Inverse problems and image deconvolution

Abstract

Founding on a physical transformation process described by aFredholm integral equation of the first kind, we first recall themain difficulties appearing in linear inverse problems in thecontinuous case as well as in the discrete case. We describe severalsituations corresponding to various properties of the kernel of theintegral equation. The need to take into account the properties of the solution notcontained in the model is then put in evidence. This leads to theregularization principles for which the classical point of view aswell as the Bayesian interpretation are briefly reminded. We then focus on the problem of deconvolution specially applied toastronomical images. A complete model of image formation isdescribed in Section 4, and a general method allowing to deriveimage restoration algorithms, the Split Gradient Method (SGM), is detailedin Section 5. We show in Section 6, that when this method is applied to thelikelihood maximization problems with positivity constraint, theISRA algorithm can be recovered in the case of the pure Gaussianadditive noise case, while in the case of pure Poisson noise, thewell known EM, Richardson-Lucy algorithm is easily obtained. Themethod is then applied to the more realistic situation typical ofCCD detectors: Poisson photo-conversion noise plus Gaussian readoutnoise, and to a new particular situation corresponding to dataacquired with Low Light Level CCD. Some numerical results areexhibited in Section 7 for these two last cases. Finally, we showhow all these algorithms can be regularized in the context of theSGM and we give a general conclusion.