Abstract

Constrained ML and MAP Image restoration algorithms Ringing effects reduction and influence of the "a priori" image in regularization

Muriel Roche
University of Marseille
Fresnel

We present the general method we have developed to
devise Maximum Likelihood (ML) and Maximum "a
posteriori" (MAP) images restoration algorithms for
constrained problems. The method we propose allows to
easily construct regularized (non-regularized), relaxed
(non-relaxed), accelerated (or not) algorithms. We
focus in this work on multiplicative (or
quasi-multiplicative) forms of the algorithms.
A constraint on the variable inferior bound of the
solution is introduced to take into account the
background that appears frequently in astronomical
images as well as a constraint for total intensity
conservation typical of deconvolution problems.

The algorithmic method is based on Kuhn-Tucker first
order optimality conditions and can be applied to the
minimization of any convex (or pseudo convex) objective
function whose definition domain contains the domains
of constraints. For ML problems, we consider Gaussian
and Poisson noises processes leading to the
minimization of convex objective functions, and
Gamma-likelihood which corresponds to the minimization
of a pseudo-convex one.

When ML iterative algorithm are used, only the
adequacy of the solution with the data is considered.
Then, because of the ill-posed nature of the problem,
the solution exhibits large instabilities corresponding
to noise amplification. They can be notably limited by
an explicit regularization of the problem using a
smoothness constraint. We then introduce the
regularization by adding a penalty term to the negative
Log-Likelihood and we show how our method can be
extended to penalized convex functions. Two types of
penalty functions are considered: a quadratic and an
entropy one and we first use a constant "a priori"
image. The results obtained are never very satisfactory
whatever the mathematical form of the penalty
function. On the contrary, when the Tikhonov
regularization with a derivative operator (Laplacian)
is used, fair results are obtained; in this case, the
"a priori" image is a smooth version of the current
solution.
This result leads us to use the same non constant "a
priori" image deduced from the Laplacian operator in
conjunction with the entropy penalty function
previously considered; we show that our method allows
to obtain effective algorithms.
We finally apply our algorithms on both simulated and
real astronomical images. The properties of the
algorithms are shown, in particular ringing effect
around bright objects is highly reduced by the only
introduction of the lower bound constraint, moreover we
show that, when smooth solutions are wanted the
improvement of the restored image by regularized
algorithms depends mainly on the choice of the "a
priori" image and to a lesser extent on the form of the
penalty function.