Jean-François Giovannelli

Regularization, maximum entropy and probabilistic methods in mass spectrometry data processing problems

Abstract 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 naive 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.

Image reconstruction in optical interferometry

Since the first multitelescope optical interferometer [1], considerable technological improvements have been achieved. Optical (visible/infrared) interferometers are now widely open to the astronomical community and provide the means to obtain unique information from observed objects at very high angular resolution (submilliarcsecond). There are numerous astrophysical applications, such as stellar surfaces, environment of premain sequence or evolved stars, and central regions of active galaxies. See [2]-[4] for comprehensive reviews about optical interferometry and recent astrophysical results. As interferometers do not directly provide images, reconstruction methods are needed to fully exploit these instruments. This article aims at reviewing image reconstruction algorithms in astronomical interferometry using a general framework to formally describe and compare the different methods. The challenging issues in image reconstruction from interferometric data are introduced in the general framework of inverse problem approach. This framework is then used to describe existing image reconstruction algorithms in radio interferometry and the new methods specifically developed for optical interferometry.

Positive deconvolution for superimposed extended source and point sources

The paper deals with the construction of images from visibilities acquired using aperture synthesis instruments: Fourier synthesis, deconvolution, and spectral interpolation/extrapolation. Its intended application is to specific situations in which the imaged object possesses two superimposed components: (i) an extended component together with (ii) a set of point sources. It is also specifically designed to the case of positive maps, and accounts for a known support. Its originality lies within joint estimation of the two components, coherently with data, properties of each component, positivity and possible sup- port. We approach the subject as an inverse problem within a regularization framework: a regularized least-squares criterion is specifically proposed and the estimated maps are defined as its minimizer. We have investigated several options for the numer- ical minimization and we propose a new efficient algorithm based on augmented Lagrangian. Evaluation is carried out using simulated and real data (from radio interferometry) demonstrating the capability to accurately separate the two components.

Bayesian estimation of regularization and point spread function parameters for Wiener–Hunt deconvolution

This paper tackles the problem of image deconvolution with joint estimation of point spread function (PSF) parameters and hyperparameters. Within a Bayesian framework, the solution is inferred via a global a posteriori law for unknown parameters and object. The estimate is chosen as the posterior mean, numerically calculated by means of a Monte Carlo Markov chain algorithm. The estimates are efficiently computed in the Fourier domain, and the effectiveness of the method is shown on simulated examples. Results show precise estimates for PSF parameters and hyperparameters as well as precise image estimates including restoration of high frequencies and spatial details, within a global and coherent approach.

Sampling High-Dimensional Gaussian Distributions for General Linear Inverse Problems

This paper is devoted to the problem of sampling Gaussian distributions in high dimension. Solutions exist for two specific structures of inverse covariance: sparse and circulant. The proposed algorithm is valid in a more general case especially as it emerges in linear inverse problems as well as in some hierarchical or latent Gaussian models. It relies on a perturbation-optimization principle: adequate stochastic perturbation of a criterion and optimization of the perturbed criterion. It is proved that the criterion optimizer is a sample of the target distribution. The main motivation is in inverse problems related to general (nonconvolutive) linear observation models and their solution in a Bayesian framework implemented through sampling algorithms when existing samplers are infeasible. It finds a direct application in myopic/unsupervised inversion methods as well as in some non-Gaussian inversion methods. An illustration focused on hyperparameter estimation for super-resolution method shows the interest and the feasibility of the proposed algorithm.

Regularized estimation of mixed spectra using a circular Gibbs-Markov model

Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such ill-posed problems. Following Sacchi et al. (see ibid., vol.46, no.1, p.32-38, 1998), we first address line spectra recovering in this framework. Then, we extend the methodology to situations of increasing difficulty: the case of smooth spectra and the case of mixed spectra, i.e., peaks embedded in smooth spectral contributions. The practical stake of the latter case is very high since it encompasses many problems of target detection and localization from remote sensing. The stress is put on adequate choices of penalty functions: following Sacchi et al., separable functions are retained to retrieve peaks, whereas Gibbs-Markov potential functions are introduced to encode spectral smoothness. Finally, mixed spectra are obtained from the conjunction of contributions, each one bringing its own penalty function. Spectral estimates are defined as minimizers of strictly convex criteria. In the cases of smooth and mixed spectra, we obtain nondifferentable criteria. We adopt a graduated nondifferentiability approach to compute an estimate. The performance of the proposed techniques is tested on the well-known Kay and Marple (1982) example.

A Bayesian method for long AR spectral estimation: a comparative study

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 parameters is addressed using both maximum likelihood (ML) and generalized cross-validation (GCV) criteria in order to automatically tune the spectral smoothness. The practical interest of the method is demonstrated by a computed simulation study in the field of Doppler spectral analysis. In a Monte Carlo simulation study with a known spectral shape, investigation of quantitative indexes such as bias and variance, but also quadratic, logarithmic, and Kullback distances shows interesting improvements with respect to the usual least squares method, whatever the window data length and the signal-to-noise ratio (SNR).

Unsupervised Bayesian Convex Deconvolution Based on a Field With an Explicit Partition Function

This paper proposes a non-Gaussian Markov field with a special feature: an explicit partition function. To the best of our knowledge, this is an original contribution. Moreover, the explicit expression of the partition function enables the development of an unsupervised edge-preserving convex deconvolution method. The method is fully Bayesian, and produces an estimate in the sense of the posterior mean, numerically calculated by means of a Monte-Carlo Markov chain technique. The approach is particularly effective and the computational practicability of the method is shown on a simple simulated example.

Data Inversion for Over-Resolved Spectral Imaging in Astronomy

We present an original method for reconstructing a 3-D object having two spatial dimensions and one spectral dimension from data provided by the infrared slit spectrograph on board the Spitzer Space Telescope. During acquisition, the light flux is deformed by a complex process comprising four main elements (the telescope aperture, the slit, the diffraction grating, and optical distortion) before it reaches the 2-D sensor. The originality of this work lies in the physical modeling, in integral form, of this process of data formation in continuous variables. The inversion is also approached with continuous variables in a semi-parametric format decomposing the object into a family of Gaussian functions. The estimate is built in a deterministic regularization framework as the minimizer of a quadratic criterion. These specificities give our method the power to over-resolve. Its performance is illustrated using real and simulated data. We also present a study of the resolution showing a 1.5-fold improvement relative to conventional methods.

Langevin and hessian with fisher approximation stochastic sampling for parameter estimation of structured covariance

We have studied two efficient sampling methods, Langevin and Hessian adapted Metropolis Hastings (MH), applied to a parameter estimation problem of the mathematical model (Lorentzian, Laplacian, Gaussian) that describes the Power Spectral Density (PSD) of a texture. The novelty brought by this paper consists in the exploration of textured images modeled by centered, stationary Gaussian fields using directional stochastic sampling methods. Our main contribution is the study of the behavior of the previously mentioned two samplers and the improvement of the Hessian MH method by using the Fisher information matrix instead of the Hessian to increase the stability of the algorithm and the computational speed. The directional methods yield superior performances as compared to the more popular Independent and standard Random Walk MH for the PSD described by the three models, but can easily be adapted to any target law respecting the differentiability constraint. The Fisher MH produces the best results as it combines the advantages of the Hessian, i.e., approaches the most probable regions of the target in a single iteration, and of the Langevin MH, as it requires only first order derivative computations.