The true object spectrum is convolved with the point spread function of
where is the time averaged intensity distribution, is the true object intensity distribution, is the time averaged intensity distribution and "*" denotes convolution. Performing Fourier transform on either side of Eq. (1), we get
where , , are the Fourier transforms of , respectively, and u, v are the spatial frequency coordinates. To recover the true object spectrum , we perform inverse filtering on the degraded image. Therefore the true object intensity distribution will be
Inverse transforming we get . In our case
is not known.
Guess psf is constructed and
inverse filtering is done. Let be the guess psf.
The Fourier transform of the guess psf is .
Using this psf we get,
The reconstructed image spectrum will be the inverse Fourier transform of .
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, is the mean wavelength of observation, is the Fried's parameter (seeing parameter) and 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 . The behaviour of the point spread function in the tail of the profile depends on and 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 and are left as free parameters.
The degraded image is deconvolved using a series of point spread functions with different and . 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 and at which the minimum occurs are the true point spread function parameters.
In the presence of noise, Eq. (1) is written as
where 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 , where
with approaching a delta function. For obtaining the parameters of the psf the above equation is Fourier transformed and inverse filtering is performed.
This equation is inverse transformed and the number of non positive pixels are found. Similarly for other values is constructed and the number of non positive pixels found. Since is always greater than , 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 and values and hence and can be found by looking for the deepest minima in N in the parameter space of and .
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.