  In this paper we propose a new Bayesian estimation method to solve linear
inverse problems in signal and image restoration and reconstruction problems
which has the property to be scale invariant. In general, Bayesian estimators
are {\em nonlinear} functions of the observed data. The only exception is the
Gaussian case. When dealing with linear inverse problems the linearity is
sometimes a too strong property, while {\em scale invariance} often remains a
desirable property. As everybody knows one of the main difficulties with using
the Bayesian approach in real applications is the assignment of the direct
(prior) probability laws before applying the Bayes' rule. We discuss here how
to choose prior laws to obtain scale invariant Bayesian estimators. In this
paper we discuss and propose a familly of generalized exponential probability
distributions functions for the direct probabilities (the prior $p(\xb)$ and
the likelihood $p(\yb|\xb)$), for which the posterior $p(\xb|\yb)$, and,
consequently, the main posterior estimators are scale invariant. Among many
properties, generalized exponential can be considered as the maximum entropy
probability distributions subject to the knowledge of a finite set of
expectation values of some knwon functions.
