# G. Demoment's research while affiliated with Etablissement Français du Sang Alsace and other places

## Publications (92)

Chapter
Regularization Criterion descent methods Choice of regularization coefficient Bibliography
Article
Full-text available
This paper is a synthetic overview of regularization, maximum entropy and probabilistic methods for some inverse problems such as deconvolution and Fourier synthesis problems which arise in mass spectrometry. First we present a unified description of such problems and discuss the reasons why simple naı̈ve methods cannot give satisfactory results. T...
Article
Dans de nombreux domaines de la physique appliquée, nous sommes confrontés au problème de la détermination de la distribution temporelle, ou bien spatiale, ou bien encore fréquentielle, d'une grandeur scalaire ou vectorielle, à partir des mesures directes ou indirectes. La caractéristique commune de tels problèmes est qu'ils sont souvent, mal-posés...
Article
Entropy-based methods are widely used for solving inverse problems, particularly when the solution is known to be positive. Here, we address linear ill-posed and noisy inverse problems of the form z=Ax+n with a general convex constraint x∈X, where X is a convex set. Although projective methods are well adapted to this context, we study alternative...
Article
Le choix de l'ordre d'un modèle peut être fait en calculant un facteur de Bayes. Le critère de Schwarz en est une approximation utile, mais grossière. Un calcul plus précis peut être sensible au choix de l'à priori sur les paramètres des différents modèles en compétition. Un nouveau critère est proposé pour les problèmes de régression linéaire, fon...
Article
In this paper we address the problem of building convenient criteria to solve linear and noisy inverse problems of the form y = Ax+n. Our approach is based on the specification of constraints on the solution x through its belonging to a given convex set C. The solution is chosen as the mean of the distribution which is the closest to a reference me...
Article
In this paper we address the problem of building convenient criteria to solve linear and noisy inverse problems of the form y = Ax + n. Our approach is based on the specification of constraints on the solution x through its belonging to a given convex set C. The solution is chosen as the mean of the distribution which is the closest to a reference...
Article
Full-text available
This paper addresses the problem of ultrasound Doppler spectral estimation when only a short observation set is available. Following the work of Kitagawa and Gersch, the spectra are described by a long autoregressive model whose coefficients are estimated in a Bayesian regularized least squares framework accounting for spectral smoothness in order...
Article
An adaptive mean frequency estimator is proposed for color flow imaging. It is based on a series expansion of the first derivative of the autocorrelation function of the Doppler signal at origin. Its bias can be reduced by shifting the integration bounds in the series expansion and its variance adjusted by adapting the coefficients of the serial de...
Article
Full-text available
We address the problem of smooth power spectral density estimation of zero-mean stationary Gaussian processes when only a short observation set is available for analysis. The spectra are described by a long autoregressive model whose coefficients are estimated in a Bayesian regularized least squares (RLS) framework accounting the spectral smoothnes...
Conference Paper
Discusses the restoration of spiky sequences distorted by a linear system and corrupted by additive noise. A (now) classical way of coping with this problem is to use a Bayesian approach with a Bernoulli-Gaussian (BG) prior model of the sequence. The authors refine this method using a Bernoulli-Gaussian plus Gaussian (BCG) prior model. This estimat...
Article
L'objet de cette communication est la résolution de problèmes inverses linéaires mal-posés. Nous proposons de résoudre ce type de problème par la Méthode du Maximum d'Entropie sur la Moyenne (MMEM). La MMEM permet de prendre en compte diverses informations a priori, contrairement à la méthode du maximum d'entropie classique. Ces informations seront...
Conference Paper
An adaptive parametric method for spectral analysis as well as an adaptive mean frequency estimator are proposed to improve the quality of flow estimation in ultrasound Doppler velocimetry. The choice of the adaptive criterion is addressed to minimize the mean square error on estimation in noisy signals. Then, two suboptimal spectral and mean-frequ...
Conference Paper
We address the problem of tracking slowly varying mean frequency and spectral width from a series of data vectors y<sub>p</sub> (p=1, 2, ... P). When the y<sub>p</sub> are small-sized, usual methods (periodogram, correlation lags...) suffer from strong variability. Moreover, the discrete nature of observed signals results in ambiguity in spectral d...
Conference Paper
This paper investigates the French system of signal processing education. At first a general description of the French system of electrical engineering education is presented. Then the particular case of signal processing studies is described by using an example of one of the French universities
Conference Paper
In this paper, we address the problem of power spectral density estimation of stationary Gaussian processes with Auto-Regressive (AR) models when only a short set of data is available for analysis. The AR coefficients are estimated through a regularized method proposed by Kitagawa and Gersch (1984). We describe an experimental study of this method...
Article
A class of adapted mean frequency estimators is proposed for color flow mapping. These estimators can be fitted to the specific characteristics of a given Doppler signal to optimize the compromise between the range of analysable frequencies and the variance of mean frequency estimation. A sub-optimal estimator is derived for real-time applications,...
Conference Paper
Full-text available
In many measurement problems, it is found that the lack of resolution of the measuring device is the consequence of some blurring of the measured signal. Under linearity and shift-invariance assumptions, the signal restoration can be performed by a linear filtering of the data implementing some minimum mean square error (MSE) deconvolution. One pos...
Article
Many linear image restoration methods minimize a compound criterion which balances some fidelity to the observed data via a least-squares measure, and some fidelity to prior information on the unknown object via a smoothing function. In the case of quadratic criteria, this regularization scheme can be interpreted as a Bayesian estimation of an obje...
Conference Paper
Full-text available
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 comp...
Conference Paper
The authors address the problem of power spectral density estimation of time series with auto-regressive (AR) models when only a short span of data is available for analysis. The AR coefficients are estimated through a regularized method proposed by G. Kitagawa and W. Gersch (1985). An experimental study of this method and a comparison with the cla...
Book
The Twelfth International Workshop on Maximum Entropy and Bayesian Methods in Sciences and Engineering (MaxEnt 92) was held in Paris, France, at the Centre National de la Recherche Scientifique (CNRS), July 19-24, 1992. It is important to note that, since its creation in 1980 by some of the researchers of the physics department at the Wyoming Unive...
Article
Contents 1 Introduction 3 2 Bayesian theory, entropy, and the Gaussian prior 4 2.1 The Bayesian theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Entropy and maximum entropy : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 The Gaussian prior : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 Positivi...
Article
Full-text available
Dans cet article, nous nous intéressons au problème de l'estimation spectrale d'un signal gaussien stationnaire par modélisation auto-régressive, (AR) à partir d'un petit nombre d'échantillons. Les coefficients AR sont estimés par une méthode initialement introduite par Kitagawa & Gersch . Nous décrivons une étude en simulation de cette méthode...
Conference Paper
An adaptive mean frequency estimator is proposed which can be fitted to the specific characteristics of a given Doppler signal by trimming a single parameter. It allows to optimize the compromise between the bias and the variance of the mean frequency estimation. The performances of this estimator are shown superior to those of the usual correlatio...
Article
Full-text available
Developments in the theory of image reconstruction and restoration in the past ten or twenty years are outlined. Particular attention is paid to Bayesian inversion methods and to the potential they offer for describing local properties in images.
Article
Full-text available
Fast recursive least squares (FRLS) algorithms have been extensively studied since the mid- 1970s for adaptive signal processing applications. Despite their large number and apparent diversity, they were almost exclusively derived using only two techniques: partitioned matrix inversion lemma or least squares geometric theory. Surprisingly, Chandras...
Article
Full-text available
Several important problems in signal processing, such as linear prediction, linear regression, or spectrum factorization, need close-to-Toeplitz matrices to be factored. To solve these problems, several fast algorithms have been derived. They differ by the kind of adaptivity (block processing or exponential weighting of the data) and by the kind of...
Article
A new algorithm for aperture synthesis in radio-astronomy is presented. It is based on the principle of the maximum entropy on the mean. The procedure jointly performs estimation of the unknown phase aberrations which composes the Fourier data and image reconstruction. It partly derives from a preexisting imaging method developed in the field of cr...
Article
Two new methods for estimating frequency-dependent attenuation are proposed which improve the compromise between the estimation variance of this parameter and the analyzed tissue volume: 1) parametric spectral estimation of the demodulated signal, based on fast Kalman filtering (ARC), 2) implementation of a new mean frequency estimator derived from...
Article
The color Doppler estimator (CE1), which is calculated from the phase of the first correlation lag of the Doppler signal, is compared to the general mean frequency estimator (CEn), which is based on a weighted summation of all the available correlation lags, for long and short Doppler data sets (typically 48 and 8 Doppler samples). A new estimator...
Article
Time-domain identification of nonlinear systems represented by functional expansions is considered. A general framework is defined for the analysis of three identification methods: the widely used cross-correlation method, Korenberg's method, and a suboptimal least-squares method based on a stochastic approximation algorithm. First, the major chara...
Article
Full-text available
On présente dans cet article un algorithme de fusion de données qui construit une représentation d'une scène observée par deux capteurs identiques. L'algorithme, utilisé comme un algorithme de type Kalman, manipule récursivement une série d'objets et utilise des règles pour la remise à jour de cette représentation. Cet algorithme, basé sur la trian...
Article
Full-text available
Les algorithmes rapides de factorisation ďune matrice proche-de-Tœplitz possèdent une structure de calcul commune du type treillis vectoriel, quel que soit le mode de récurrence adopté. Les auteurs étudient dans cet article les problèmes que pose leur mise en œuvre en parallèle sur des processeurs de traitement du signal qui ne permettent pas de co...
Article
A new algorithm for aperture synthesis in radio-astronomy is presented. It is based on the principle of the maximum entropy on the mean. The procedure jointly performs estimation of the unknown phase aberrations which composes the Fourier data and image reconstruction. It partly derives from a preexisting imaging method developed in the field of cr...
Conference Paper
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 a...
Conference Paper
The restoration of a 2-D discrete object from its degraded image observed on a finite lattice is considered. The solution, interpreted as a Bayesian estimate of the original object modeled as a Gaussian random field, can be computed using Kalman filtering techniques. The problem of choosing the value of the smoothing parameter is addressed. Two met...
Conference Paper
Full-text available
A class of discrete image-reconstruction and restoration problems is addressed. A brief description is given of the maximum a posteriori (MAP) Bayesian approach with maximum entropy (ME) priors to solve the linear system of equations which is obtained after the discretization of the integral equations which arises in various tomographic image resto...
Article
Developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined. Particular attention is paid to common estimation structures and to practical problems not properly solved yet. The problem of image reconstruction and restoration is first formulated. Some of the current regularization approaches used to s...
Conference Paper
The extension to 2-D of three statistical methods successfully used in the 1-D problem has been studied, namely: (1) Lagrange multiplier techniques using properties of the residuals; (2) ordinary and generalized cross-validation techniques using prediction errors; and (3) maximum-likelihood estimation. Particular attention has been paid to implemen...
Article
The authors deal with the problem of deconvolution of Bernoulli-Gaussian random processes observed through linear systems. This corresponds to situations frequently encountered in areas such as geophysics, ultrasonic imaging, and nondestructive evaluation. Deconvolution of such signals is a detection-estimation problem that does not allow purely li...
Article
In this paper we propose a Bayesian approach with Maximum Entropy (ME) priors to solve an integral equation which arises in various image restoration and reconstruction problems. Our contributions in this paper are the following: i) We discuss the a priori probability distributions which are deduced from different a priori constraints when the prin...
Article
Full-text available
Le problème du calcul rapide des facteurs de Cholesky d'une matrice proche-de Toeplitz se rencontre fréquemment en traitement du signal. Sa résolution fait appel à des algorithmes de Levinson, de Cybenko, ou de Schur généralisés dont les propriétés sont maintenant bien connues. Nous proposons une approche différente, fondée sur l'emploi d'équations...
Conference Paper
A method is described for the numerical analysis of nonstationary signals whose spectral distributions vary slowly enough with time for an adaptive method to be used. The data is processed by blocks. The theory describes the signal by an autoregressive series of high order with stochastic coefficients having a Gaussian distribution. The coefficient...
Article
The authors propose a Bayesian approach with maximum-entropy (ME) priors to reconstruct an object from either the Fourier domain data (the Fourier transform of diffracted field measurements) in the case of diffraction tomography, or directly from the original projection data in the case of X-ray tomography. The objective function obtained is compos...
Chapter
A method for acquisition and processing of echographic B-scans was developed. It affords high definition images and a number of advantages with respect to 2D deconvolution. The process works as follows: a) acquisition of radio-frequency B-scans, b) line-by-line deconvolution of each scan in the axial direction, c) video detection, d) reconstruction...
Conference Paper
Adaptive spectrum estimation is based on a local stationarity assumption for the studied process, and uses methods of the stationary case with data windows of reduced length. But conventional least squares methods and parsimony principle (for example Akaïke's criterion) preclude use of long AR models necessary for a good spectral resolution. We dev...
Article
In diffraction tomography, the generalized Radon theorem relates the Fourier transform (FT) of the diffracted field to the two-dimensional FT of the diffracting object. The relationship stands on algebraic contours, which are semicircles in the case of Born or Rytov first-order linear approximations. But the corresponding data are not sufficient to...
Article
Full-text available
In diffraction tomography, the generalized Radon theorem relates the Fourier Transform (FT) of the diffracted field to the two-dimensional FT of the diffracting object. The relation stands on algebraic contours, which are semi-circles in the case of Born or Rytov first order linear approximations. To increase the insufficient resolution of generall...
Conference Paper
Adaptivity, stability, fast initial convergence, and low complexity are contradictory exigences in adaptive filtering. The least-mean-squares (LMS) algorithms suffer from a slow initial convergence, and the fast recursive least-squares (RLS) ones present numerical stability problems. In this paper we address this last-mentioned problem and perform...
Conference Paper
Restoration of an image distorted by a linear shift-invariant system is a 2-D deconvolution problem which is treated here in a Bayesian framework to stabilize the solution. The usual introduction of dynamics into the state equation to reduce the problem dimensions requires an artificial causality assumption. Thus we propose state-space models where...
Conference Paper
In diffraction tomography, the generalized Radon theorem relates the Fourier Transform (FT) of the diffracted field to the two-dimensional FT of the diffracting object. The relation stands on algebraic contours, which are semi-circles in the case of Born or Rytov first order linear approximations. We propose a Maximum Entropy method to reconstruct...
Article
Full-text available
We propose a statistical estimation approach to the deconvolution problem which is optimal in the minimum variance sense when the a priori knowledge on the signal to be restored is strictly limited to its first two moments. By viewing the estimation problem as a degenerate case of a Kalman filter applied to a static system with time-varying measure...
Article
On s'intéresse au calcul du vecteur gain de Kalman asymptotique dans un problème d'estimation avec un modèle invariant et bruits stationnaires. Les dimensions de l'état étant élevées, il faut trouver des méthodes rapides pour résoudre l'équation de Riccati algébrique. On compare deux méthodes possibles: une méthode de factorisation du type Chandras...
Article
Full-text available
Dans cette deuxième partie, nous analysons les méthodes de reconstruction tomographiques utilisées afin de mettre en évidence leurs hypothèses implicites et les limitations qu'elles entraînent. Il s'agit d'un problème inverse, celui de la résolution d'une équation intégrale de première espèce à partir de mesures discrètes, et limitées en nombre, du...
Chapter
Resolution of B-scan echographic images is basically limited in both axial and lateral directions by: (i) the dynamic characteristics and the radiation pattern of the transducer; (ii) the measurement noise. Many methods have been proposed to improve the resolution of these images: tracking focus, synthetic aperture, echotomography… But they all ign...
Article
De nombreux problèmes de restauration d'image se traduisent par une équation de convolution bi-dimensionnelle. Pour résoudre ce problème mal posé, on se place dans un cadre statistique bayésien pour prendre en compte simultanément le bruit sur les observations et l'information a priori nécessaire à la stabilisation de la solution. Le filtre de Kalm...
Chapter
Full-text available
In this communication we describe the results obtained with the solution we developed to improve the range resolution of existing echographic appliances. This solution consists in a fast minimum variance estimator implemented with a specialized signal processor that can be inserted in a conventional echographic system. This solution is well dedicat...
Article
Full-text available
Range resolution improvement in ultrasonic echography is considered as an estimation problem which is solved using a new fast minimum variance deconvolution algorithm specially designed for a microprocessor-based on-line processing. This method is used to accurately study the lenses and fundus of the eye and to follow variations of an arterial wall...
Conference Paper
Restoration of an image distorted by a linear spatially invariant system can be viewed as a 2-D deconvolution problem. The major difficulties lie in stabilizing the solution of such an ill-posed problem and in the computational burden inherent to the large amount of data involved in realistic image processing. The iterative and recursive Kaczmarz m...
Article
It is shown how a Kalman filter approach can solve an orthogonalisation problem that occurs when accelerating the convergence of the Kaczmarz algorithm for real-time deconvolution. A new suboptimal estimator is developed in the case of noise and input stationary statistics. Simulation results in reflection seismology are presented.
Article
Information loss due to strong filtering by the echograph is considered. Two deconvolution methods have been designed to correct partly the echograph characteristics. The first one, which is sub-optical Kalman filter, is easy to implement and others short calculation time. It is consequently well dedicated to quasi real time imaging. The second one...
Article
Linearity and stationarity of pressure-flow relationship in the circumflex coronary artery are fundamental properties which rule any mathematical description of this system. They are studied by evaluating through a quantitative criterion the error done in a linear approximation. Parameters of two different impulse response models are estimated, fro...
Article
A new least-mean-squares (LMS) adaptive algorithm is developed in the letter. This new algorithm solves a specific variance problem that occurs in LMS algorithms in the presence of high noise levels and when the input signal is bandlimited. Quantitative results in terms of an accuracy measure of a finite impulse response (FIR) system identification...
Article
Full-text available
La principale limitation des techniques traditionnelles d'analyse spectrale haute résolution réside dans leur faible robustesse vis-à-vis d'une méconnaissance du nombre de sources. Une alternative consiste à reformuler le problème en termes de détection et d'estimation conjointes dans le domaine de Fourier. La transformée de Fourier discrète du sig...
Article
Le problème posé est celui de l'amélioration de la résolution d'images obtenues par tomographie micro-ondes active. Ces images sont le résultat de la convolution bi-dimensionnelle des densités de courant complexes équivalentes dans l'objet étudié,par une fonction d'appareil. L'étalement du noyau est une des causes de la limitation de la résolution...
Article
Full-text available
Ce texte présente deux approches du problème de la déconvolution de signaux de type multi-impulsionnel ou Bemouilli-Gaussien, filtrés linéairement. La première approche suppose que le filtre est à réponse impulsionnelle finie (filtre MA) tandis que la seconde utilise un modèle de type autorégressif (filtre AR).
Article
On s'intéresse à la restauration d'images dégradées par un processus linéaire et considérées comme des fonctions de R dans C. Dans ce problème de déconvolution bi dimensionnelle, les difficultés proviennent de son caractère mal posé et du nombre important de données à manipuler. La méthode itérative et récursive de Kaczmarz pour inverser un système...
Article
On présente dans cette communication un algorithme rapide pour le calcul des m coefficients d'un filtre transverse adaptatif. La méthode proposée se distingue des solutions habituelles (moindres carrés récursifs rapides, LMS) par l'incorporation d'une information à priori permettant de stabiliser la solution de ce problème, tout en gardant un volum...
Article
Cet article présente une méthode itérative de déconvolution de processus Bernoulli-gaussiens par maximum a posteriori. Ce problème de détection-estimation est reformulé comme celui d'un changement de conditions initiales dans un problème de moindres carrés. Ceci conduit à un algorithme de structure très simple et qui permet le calcul exact du critè...
Article
Full-text available
Le problème traité est celui du lien entre les différentes approches (internes, externes, algébriques, géométriques,...) pour l'estimation de processus non-stationnaires par des algorithmes rapides. Ce lien entre les algorithmes du type Levinson ou Schur et ceux du type Chandrasekhar n'a été clairement établi que dans le cas où l'on dispose d'un mo...
Article
Cette première journée de rencontre entre des chercheurs qui relient l'art au traitement du signal et de l'image a été très apprécié de tous les participants, nous étions 31. Un CD a été édité, il est à la disposition de toute personne qui en fera la demande à Pierre Bonton.
Article
Nous présentons un algorithme pour la synthèse d'ouverture en radio-astronomie. Cet algorithme effectue la reconstruction d'une l'image à partir de données de Fourier, et corrige simultanément les données des aberrations de phases apparues lors de la mesure. Ces deux objectifs sont atteints à travers l'optimisation d'un même critère, contrairement...
Article
De nombreuses méthodes ont été proposées pour la restauration d'un signal dégradé par un système linéaire, à partir de la mesure de sa sortie bruitée, de la connaissance du système, et d'une information a priori sur les propriétés respectives du signal et du bruit. Mais ces méthodes nécessitent le plus souvent d'importants calculs en temps différé....
Article
Full-text available
Une extension stochastique du problème de la déconvolution est faite afin de résoudre une difficulté d'orthogonalisation rencontrée quand on accélère la convergence de la méthode de Kaczmarz pour effectuer une déconvolution en temps réel. La formulation choisie s'applique aux problèmes mono ou bi-dimensionnels et un algorithme rapide est utilisé po...

## Citations

... La théorie de l'inversion généralisée (dont fait partie la célèbre solution des Moindres Carrés) offre une alternative permettant de bien-poser le problème [DEMOMENT et al. (2001)]. Le mau- vais conditionnement du problème peut être réglé par des méthodes de régularisation. ...
... In this method, a series of functions G N (x) needs to be established that satisfies both of these two conditions: it minimizes the entropy, S, of the system and converges to a real distribution as the number of the distribution moments, N, approaches infinity.  (1) (2) To determine the complete orientation picture of interfacial proteins, we adopt the same methodology and solve for the average cosines using Equation 3, where λ i is calculated from the experimental results of the average cosines. (3) Due to the limited number of independent measurements, the series needs to be truncated. ...
... Regularization theory and the Bayesian inversion have been successfully used for this task. See for example [33,35,36,48] for quadratic and Tikhonov regularization, [27,42,47] for Total variation, 34] for different entropy based regularization, [32,40] for L p and sparsity enforcing, 43] for blind deconvolution and applications, [31,41] for Cross Validation (CV) and generalized CV methods for determining the regularization parameter, and [26,54] for Bernoulli-Gaussian models,  for Compress Sensing approach, [45,46] for multichannel blind deconvolution, [49,50] for nonlinear and space variant PSF, [51,52] for document image restoration,  for joint restoration and segmentation. ...
... Le cas de la représentation MA est traitée dans [GD89] et est affinée dans [IG90] avec une résolution sous-optimale plus rapide avec un algorithme de Viterbi. La généralisation aux modèles ARMA est donnée dans [GD87]. Une autre estimation sous optimale du problème MAP est possible à l'aide d'une méthode comportant une fenêtre glissante [Kaa98]. ...
... For example, when our prior knowledge are the moments of the image to be restored, application of maximum entropy principle leads Djafari & Demoment  to exact determination of the prior, including its parameters. Knowledge of the bounds (a gabarit) and the choice of a reference measure leads LeBesnerais [5,6] to the construction of a model accounting for human shaped prior in the context of astronomic deconvolution. ...
... The strength of this approach derives from the fact that it avoids the resolution of the standard Riccati equation of the Kalman filter. The derivation of fast adaptive algorithms based on Chandrasekhar fast equations using a state space model was presented in  and  for MA and ARMA linear filtering, respectively. It has been extended to the multichannel and non-linear filtering in . ...
... Estimating hyperparameters within the regularization framework is generally a delicate problem. It has been extensively studied, several techniques have been proposed and compared  and the preferred strategy is founded on ML. ...
... C'est le cas en particulier des critères contenant un terme de barrière, c'est à dire un terme à gradient non borné. Cette singularité se rencontre fréquemment dans le contexte du traitement du signal par exemple dans les critères de maximum d'entropie (Mohammad-Djafari et al., 1988), de vraisemblance Poissonienne (Bertero et al., 2009 ) et est également utilisée comme outil d'optimisation sous contrainte, par exemple dans les algorithmes de points intérieurs (Fiacco et al., 1967). Lorsqu'un algorithme de descente itérative est utilisé pour la minimisation d'un tel critère, la présence de la barrière rend inefficace les procédures standards de recherche de pas (Murray et al., 1994; Wright, 1995). ...
... This approach is based on a maximum a posteriori (MAP) method to linearize image reconstruction. A complete mathematical description of the MAP approach can be found in . This method can be seen as a simplified version of the iterative GN method, where only the first step of the nonlinear method is calculated. ...
... Regularization theory and the Bayesian inversion have been successfully used for this task. See for example [33,35,36,48] for quadratic and Tikhonov regularization, [27,42,47] for Total variation, 34] for different entropy based regularization, [32,40] for L p and sparsity enforcing, 43] for blind deconvolution and applications, [31,41] for Cross Validation (CV) and generalized CV methods for determining the regularization parameter, and [26,54] for Bernoulli-Gaussian models,  for Compress Sensing approach, [45,46] for multichannel blind deconvolution, [49,50] for nonlinear and space variant PSF, [51,52] for document image restoration,  for joint restoration and segmentation. ...