Amrane Houacine a obtenu le Diplôme d'Études Supérieures de Physique, spécialité électronique, en 1982 à l'Université des Sciences et de la Technologie d'Alger, le Diplôme d'Études Approfondies en Automatique et Traitement du Signal en 1983, et le Doctorat en Sciences Informatiques en janvier 1987 à l'Université d'Orsay . Ses principaux domaines de recherche scientifique portent sur les algorithmes rapides, les problèmes de stabilité numérique, le filtrage et l'analyse spectrale adaptatifs.

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. Then we briefly present the main classical deterministic regularization methods, maximum entropy-based methods and the probabilistic Bayesian estimation framework for such problems. The main idea is to show how all these different frameworks converge to the optimization of a compound criterion with a data adequation part and an a priori part. We will however see that the Bayesian inference framework gives naturally more tools for inferring the uncertainty of the computed solutions, for the estimation of the hyperparameters or for handling the myopic or blind inversion problems. Finally, based on Bayesian inference, we present a few advanced methods particularly designed for some mass spectrometry data processing problems. Some simulation results illustrate mainly the effect of the prior laws or equivalently the regularization functionals on the results one can obtain in typical deconvolution or Fourier synthesis problems arising in different mass spectrometry technique.

Dans de nombreux domaines de la physique appliquee, tels que l’optique, le radar, la thermique, la spectroscopie, la geophysique, l’acoustique, la radioastronomie, le controle non destructif, le genie biomedical, l’instrumentation et l’imagerie en general, se pose le probleme de la determination de la distribution spatiale d’une grandeur scalaire ou vectorielle, souvent appelee l’ objet , a partir de mesures. Selon les cas, ces mesures de l’objet sont directes – on parle alors d’ image – ou indirectes – on parle alors de projection dans le cas de la tomographie, ou de visibilite en astronomie, par exemple. La resolution d’un tel probleme d’imagerie peut etre habituellement decomposee en trois etapes : un probleme direct ou, connaissant l’objet et le mecanisme d’observation, on etablit une description mathematique des donnees observees. Ce modele doit etre assez precis pour fournir une description correcte du phenomene physique d’observation, et assez simple cependant pour se preter a un traitement numerique ulterieur ; un probleme d’instrumentation ou l’on doit recueillir des donnees le plus informatives possible afin de resoudre le probleme d’imagerie dans les meilleures conditions ; un probleme inverse ou l’on doit calculer une image acceptable de l’objet a partir du modele et des donnees precedents. Une bonne estimation de l’objet necessite evidemment que ces trois sous- problemes soient etudies de maniere coordonnee. Or, la caracteristique commune de ces problemes de reconstruction ou de restauration d’image est qu’ils sont souvent mal-poses ou mal-conditionnes. Les problemes de plus haut niveau que l’on rencontre en vision par ordinateur, tels que la segmentation d’image, le traitement du flot optique, la reconstruction de formes a partir d’ombrages, sont aussi des problemes inverses et ils souffrent des memes difficultes. Il existe, schematiquement, deux grandes communautes scientifiques qui s’interessent a ces problemes inverses, d’un point de vue methodologique : celle de la physique mathematique , que l’on peut rattacher aux travaux fondateurs de Phillips, Twomey et Tikhonov dans les annees 1960, dont P.C. Sabatier fut un des pionniers en France (avec son action thematique programmee du meme nom), et dont une revue representative est « Inverse problems » ; celle du traitement statistique des donnees , que l’on peut rattacher aux travaux de Franklin a la fin des annees 1960, dont les freres Geman ont constitue les accelerateurs en traitement d’image, et dont une revue representative est « IEEE Transactions on Image Processing ». On peut dire, grossierement, que les uns abordent le probleme en dimension infinie, avec les questions d’existence, d’unicite et de stabilite qui deviennent tres compliquees avec des problemes directs non lineaires, et le resolvent numeriquement en dimension finie ; alors que les autres partent d’un probleme dont la discretisation est deja faite a la resolution souhaitee, et non remise en cause, et profitent du caractere fini du probleme pour introduire une information a priori elaboree au travers de modeles probabilistes. Nous nous proposons d’indiquer brievement dans la suite quels sont, de notre point de vue, l’etat de l’art et les questions ouvertes dans le domaine de la resolution des problemes inverses, en accordant une place importante aux approches probabilistes.

Many estimation problems in signal or image processing lead to a final step, being the minimization of a cost function. When this function is strictly convex, the usual stopping criterion is the nullity of the gradient: the solution is reached when the gradient vanishes. As this is numerically out of reach, the actual stopping test consists of comparing a norm of the gradient with a threshold. In doing so, the iterations may be stopped as the current point is still far away from the solution or as the solution has long been reached. Whatever the value of the threshold, one has no idea of the discrepancy between the current point and the effective solution. Taking advantage of the Fenchel dual formulation of the problem, here we propose a much more sensitive stopping test. This test overcomes the problems caused by the comparison of a norm of the gradient to a threshold. To illustrate our claim, the test is implemented on a signal processing simulation example.

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/spl isin/X, where X is a convex set. Although projective methods are well adapted to this context, we study alternative methods which rely highly on some "information-based" criteria. Our goal is to clarify the role played by entropy in this field, and to present a new point of view on entropy, using general tools and results coming from convex analysis. We present then a new and broad scheme for entropic-based inversion of linear-noisy inverse problems. This scheme was introduced by Navaza in 1985 in connection with a physical modeling for crystallographic applications, and further studied by Dacunha-Castelle and Gamboa (1990). Important features of this paper are: (i) a unified presentation of many well-known reconstruction criteria, (ii) proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics, (iii) a description of practical inversion of data using the aforementioned criteria, and (iv) a presentation of some reconstruction results.

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 to avoid too spiky spectra. The critical computation of the tradeoff parameters is addressed using both maximum likelihood and generalized cross validation criteria in order to automatically tune the smoothness constraint. The practical potential of the method is demonstrated using both simulated and in vitro signals. In a Monte-Carlo simulation study, investigation of quantitative indices such as quadratic distances shows interesting improvements with respect to the usual least squares method whatever the window data length and the signal to noise ratio. When applied to actual Doppler signals, the proposed method offers better description of the Doppler spectrum morphology than the usual least squares one.

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 smoothness prior. The critical computation of the tradeoff ...

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-development. This estimator can be fitted to the specific characteristics of the clutter rejection filter using the signal-to-noise ratio (SNR) of the Doppler signal as an adaptive parameter. Its performance is compared to that of the usual correlation angle estimator, and its thresholded version, as well as that of the general mean frequency estimator, using a model of Doppler signal. The detection of low frequencies was significantly improved. The mean square error (MSE) was reduced an average 15 fold over a 25-dB range on the SNR, compared to the correlation angle estimator (CAE) or the general mean frequency estimator. A two-fold reduction in the MSE was obtained compared to the thresholded correlation angle estimator.

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 measure μ on C with respect to the Kullback divergence, or cross-entropy. This is therefore called the Maximum Entropy on the Mean Method (MEMM). This problem is shown to be equivalent to the convex one x = arg min x F(x) submitted to y = Ax (in the noiseless case). Many classical criteria are found to be particular solutions with different reference measures μ. But except for some measures, these primal criteria have no explicit expression. Nevertheless, taking advantage of a dual formulation of the problem, the MEMM enables us to compute a solution in such cases. This indicates that such criteria could hardly have been derived without the MEMM. In order to integrate the presence of additive noise in the MEMM scheme, the object and noise are searched simultaneously for in an appropriate convex C′. The MEMM then gives a criterion of the form x = arg min x F(x) + G(y - Ax), where F and G are convex, without constraints. The functional G is related to the prior distribution of noise, and may be used to account for specific noise distributions. Using the regularity of the criterion, the sensitivity of the solution to variations of the data is also derived.

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-frequency estimators are derived for real-time applications. Finally, these two estimators are compared to the periodogram and the correlation angle estimator, respectively

We address the problem of tracking slowly varying mean frequency and spectral width from a series of data vectors yp (p=1, 2, ... P). When the yp are small-sized, usual methods (periodogram, correlation lags...) suffer from strong variability. Moreover, the discrete nature of observed signals results in ambiguity in spectral domain. First in a Gaussian setting both for the shape of the spectra and for the statistics of observed signals, a maximum likelihood (ML) technique is provided for spectral moments estimation. Second, we propose a maximum a posteriori (MAP) method in order to account also for the slowly varying character of the spectral moments. Noticeable improvement are obtained with respect to usual methods: strong variance reduction in the field of ultrasound attenuation measurement (AM), and correct estimation beyond the Shannon limit in the field of color flow mapping (CFM)

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 classical least squares (LS) method are outlined. The principles of the statistical study and computation results are presented.

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 possible solution to the problem involves the use of a Kalman filter. If all the processes are stationary and the measurement noise is white, the steady-state Kalman filter and the infinite impulse response (IIR) Wiener filter are identical. The recursiveness of the Kalman filter algorithm is very amenable to VLSI implementation. The aim is to discuss the problems inherent in the implementation of a Kalman filter structure on specialized VLSI devices such as discrete signal processors (DSP). To this end, the basic algorithm is split into elementary operations involving functional units with a high degree of internal parallelism such as a multiplier-accumulator unit. Due to the real-time processing constraint, special attention is paid to rounding effects, and a comparison is made between fixed point and floating point arithmetics

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.<>

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, and an adaptive criterion based on the Doppler signal variance is developed for color flow mapping applications. Its performance is compared to that of the usual correlation phase estimator on simulated Doppler signals and on synthetic Doppler images. An improvement in image quality is achieved, mainly for low signal-to-noise ratio Doppler signals.<>

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 and a comparison with the classical least squares (LS) method. This work is motivated by the excellent paper by Vaitkus et al (1988) whose purpose was to compare and assess classical spectral estimation methods - whether parametric or not - for biomedical applications in the field of Doppler ultrasound velocimetry