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2. Technique for recovering the point spread function

The true object spectrum is convolved with the point spread function of the medium.
where tex2html_wrap_inline1597 is the time averaged intensity distribution, tex2html_wrap_inline1599 is the true object intensity distribution, tex2html_wrap_inline1601 is the time averaged intensity distribution and "*" denotes convolution. Performing Fourier transform on either side of Eq. (1), we get

where tex2html_wrap_inline1603 , tex2html_wrap_inline1605, tex2html_wrap_inline1607 are the Fourier transforms of tex2html_wrap_inline1597, tex2html_wrap_inline1599 tex2html_wrap_inline1601 respectively, and u, v are the spatial frequency coordinates. To recover the true object spectrum tex2html_wrap_inline1599, we perform inverse filtering on the degraded image. Therefore the true object intensity distribution will be

Inverse transforming tex2html_wrap_inline1605 we get tex2html_wrap_inline1599. In our case tex2html_wrap_inline1625 is not known. Guess psf is constructed and inverse filtering is done. Let tex2html_wrap_inline1627 be the guess psf. The Fourier transform of the guess psf is tex2html_wrap_inline1629. Using this psf we get,
The reconstructed image spectrum tex2html_wrap_inline1631 will be the inverse Fourier transform of tex2html_wrap_inline1633.

The point spread function of the atmosphere which blurs the object intensity distribution is (Tatarski 1961; Fried 1966)
where u is the spatial frequency vector, tex2html_wrap_inline1637 is the mean wavelength of observation, tex2html_wrap_inline1639 is the Fried's parameter (seeing parameter) and tex2html_wrap_inline1641 the power index which was derived to have a value of 5/3 in the case of astronomical observations. In practice there could be deviations in the value of tex2html_wrap_inline1641. The behaviour of the point spread function in the tail of the profile depends on tex2html_wrap_inline1641 and tex2html_wrap_inline1639 is a measure of the core of the point spread function profile. In our proposed technique we use the Fried's coherence function in its functional form but both tex2html_wrap_inline1641 and tex2html_wrap_inline1639 are left as free parameters.

The degraded image is deconvolved using a series of point spread functions with different tex2html_wrap_inline1639 and tex2html_wrap_inline1641. The number of elements N, equal to and less than zero is found in each reconstruction. In this two parameter space we search for the minimum of number of zeros and negative values. The corresponding tex2html_wrap_inline1639 and tex2html_wrap_inline1641 at which the minimum occurs are the true point spread function parameters.

In the presence of noise, Eq. (1) is written as
where tex2html_wrap_inline1663 is the noise in the image plane which gets added to the blurred object intensity distribution. Since noise is additive, it is not convolved with the atmospheric psf, but is effectively convolved with a delta function, which in turn, can be considered as a psf with very large Fried's parameter, say tex2html_wrap_inline1665, where tex2html_wrap_inline1667
with tex2html_wrap_inline1669 approaching a delta function. For obtaining the parameters of the psf the above equation is Fourier transformed and inverse filtering is performed.

Inverse filtering,

This equation is inverse transformed and the number of non positive pixels are found. Similarly for other tex2html_wrap_inline1639 values tex2html_wrap_inline1629 is constructed and the number of non positive pixels found. Since tex2html_wrap_inline1665 is always greater than tex2html_wrap_inline1639, the number of non-positive pixels N contributed by the second term is not expected to go through a minimum. Therefore even in the presence of noise the minima in N is expected to occur when the guess psf parameters matches with the true tex2html_wrap_inline1639 and tex2html_wrap_inline1641 values and hence tex2html_wrap_inline1639 and tex2html_wrap_inline1641 can be found by looking for the deepest minima in N in the parameter space of tex2html_wrap_inline1641 and tex2html_wrap_inline1639.

This makes the proposed technique more general and could be used when the functional form of the point spread function of the intervening medium is of the Fried's coherence function type.

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