Scale Invariant Markov Models for Bayesian Inversion of Linear Inverse Problems

Abstract

In a Bayesian approach for solving linear inverse problems one needs to specify the prior laws for calculation of the posterior law. A cost function can also be defined in order to have a common tool for various Bayesian estimators which depend on the data and the hyperparameters. The Gaussian case excepted, these estimators are not linear and so depend on the scale of the measurements. In this paper a weaker property than linearity is imposed on the Bayesian estimator, namely the scale invariance property (SIP).First, we state some results on linear estimation and then we introduce and justify a scale invariance axiom. We show that arbitrary choice of scale measurement can be avoided if the estimator has this SIP. Some examples of classical regularization procedures are shown to be scale invariant. Then we investigate general conditions on classes of Bayesian estimators which satisfy this SIP, as well as their consequences on the cost function and prior laws. We also show that classical methods for hyperparameters estimation (i.e., Maximum Likelihood and Generalized Maximum Likelihood) can be introduced for hyperparameters estimation, and we verify the SIP property for them. Finally we discuss how to choose the prior laws to obtain scale invariant Bayesian estimators. For this, we consider two cases of prior laws: {\em entropic prior laws} and {\em first-order Markov models}. In related preceding works [Mohammad-Djafari90,Mohammad-Djafari93], the SIP constraints have been studied for the case of entropic prior laws. In this paper extension to the case of first-order Markov models is provided. KEYWORDS: Bayesian estimation, Scale invariance, Markov modelling, Inverse Problems, Image reconstruction, Prior model selection