Scale Invariant Markov Models for Bayesian Resolution of Inverse Problems

Stéphane Brette, Jérôme Idier and Ali Mohammad-Djafari
Laboratoire des Signaux et Systèmes (CNRS-ESE-UPS)
École Supérieure d'Électricité,
Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France

Abstract

In a Bayesian estimation approach for solving inverse problems we need to specify the prior law p(x;\theta) and the conditional law p(y|x;\theta) for calculation of the posterior law p(x|y;\theta). We need also a cost function C(\widehat{x},x) to define an estimator \widehat{x}(y,\theta) depending on the data y and the hyperparameters \theta. Except in the linear Gaussian case where all the classical Bayesian estimators become a unique linear function of the data, most Bayesian estimators are otherwise nonlinear functions of the data, and so dependent on the measurement scale. When dealing with linear inverse problems linearity is sometimes too strong a property, while scale invariance property (SIP) often remains a desirable one. In this paper, first we investigate general conditions on classes of Bayesian estimators which satisfy the SIP and their consequences on the cost function and prior laws. Then we show that the cost functions of the three classical Bayesian estimators satisfy the SIP constraints. Finally we discuss how to choose the prior laws to obtain scale invariant Bayesian estimators. For this, we consider two cases of prior laws: entropic prior laws and first-order Markov models. In related preceding works MaxEnt91, MaxEnt93, the SIP constraints have been studied for the case of entropic prior laws. In this paper we extend that work to the case of first-order Markov models and show that the SIP constrains the potential functions of the posterior laws to be in one of the following forms:

We also investigate the application of further constraints such as symmetry, unimodality and convexity of the clique potentials \phi(x_s, x_r) and show that only some of the Markov models in use for modern imaging purposes belong to the exhibited classes.

Keywords

Bayesian estimation, Scale invariance, Inverse Problems, Entropic prior laws, Markov models, Image reconstruction
MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk