As is well known the Hubble Space Telescope (HST) suffers from, amongst other defects, severe spherical aberration which takes light away from the central core of the Point Spread Function (PSF). A partial correction of the aberration effects can be attempted since the PSF of the instrument is known. One knows the linear integral relationship between the object and the image and therefore one can try to restore the object by solving this equation. This is a Fredholm equation of the first kind, which is a classical example of an ill-posed problem, namely a problem whose solution is affected by numerical instability and therefore is strongly noise dependent. Approximate and stable solutions can be obtained using suitable techniques (regularized inversion methods). The accuracy of these solutions depends essentially on the signal-to-noise ratio. For the HST, this is not very high.
The isoplanatic (deconvolution) problem, assuming the scalar diffraction theory which applies well to these low-aperture systems, reduces to one of inverting the first-kind Fredholm convolution integral equation
where is the image from some distribution
via the integral operator
.
represents the PSF of the telescope. The problem is to estimate
for a given
, assuming
is known.
Current efforts at reconstructing aberration-degraded images are, in the main,
based upon conventional Fourier methods such as iterative deconvolution (here
we use the term deconvolution in this more restricted sense rather than the
more general term ``reconstruction'' which it sometimes erroneously replaces)
as well as other techniques such as Maximum Entropy (ME) (Gull and Skilling,
1984) and so on. The favored data inversion method currently employed by the
astronomy community is that of the Richardson-Lucy iterative deconvolution
where the 'th iteration approximating the solution is given by
(Richardson 1972, Lucy 1974)
where the symbol denotes convolution. It may be seen that this method
retains the positivity expected of incoherent-image data since
and
also
. This is not necessarily true of other matrix and projection
methods which can give negative values also in the case of a noiseless image.
In the case of iterative methods, positivity may be imposed at each iteration
stage or simply at the end of the reconstruction, depending on the method used
and, in our work, was not found to limit the efficiency of such routines.
Deconvolution techniques become suspect when the transfer function (TF) (i.e.,
the Fourier transform of the PSF)
becomes small, as noise
on the image in this region would result in anomalous reconstruction,
exaggerated by the vanishing modulus of the TF. Thus, for conventional
techniques that do not have an adequate stopping criterion, small amounts of
error on the data manifests itself as a much greater error on the solution.
This is the source of the well known ill-conditioning associated with the
inversion of such integral equations. In § 2 we briefly describe the inversion
methods we have used and in we give the numerical results we have
obtained.