The deconvolution problem is basically an ill-posed problem (Tikhonov & Arsenin 1977): it does not fulfill the three Hadamard conditions of existence, uniqueness and stability of the solution. The last condition of stability may cause the main problems because if it is not fulfilled, a slight error in the data may lead to a very large error in the solution.
To circumvent the ill-conditionedness nature of this problem, one is led to postulate that the properties of the solution are not entirely contained in the equation to be solved. Therefore, one has to introduce a priori information on the solution to regularize the deconvolution process.
To a first approximation, the experimental data
are related to the ``original object'' 
- the intensity of the source at some high level of resolution -
by an experimental Point-Spread Function (PSF)
in the following way:
where e is an additive term including
random or systematic errors
(i.e. errors on the determination of the PSF,
linearity assumption, image
sampling...) and signal-uncorrelated random noise
(telescope, detectors, atmosphere, guiding...).
The support of 
is contained in some finite region
V whose size and shape, determined in an interactive manner,
will prove to play an essential role.
As the support H of the transfer function h ,
Fourier transform of
, determines the experimental spatial-frequency
aperture, one defines a centrosymmetric synthetic aperture
, including H , and regularizing it.
The choice of the diameter of this synthetic aperture
defines the best compromise possible between
the resolution to reach and the stability of the solution.
Of course, it is preferable to give up trying to determine
at its highest level of resolution,
and one defines the ``object to be reconstructed''
as a smoothed version of 
by a relation
of the form 
,
where s(u) is a Prolate Spheroidal Function
(Slepian & Pollak 1961) whose energy is concentrated in 
.
The ratio of the amplitude spectrum of the image to the
amplitude spectrum of the noise defines the pointwise
signal-to-noise ratio SNR in the frequency space:
where 
is an image-error bound such that
.
Let us define a threshold value 
(in practice of the order of 1 or greater)
under which SNR is considered as
poor. In these conditions, one can give a first
approximation 
of the spectrum
of the object to be reconstructed:
Note that information
contained in the spectrum of 
when 
is less than 
is lost.
In the case of a deterministic procedure based on a least-squares minimization one defines the reconstructed object as the function which minimizes the functional:
where g(u) is a weight function bounded by 1, to be defined
in relation to 
and 
. In particular,
g(u) = 0 in the parts of 
where
(characterizing the frequency
gaps in 
), and g(u) = 1 outside 
(regularization principle). In the parts of 
where the
information must be taken into consideration, g(u) is defined
as an increasing function of 
.
Before implementing a numerical iterative method for solving the problem, it is preferable to verify that the synthetic aperture (i.e. the resolution) has been well chosen, to avoid long and expensive computations leading in any case to an unstable solution.
The smallest eigenvalue 
of the imaging
operator 
conditions the stability of the reconstruction
problem (U and 
are the direct and inverse Fourier
Transform operators).
It can be analytically estimated by examining some
physical parameters of the problem: the functions v and
, characteristic functions of V and 
.
Indeed, this eigenvalue is a function of the
``interpolation parameter'':
characterizing the amount of interpolation to be performed both in real and Fourier spaces. One has the following relation:
where the 
' s depend on v and w and are of
the kind ``moment of inertia'' relatively to 
.
This equation provides useful approximation to the minimum
eigenvalue 
of the imaging operator occuring in
the expression of an upper limit
of the quadratic reconstruction error:
Then, by suitably choosing 
( V being quasi imposed),
the size of the error can be acceptably small. So,
we have an idea of the stability of the problem before
its implementation.
If this error remains too large,
must be redefined at a lower level of
resolution by reducing 
. This operation, executable
in an interactive manner, leads to a compromise
between resolution and reliability.
A more exhaustive error analysis can be conducted after
the reconstruction process to obtain a better estimate of
the upper-bound of the error (see Lannes et al. 1996).
The multiresolution analysis (MRA) provides a representation intermediate between the spatial and the Fourier ones: the data are represented as a superposition of wavelets at some scale level, each level being further decomposed into a lower scale level. The reader unfamiliar with this analysis is refered to Mallat (1989) or Daubechies (1992). Here, the MRA plays a decisive role in the control of the stability of the deconvolution (in particular when the field to resolution ratio increases).
Indeed, the MRA is very useful for the definition of the mathematical space in which the image has to be reconstructed: it can be decomposed in a collection of orthogonal sub-spaces corresponding to different resolutions. It is then usually easier to solve several sub-problems separately in each sub-space and to reconstruct the global solution afterwards. For instance, the deconvolution of SN 1987A makes use of a MRA for choosing the multiresolution support V of the restored object (Fig. 1 (click here)).
Figure 1: Multiresolution support of SN 1987A