The true object spectrum is convolved with the point spread function of
the medium.

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.

Inverse filtering,

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.

web@ed-phys.fr