Céline Theys

Restoration of astrophysical images: the case of poisson data with additive Gaussian noise

We consider the problem of restoring astronomical images acquired with charge coupled device cameras. The astronomical object is first blurred by the point spread function of the instrument-atmosphere set. The resulting convolved image is corrupted by a Poissonian noise due to low light intensity, then, a Gaussian white noise is added during the electronic read-out operation. We show first that the split gradient method (SGM) previously proposed can be used to obtain maximum likelihood (ML) iterative algorithms adapted in such noise combinations. However, when ML algorithms are used for image restoration, whatever the noise process is, instabilities due to noise amplification appear when the iteration number increases. To avoid this drawback and to obtain physically meaningful solutions, we introduce various classical penalization-regularization terms to impose a smoothness property on the solution. We show that the SGM can be extended to such penalized ML objective functions, allowing us to obtain new algorithms leading to maximum a posteriori stable solutions. The proposed algorithms are checked on typical astronomical images and the choice of the penalty function is discussed following the kind of object.

Polynomial-phase signal analysis using stationary moments

Abstract This article proposes a method for polynomial-phase signals analysis relying on their time-invariant high-order moments. The set of these moments is first characterized in the noiseless and noisy case. It is demonstrated that the only identifiable phase parameter is the highest-order coefficient, the estimation requiring moments of order at least the double of the phase degree. Next, the consistency of the estimated moments calculated by sample averages is proved. Finally, an estimation algorithm based on the moments having for lags the multiples of a given ‘root’ is developed. Performances, evaluated by computer simulations, show the influence of the different parameters and the efficiency of the method.

Hyperspectral image unmixing using manifold learning methods derivations and comparative tests

In hyperspectral image analysis, pixels are mixtures of spectral components associated to pure materials. Although the linear mixture model is the mostly studied case, nonlinear techniques have been proposed to overcome its limitations. In this paper, a manifold learning approach is used as a dimensionality-reduction step to deal with non-linearities beforehand, or is integrated directly in the endmember extraction and abundance estimation steps using geodesic distances. Simulation results show that these methods improve the precision of estimation in severely nonlinear cases.

Linear unmixing of hyperspectral images using a scaled gradient method

This paper addresses the problem of linear unmixing for hyperspectral imagery. This problem can be formulated as a linear regression problem whose regression coefficients (abundances) satisfy sum-toone and positivity constraints. Two scaled gradient iterative methods are proposed for estimating the abundances of the linear mixing model. The first method is obtained by including a normalization step in the scaled gradient method. The second method inspired by the fully constrained least squares algorithm includes the sum-to-one constraint in the observation model with an appropriate weighting parameter. Simulations on synthetic data illustrate the performance of these algorithms.

Accuracy and performance of linear unmixing techniques for detecting minerals on OMEGA/Mars express

Detecting minerals on a huge hyperspectral dataset (> To) is a difficult task that we proposed to address using linear unmixing techniques. We test different algorithms with positivity constrains on a typical Martian hyperspectral image of the Syrtis Major volcanic complex. The usefulness of additional constrains, such as sparsity and sum-to-one constrains are discussed. We compare the results with a supervised detection technique based on band ratio.

Regularized split gradient method for nonnegative matrix factorization

This article deals with a regularized version of the split gradient method (SGM), leading to multiplicative algorithms. The proposed algorithm is available for the optimization of any divergence depending on two data fields under positivity constraint. The SGM-based algorithm is derived to solve the nonnegative matrix factorization (NMF) problem. An example with a Frobenius norm on both the data consistency and the penalty term is developped and applied to hyperspectral data unmixing.

Marginal Bayesian analysis of polynomial-phase signals

Abstract The aim of this paper is the estimation of the parameters of a noisy polynomial phase signal with deterministic time-varying amplitude. This problem has been recently addressed using polynomial phase transformations which give satisfactory results for low degree signals. Nevertheless, they do not answer the polynomial-order estimation problem, which is usually assumed to be known. In order to achieve a simultaneous estimation of the coefficients and the order, a Bayesian approach is proposed. Markov chain Monte Carlo (MCMC) algorithms are used to simulate posterior distributions. Estimation of the order implying a change in dimensionality, a reversible jump MCMC sampler is built in association with the “one variable at a time” Metropolis Hastings algorithm. The polynomial phase order, coefficients and the amplitude parameter are then computed from the simulated Markov chains. In order to evaluate the algorithm performances, “marginal” bounds are derived and compared to the variance of the estimated values.

A gradient based method for fully constrained least-squares unmixing of hyperspectral images

Linear unmixing of hyperspectral images is a popular approach to determine and quantify materials in sensed images. The linear unmixing problem is challenging because the abundances of materials to estimate have to satisfy non-negativity and full-additivity constraints. In this paper, we investigate an iterative algorithm that integrates these two requirements into the coefficient update process. The constraints are satisfied at each iteration without using any extra operations such as projections. Moreover, the mean transient behavior of the weights is analyzed analytically, which has never been seen for other algorithms in hyperspectral image unmixing. Simulation results illustrate the effectiveness of the proposed algorithm and the accuracy of the model.

Bayesian analysis for the fault detection of three-phase induction machine

One of the most widely used techniques for obtaining information on the state of health of three-phase induction machines is based on the processing of stator current. In fact, in the case of steady state operations, anomalous current spectral components, that increase if a fault occurs, allow diagnosis of the presence and, in some case, the type of fault. In this paper, a Bayesian approach is proposed using a simulation technique, the Markov chain Monte Carlo (MCMC), to estimate the amplitude of some spectral components modified by machine faults and the slip, a parameter related to the load conditions, with a view to automatically detecting faults. Results on real stator current waveform are given.

Reconstructing Images in Astrophysics, an Inverse Problem Point of View

After a short introduction, a first section provides a brief tutorial to the physics of image formation and its detection in the presence of noises. The rest of the chapter focuses on the resolution of the inverse problem . In the general form, the observed image is given by a Fredholm integral containing the object and the response of the instrument. Its inversion is formulated using a linear algebra. The discretized object and image of size N × N are stored in vectors x and y of length N2. They are related one another by the linear relation y = Hx, where H is a matrix of size N2 × N2 that contains the elements of the instrument response. This matrix presents particular properties for a shift invariant point spread function for which the Fredholm integral is reduced to a convolution relation. The presence of noise complicates the resolution of the problem. It is shown that minimum variance unbiased solutions fail to give good results because H is badly conditioned, leading to the need of a regularized solution. Relative strength of regularization versus fidelity to the data is discussed and briefly illustrated on an example using L-curves. The origins and construction of iterative algorithms are explained, and illustrations are given for the algorithms ISRA , for a Gaussian additive noise, and Richardson–Lucy , for a pure photodetected image (Poisson statistics). In this latter case, the way the algorithm modifies the spatial frequencies of the reconstructed image is illustrated for a diluted array of apertures in space. Throughout the chapter, the inverse problem is formulated in matrix form for the general case of the Fredholm integral, while numerical illustrations are limited to the deconvolution case, allowing the use of discrete Fourier transforms, because of computer limitations.