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The reader will find extensive studies in Lannes et al. (1987a-b)
where the method summarized here is presented in a closed form,
and some astrophysical applications in
Fraix-Burnet et al. (1989), Roques et al. (1993) and Bouyoucef
et al. (1994).
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 transfer relation:
where 
is the Point Spread Function. The error term includes
random or systematic errors
(e.g., errors on the determination of the PSF,
linearity assumption of the imaging relation, image
sampling) and signal-uncorrelated random noise
(telescope, detectors, atmosphere, guiding, etc.).
The accuracy of the approximation considered when writing the convolution
equation is controlled in Fourier space by a pointwise image-error bound
such that
.
Moreover, can be regarded as a member of a
certain family of objects. For each spatial frequency 
, it is
possible to exhibit a suitable upper bound of the Fourier transform
of : 
.
Due to the noise-type and systematic errors characterized by the
pointwise Signal-to-Noise Ratio (SNR) in the frequency space:
it is preferable to give up trying to determine at its highest
level of resolution. One then defines the ``object to be reconstructed''
as a smoothed version of by a relation of the form
, where
is some synthetic transfer function
small in the mean square sense outside
some extent 
. This domain is the frequency coverage
to be synthesized and regularizes the effective support
of the transfer function 
.
The choice of its diameter is of fundamental
importance because it is closely related to the resolution limit
of the reconstruction process. Intuitively, the
greater is 
with respect to , the greater is the gain in
resolution, but at the same time, the less stable is the
deconvolution process. The problem is then to
define the best compromise between
the resolution to reach and the stability of the solution.
The support of is contained in some finite region 
a priori known, which size and shape, determined in an interactive
manner, will prove to play, together with , an essential part.
This filter 
fulfills three conditions:
-
As the support of
is essentially in , the
support of 
is essentially in 
(say for instance 95%of
its energy in 
). One imposes a condition of the form:
-
The support of
must be as small as possible (with respect to the choices of
and 
) to ensure the best possible resolution
for 
.
-
what secures flux conservation.
A first approximation 
of
of the object to be reconstructed is:
where 
is some threshold value of the order of unity.
In the frequency domain where the SNR is considered as
good (
),
is more or less reliable. One then assigns
to the data a weight depending on the
values of SNR. Let 
be the threshold value beyond
which the whole information contained in 
is considered as entirely reliable. Let 
be the ``weight function''
characterizing the weight attached to this information.
The function is defined as follows:
One supposes that 
outside 
.
So defined, 
characterizes
the way of grounding the reconstruction on the information.
Our deterministic procedure based on a least squares
minimization defines the reconstructed object as the function
which minimizes the functional:
The stability of the reconstruction problem is conditioned by
the smallest eigenvalue 
of the imaging operator.
It can be analytically estimated by examining some
physical parameters: 
, characteristic functions of , and
related to the choice of and to SNR.
Indeed, this eigenvalue is a function of the
``interpolation parameter'' :
characterizing the amount the interpolation to be performed
both is real and in Fourier spaces.
One has the following relation:
where the 
' depend on and 
and are of
the kind ``moment of inertia'' relatively to and 
.
This equation provides useful approximation of the minimum
eigenvalue of the imaging operator occuring in the expression of
an upper limit 
of the reconstruction error:
When this analysis is implemented before the reconstruction,
one has only an estimation 
of 
and the
estimation 
of 
.
The upper bound of the quadratic reconstruction error is given by:
In these two expressions, 
and 
can be regarded as errors resulting from the
noise analysis and from systematic errors related
to the choice of 
.
Note that these majorants correspond to the case where the overall error
would be concentrated in the eigenspace associated with the
minimum eigenvalue, what is not the case in practice
(the error is generally distributed amongst all the eigenspaces).
Once the stability conditions are fulfilled,
the solution can then be obtained by several well known
minimization iterative methods.
We apply the conjugate gradients method (Hestenes et al., 1952) because a
particular implementation allows to compute the spectral
decomposition of the imaging operator (then a more precise error
can be calculated). The convergence is superlinear,
and the least squares solution is reached in a number
of iterations of the order of the number of degrees of freedom
of the reconstruction process.
Next: Example of Deconvolution:
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