Jérôme Idier

Marginal estimation of aberrations and image restoration by use of phase diversity

We propose a novel method called marginal estimator for estimating the aberrations and the object from phase-diversity data. The conventional estimator found in the literature concerning the technique first proposed by Gonsalves has its basis in a joint estimation of the aberrated phase and the observed object. By means of simulations, we study the behavior of the conventional estimator, which is interpretable as a joint maximum a posteriori approach, and we show in particular that it has undesirable asymptotic properties and does not permit an optimal joint estimation of the object and the aberrated phase. We propose a novel marginal estimator of the sole phase by maximum a posteriori. It is obtained by integrating the observed object out of the problem. This reduces drastically the number of unknowns, allows the unsupervised estimation of the regularization parameters, and provides better asymptotic properties. We show that the marginal method is also appropriate for the restoration of the object. This estimator is implemented and its properties are validated by simulations. The performance of the joint method and the marginal one is compared on both simulated and experimental data in the case of Earth observation. For the studied object, the comparison of the quality of the phase restoration shows that the performance of the marginal approach is better under high-noise-level conditions.

Multichannel seismic deconvolution

Deals with Bayesian estimation of 2D stratified structures from echosounding signals. This problem is of interest in seismic exploration, but also for nondestructive testing or medical imaging. The proposed approach consists of a multichannel Bayesian deconvolution method of the 2D reflectivity based upon a theoretically sound prior stochastic model. The Markov-Bernoulli random field representation introduced by Idier et al. (1993) is used to model the geometric properties of the reflectivity, and emphasis is placed on representation of the amplitudes and on deconvolution algorithms. It is shown that the algorithmic structure and computational complexity of the proposed multichannel methods are similar to those of single-channel B-G deconvolution procedures, but that explicit modeling of the stratified structure results in significantly better performances. Simulation results and examples of real-data processing illustrate the performances and the practicality of the multichannel approach. >

A new algorithm for iterative deconvolution of sparse spike trains

An iterative algorithm for deconvolution of Bernoulli-Gaussian processes is presented. This detection-estimation problem is formulated as that of a change of initial conditions in linear least-squares estimation. An algorithm with a very simple structure is obtained. It allows the evaluation of either marginal or joint likelihood criteria without any approximation; the resulting method is easy to implement and computationally inexpensive and remains nearly optimal.<<ETX>>

Inversion of large-support ill-conditioned linear operators using a Markov model with a line process

We propose a method for the reconstruction of an image, only partially observed through a linear integral operator. As such an inverse problem is ill-posed, prior information must be introduced. We consider the case of a compound Markov random field with a non-interacting line process. In order to maximise the posterior likelihood function, we propose an extension of the graduated non convexity principle pioneered by Blake and Zisserman (1987) which allows its use for ill-posed linear inverse problems. We discuss the role of the observation scale and some aspects of the implemented algorithm. Finally, we present an application of the method to a diffraction tomography imaging problem.<<ETX>>

Deconvolution of sparse spike trains accounting for wavelet phase shifts and colored noise

The problem of the restoration of spiky sequences when the usual convolution model is corrupted by nonstationary wavelet phase-shifts is addressed. To this end, an extended convolution model driven by a Bernoulli-Gaussian (BG)-like process is introduced. This setting lends itself to easy extension of algorithms designed for BG deconvolution. A comparison of practical results obtained with this new method and BG deconvolution is provided. Numerical experiments indicate an increased robustness compared with standard BG methods.<<ETX>>

Markov modeling for Bayesian restoration of two-dimensional layered structures

Bayesian estimation of two-dimensional stratified structures is described. The major point addressed is the derivation of a statistical prior model that adequately describes such layered media. This problem is of interest in applications in which the data are generally processed in one dimension only. In order to take local interactions into account, a Markovian description is used. The model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements. The resulting class of Markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process. In addition, it is shown to be an extension of one-dimensional models already in use. The properties of the model are investigated, and its practicality is demonstrated by an application to seismic deconvolution. Simulation results show significant improvements with respect to the usual one-dimensional methods. >

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 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 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 family of generalized exponential probability distributions functions for the direct probabilities (the prior p(x) and the likelihood p(y|x)), for which the posterior p(x|y), and, consequently, the main posterior estimators are scale invariant. Among many properties, generalized exponentials can be considered as the maximum entropy probability distributions subject to the knowledge of a finite set of expectation values of some known functions.

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).

Stack algorithm for recursive deconvolution of Bernoulli-Gaussian processes (seismic exploration)

The deconvolution of pulse trains modeled as Bernoulli-Gaussian processes is addressed. The detector presented implements a stack Viterbi algorithm with a parametric level of suboptimality. Simulation results are satisfactory, even in the difficult case of a poor spectral content of the wavelet. The recursive and parallel structure of the method allows fast data processing on modern architectures. >

Novel estimator for the aberrations of a space telescope by phase diversity

In this communication, we propose a novel method for estimating the aberrations of a space telescope from phase diversity data. The images recorded by such a telescope can be degraded by optical aberrations due to design, fabrication or misalignments. Phase diversity is a technique that allows the estimation of aberrations. The only estimator found in the relevant literature is based on a joint estimation of the aberrated phase and the observed object. By means of simulations, we study the behavior of this estimator. We propose a novel marginal estimator of the sole phase by Maximum Likelihood. It is obtained by integrating the observed object out of the problem; indeed, this object is a nuisance parameter in our problem. This reduces drastically the number of unknown and provides better asymptotic properties. This estimator is implemented and its properties are validated by simulation. Its performance is equal or even better than that of the joint estimator for the same computing cost.