Ringing reduction in image restoration algorithms using a constraint on the inferior bound of the solution

For many image reconstruction (deconvolution) problems, particularly in the field of astrophysics, the "ringing" effect around bright point sources (or more generally in the vicinity of abrupt variations of the intensity) is a prominent artifact. This effect appears when point sources are superimposed on a background and when the reconstruction algorithm do not take into account such background. For astronomical images, difficulties due to this artifact arise during photometric measurements of point sources or in estimating low intensity structures around point sources. We propose a general method allowing to devise maximum-likelihood image restoration algorithms with a general constraint on the inferior bound of the solution, this bound can be different at each point of the signal; ringing phenomena are then reduced. The method is founded on the use of the Kuhn-Tucker first order optimality conditions and can be applied to the minimization of any convex objective function whose definition domain contains the domain of the constraints. We consider Gaussian and Poisson noise processes leading to the minimization of convex objective functions. Moreover, the method is extended to Gamma likelihood as proposed recently by Cao et al. (IEEE Trans. Image, Process. 8(2) (1999) 286). This correspond to the minimization of a quasi-convex objective function: the Itakura-Saito distance. In each case, we develop non-relaxed algorithms, written in a "quasi multiplicative" form, very easy to use, and whose convergence is formally demonstrated in the Gaussian case. The classical EM and ISRA algorithms as well as those by Cao et al. (IEEE Trans. Image, Process. 8(2) (1999) 286) appear as particular cases of the proposed algorithms. The relaxed and accelerated forms of the algorithms are given. In that case, the convergence is ensured, and we show that "economic methods" for the stepsize computation, allow to obtain fast algorithms. This permits to avoid time expensive computations of the "precision" line search methods generally used. Accelerated forms deduced from the method proposed in (Signal processing 81(5) (2001) 945) are exhibited. The algorithms are tested on simulated images and applied to real cases. The properties of the algorithms are shown, in particular their ability to suppress the "ringing" phenomena.