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# 3. Simulations

## 3.1. Object intensity distribution

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.

## 3.2. Search for minima in N in the absence of noise

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 .

## 3.3. Parametric search in the presence of noise

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

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