Divergences. Scale invariant Divergences. Applications to linear inverse problems. N.M.F. Blind deconvolution

Abstract

This book deals with functions allowing to express the dissimilarity (discrepancy) between two data fields or ''divergence functions'' with the aim of applications to linear inverse problems. Most of the divergences found in the litterature are used in the field of information theory to quantify the difference between two probability density functions, that is between positive data whose sum is equal to one. In such context, they take a simplified form that is not adapted to the problems considered here, in which the data fields are non-negative but with a sum not necessarily equal to one. In a systematic way, we reconsider the classical divergences and we give their forms adapted to inverse problems. To this end, we will recall the methods allowing to build such divergences, and propose some generalizations. The resolution of inverse problems implies systematically the minimisation of a divergence between physical measurements and a model depending of the unknown parameters. In the context image reconstruction, the model is generally linear and the constraints that must be taken into account are the non-negativity as well as (if necessary) the sum constraint of the unknown parameters. To take into account in a simple way the sum constraint, we introduce the class of scale invariant or affine invariant divergences. Such divergences remains unchanged when the model parameters are multiplied by a constant positive factor. We show the general properties of the invariance factors, and we give some interesting characteristics of such divergences.An extension of such divergences allows to obtain the property of invariance with respect to both the arguments of the divergences; this characteristic can be used to introduce the smoothness regularization of inverse problems, that is a regularisation in the sense of Tikhonov.We then develop in a last step, minimisation methods of the divergences subject to non-negativity and sum constraints on the solution components. These methods are founded on the Karush-Kuhn-Tucker conditions that must be fulfilled at the optimum. The Tikhonov regularization is considered in these methods.Chapter 11 associated with Appendix 9 deal with the application to the NMF, while Chapter 12 is dedicated to the Blind Deconvolution problem.In these two chapters, the interest of the scale invariant divergences is highlighted.