objects with intensity distribution with smooth edges as well as on objects with sharp edge. Here we discuss the case of an object with multiple sources and sharp edges.
The true object consists of two sources with unequal intensities.
The object plane has a plate scale of 
per pixel.
The angular size of source A and B is 
and 
respectively.
The two sources are seperated by a angular distance of around 
.
Figure 1 (click here) shows the true object intensity distribution and Fig. 2 (click here) is the
atmospheric psf. The convolution of the object intensity distribution
with the atmospheric psf yields the degraded image. Figure 3 (click here) is the
convolved image.
Fried derived the
expression for the coherence function H(u,v) and obtained
a value of 5/3 for 
.
No departures are seen in the value of = 5/3
under the conditions of astronomical observations. Assuming
is known, for each values of 
the degraded image is
deconvolved and N is obtained. The plot of N for various is given in
Fig. 4 (click here). We see that when the deconvolving psf's is equal to the
true , N is a minimum.
However experimental evidence for departures from 5/3 power law has been reported in the case of horizontal propogation near the ground (Bouricius & Clifford 1970; Clifford et al. 1971; Buser 1971).
Therefore
assuming that is unknown, we do the following. For different values
of , the reconstruction is done for a range of .
Figure 5 (click here)
is a plot of N vs. at a given . Here again we see that
when the deconvolving psf's becomes equal to the true psf's
, N goes to a minimum.
Now assuming both and are unknown,
N is found at each value of and .
Figure 6 (click here) is the surface plot of N as a function of
and .
Wrong estimation of one of the parameters,
or leads to inaccurate photometric values and also generates
spurious features in the reconstructed
image. Figure 7 (click here) is an example of an image degraded by a psf with
, but
reconstructed using 
, 
cm and
Fig. 8 (click here) is reconstruction
done with correct value of but with a = 5.5 cm instead
of the true cm. We can
clearly see spurious features in both the reconstructions.
In the presence of noise we need to reduce the grid size of the parameters
in order to
get the correct or .
The above simulations were repeated with different kinds of noise added to the blurred image.
Figure 9 (click here) is the
image degraded by the atmospheric psf with cm and . A uniform distribution noise is added to the degraded image.
The signal to noise ratio in all the noisy images is around 5.0.
Figure 10 (click here) is the plot of the number of
non positive pixels N for various values of Fried's
parameter in the presence of additive noise and
Fig. 11 (click here) is the plot of the number of non positive pixels N in each
reconstructed image for different values of . In both
the plots we see that at the minima in N occurs at
the true and .
Figure 12 (click here) is
the surface plot of N as a function of and .
A search for the minima in N gives the exact values of and
of the true psf.
Figure 13 (click here) is the convolved image to which zero mean
Gaussian white noise has been added.
Figure 14 (click here) is the plot of the number of non positive pixels N in
each reconstructed image for different values of Fried's parameter
. Figure 15 (click here) and Fig. 16 (click here) are the plots of N against
for reduced grid size of , 0.5cm and 0.1 cm respectively.
It is seen that the minima in N occurs at a close
to the true .
Figure 17 (click here) is the surface plot of the number of non positive pixels N
in the , parameter space. The minima in N gives the true
and .
Figure 18 (click here) is the image intensity distribution with additive
Poisson noise. Figures 19 (click here), 20 (click here) and 21 (click here) are the plots
of the number of non positive pixels N against the Fried's
parameter with grid size in equal to 1.0 cm, 0.5 cm
and 0.1 cm respectively. We can see that for a grid size in equal
to 1.0 cm the minima in N occurs at 2.0 cm. When the grid size is reduced
to 0.5 cm the minima in N shifts to 5.5 cm and the minima in N stays at 5.5 cm
when the grid size is reduced to 0.1 cm. Hence in the presence of noise
reduction of grid size helps to identify the parameters
and
more accurately.
Figure 22 (click here) is the surface plot of N in the parameter space of
and . The minima in N occurs at the true and .
However, any deconvolution performed on these images will not give the correct reconstruction unless the noise is filtered.
Figure 1: True object intensity distribution. Two sources
with unequal intensities.
Plate scale in the image plane is per pixel.
Angular seperation between the two sources is 
.
Angular size of stronger source is 
and the angular size of
the weaker sfource is 
Figure 2: The atmospheric point spread function (psf) with
cm corresponds
to 
,
Figure 3: The true object intensity distribution (Fig. 1) convolved with the atmospheric
psf (Fig. 2) and produces this degraded image.
Since the atmospheric psf is wider the angular separation between the
two sources the blurred image is seen as a broadened single source
Figure 4: Plot of number of non positive pixels N (including zeros in the image plane)
with varying Fried's parameter . is fixed at 1.67 which is the
true .
True is 5.0 cm. Minima in N occurs at cm
Figure 5: Plot of number of non positive pixels N (including zeros in the image plane)
for different values of the power index . The Fried's parameter
cm. True . Minima in N at
Figure 6: Surface plot of N for different values of the power index and
the Fried's parameter . The and corresponding to the
minmum of N gives the true
and values
Figure 7: Image intensity distribution reconstructed using a Fried
parameter
of 5.0 cm but instead of ,
has been used.
Spurious features can be
seen in the reconstruction
Figure 8: Image reconstructed using of 1.67. The
value used in this reconstruction is 
cm
instead of 5.0 cm which is the true .
Spurious features can be seen in the reconstructed image
Figure 9: Object intensity distribution
degraded by the atmospheric psf. An
uniform noise is added to the blurred image. The signal to
noise in the image is 5.0
Figure 10: Plot of number of non positive pixels N
(including zeros in the
image plane) in a blurred image with
additive uniform distribution noise
against varying Fried's parameter .
is fixed at 1.67
which is the
true .
True cm. Minima in N at cm
Figure 11: Plot of number of non positive pixels N
(including zeros in the image plane)
for different values of the power index . The Fried's parameter
fixed at 5.0 cm which is the true value of .
True . Minima in N at
Figure 12: Surface plot of N for different values of the power
index and
the Fried's parameter for
an image with uniform distribution
noise added. The and corresponding to the
minmum of N gives the true
and values
Figure 13: Object intensity
distribution degraded by the atmospheric psf. Zero mean
Gaussian white
noise is added to the blurred image. The signal to
noise in the image is 5.0
Figure 14: Plot of number of non positive pixels N
(including zeros in the image plane)
in a blurred image with
additive Zero mean Gaussian white noise
against varying Fried's parameter .
is fixed at 1.67 which is the
true .
The grid size of is 1.0 cm.
True cm. Minima in N at cm
Figure 15: Plot of number of non positive pixels N
(including zeros in the image plane)
in a blurred image with
additive Zero mean Gaussian white noise
against varying Fried's parameter .
is fixed at 1.67 which is the
true .
The grid size of is 0.5 cm.
True cm. Minima in N at cm
Figure 16: Plot of number of non positive pixels N (including zeros in the image plane)
in a blurred image with
additive Zero mean Gaussian white noise
against varying Fried's parameter .
is fixed at 1.67 which is the
true .
The grid size of is 0.1 cm. True cm.
Minima in N at cm
Figure 17: Surface plot of N for different values of the power index and
the Fried's parameter for an image with zero mean Gaussian white
noise. The and corresponding to the
minmum of N gives the true
and values
Figure 18: Object intensity
distribution degraded by the atmospheric psf. Poisson
noise is added to the blurred image. The signal to
noise in the image is 5.0
Figure 19: Plot of number of non positive pixels N (including zeros in the image plane)
in a blurred image with
additive Poisson noise
against varying Fried's parameter .
is fixed at 1.67 which is the
true .
The grid size of is 1.0 cm.
True cm. Minima in N at 
cm
Figure 20: Plot of number of non positive pixels N
(including zeros in the image plane)
in a blurred image with
additive Poisson noise
against varying Fried's parameter
. is fixed at 1.67 which is the
true .
The grid size of ro is 0.5 cm.
True cm. Minima in N at cm
Figure 21: Plot of number of non positive pixels N
(including zeros in the image plane)
in a blurred image with
additive Poisson noise
against varying Fried's parameter
. is fixed at 1.67 which is the
true .
The grid size of is 0.1 cm.
True cm. Minima in N at ro = 5.5 cm
Figure 22: Surface plot of N
for different values of the power index and
the Fried's parameter for an image with Poisson
noise. The and corresponding to the
minmum of N gives the true
and values