arXiv subjects

Jérôme Idier

Publications and source records attributed to Jérôme Idier.

21 records · Page 2Linked to original sources

arXiv2009

On Algorithms Based on Joint Estimation of Currents and Contrast in Microwave Tomography

This paper deals with improvements to the contrast source inversion method which is widely used in microwave tomography. First, the method is reviewed and weaknesses of both the criterion form and the optimization strategy are underlined. Then, two new algorithms are proposed. Both of them are based on the same criterion, similar but more robust than the one used in contrast source inversion. The first technique keeps the main characteristics of the contrast source inversion optimization scheme but is based on a better exploitation of the conjugate gradient algorithm. The second technique is based on a preconditioned conjugate gradient algorithm and performs simultaneous updates of sets of unknowns that are normally processed sequentially. Both techniques are shown to be more efficient than original contrast source inversion.

math.NA↗

arXiv2001

Scale Invariant Markov Models for Bayesian Inversion of Linear Inverse Problems

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

physics.data-an↗

arXiv2001

A scale invariant Bayesian method to solve linear inverse problems

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.

physics.data-an↗