  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.
