# The maximum entropy on the mean method, noise and sensitivity

## J.-F. Bercher, G. Le Besnerais and G. Demoment Laboratoire des signaux et systèmes, Plateau de Moulon 91190 Gif-sur-Yvette, France

### Abstract

In this paper, we discuss some developments in the Maximum Entropy on the Mean Method (MEMM), which was introduced by J. Navaza (1985), and further studied by D. Dacunha-Castelle and F. Gamboa (1989). They concern accounting for noisy data and the derivation of error bars on the reconstruction.

The MEMM enables one to solve the linear ill-posed problem y=Ax when prior knowledge or constraints are added in the form x \in {\cal C}, where {\cal C} is a convex set endowed with a measure \nu. It may be shown, using tools of convex analysis, that the solution \hat{x} can be obtained as \hat{x}= \argmin{x}{{\cal F}(x)} submitted to y=Ax, where {\cal F} is a convex functional built from the knowledge of {\cal C} and \nu. With different reference measures \nu, classical criteria are obtained, such as least-squares, Shannon entropy, Burg entropy, Fermi-Dirac entropy... In some cases, however, {\cal F} has no explicit form and the problem has to be solved from its dual formulation. These basics of the MEMM are summarized in the first part of the paper; further details may be found in the references therein.

The presence of additive noise on the data is considered in a second part. After having reviewed some common approaches, we show that additive noise may be properly integrated in the MEMM scheme. The object and noise are searched for simultaneously in a new convex {\cal C}', which includes the previous convex {\cal C}. The reference measure of this new set is built from the prior knowledge on the solution and from the noise distribution. The MEMM solution is now obtained as \hat{x}=\argmin{x}{\left({\cal F}(x) + {\cal G}(y-Ax)\right)}, where {\cal F} and {\cal G} are convex functionals. The functional {\cal G} is related to the prior distribution of noise, and may be used to take into account specific non gaussian noise distributions.

Thanks to the regularity of the criteria, the sensitivity of the solution to variations of the data may be obtained from second order derivatives of the criterion at the stationary point.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk