Founding on a physical transformation process described by a
Fredholm integral equation of the first kind, we first recall the
main difficulties appearing in linear inverse problems in the
continuous case as well as in the discrete case. We describe several
situations corresponding to various properties of the kernel of the
integral equation.
The need to take into account the properties of the solution not
contained in the model is then put in evidence. This leads to the
regularization principles for which the classical point of view as
well as the Bayesian interpretation are briefly reminded.
We then focus on the problem of deconvolution specially applied to
astronomical images. A complete model of image formation is
described in Section 4, and a general method allowing to derive
image restoration algorithms, the Split Gradient Method (SGM), is detailed
in Section 5.
We show in Section 6, that when this method is applied to the
likelihood maximization problems with positivity constraint, the
ISRA algorithm can be recovered in the case of the pure Gaussian
additive noise case, while in the case of pure Poisson noise, the
well known EM, Richardson-Lucy algorithm is easily obtained. The
method is then applied to the more realistic situation typical of
CCD detectors: Poisson photo-conversion noise plus Gaussian readout
noise, and to a new particular situation corresponding to data
acquired with Low Light Level CCD. Some numerical results are
exhibited in Section 7 for these two last cases. Finally, we show
how all these algorithms can be regularized in the context of the
SGM and we give a general conclusion.