Inverse problems and image deconvolution

Abstract

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.