Henri Lantéri

A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints

Abstract The aim of the present paper is to give a general method allowing us to devise maximum-likelihood multiplicative algorithms for inverse problems, and particularly for signal and image restoration with non-negativity constraint. We consider the case of a Gaussian additive noise and that of a Poisson process. The method is founded on the Kuhn–Tucker first-order optimality conditions and the algorithms are developed to satisfy these conditions. The proposed method can be used for any convex function whose definition range includes the domain of constraints. It allows to obtain generalized forms of classical algorithms (ISRA and RLA) and to unify the method for obtaining these algorithms. We give relaxed forms of the algorithms to increase the convergence speed; moreover, the effect of the constraints is clearly shown. For a better understanding of the method to take into account the constraints, we express the non-negativity constraint using different functions and we reach a large class of algorithms that can be analyzed as descent algorithms. Then, we can justify and analyze the behavior of several algorithms suggested in the literature. The particular displacement directions appearing in such algorithms are evidenced and the convergence speed is analyzed. The algorithms are applied for simulated data, to a two-dimensional deconvolution problem, to show their performance and effectiveness. A support constraint is taken into account implicitly in the algorithms. Our method can be extended to more general hard constraints on the extreme values or on the support of the solution and a regularization of the problem can be easily introduced in the method.

The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise

In 1993, Snyder et al investigated the maximum-likelihood (ML) approach to the deconvolution of images acquired by a charge-coupled-device camera and proved that the iterative method proposed by Llacer and Nunez in 1990 can be derived from the expectation-maximization method of Dempster et al for the solution of ML problems. The utility of the approach was shown on the reconstruction of images of the Hubble space Telescope. This problem deserves further investigation because it can be important in the deconvolution of images of faint objects provided by next-generation ground-based telescopes that will be characterized by large collecting areas and advanced adaptive optics. In this paper, we first prove the existence of solutions of the ML problem by investigating the properties of the negative log of the likelihood function. Next, we show that the iterative method proposed by the above-mentioned authors is a scaled gradient method for the constrained minimization of this function in the closed and convex cone of the non-negative vectors and that, if it is convergent, the limit is a solution of the constrained ML problem. Moreover, by looking for the asymptotic behavior in the regime of high numbers of photons, we find an approximation that, as proved by numerical experiments, works well for any number of photons, thus providing an efficient implementation of the algorithm. In the case of image deconvolution, we also extend the method to take into account boundary effects and multiple images of the same object. The approximation proposed in this paper is tested on a few numerical examples.

Reduction of boundary effects in multiple image deconvolution with an application to LBT LINC-NIRVANA

Our approach proposed in a previous paper for the reduction of boundary effects in the deconvolution of astronomical images by the Richardson-Lucy method (RLM) is extended here to the problem of multiple image deconvolution and applied to the reconstruction of the images of LINC-NIRVANA, the German-Italian beam combiner for the Large Binocular Telescope (LBT). We investigate the multiple image RLM, its accelerated version ordered subsets expectation maximization (OSEM), and the regularized versions of these two methods. In addition we show how the approach can be extended to the iterative space reconstruction algorithm (ISRA), which is an iterative method converging to non-negative least squares solutions. Numerical simulations indicate that the approach can provide excellent results with a considerable reduction of the boundary effects.

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.

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.

Deconvolution of multiple images with high dynamic range and an application to LBT LINC-NIRVANA

Context. The standard Richardson-Lucy method (RLM) does not work well in the deconvolution of astronomical images containing objects with very different angular scales and magnitudes. Therefore, modifications of RLM, applicable to this kind of objects, must be investigated. Aims. We recently proposed a regularization of RLM which provides satisfactory results in the case of particular test objects with high dynamic range. In this paper we extend this method to the case of multiple image deconvolution, having in mind application to the reconstruction of the images provided by Fizeau interferometers such as LINC-NIRVANA, the German-Italian beam combiner for the Large Binocular Telescope. Methods. RLM is an iterative method for the minimization of the Csiszar divergence, a problem equivalent to maximum likelihood estimation in the case of photon noise. In our approach, the problem is regularized by adding a suitable penalization term to the Csiszar divergence and an iterative method converging to the minimum of the resulting functional is derived from the so-called split gradient method (SGM). Results. The method is tested on a model of young binary star consisting of a core binary surrounded by a dusty circumbinary ring. The results are quite good in the case of exact knowledge of the point spread functions (PSF). However, in the case of approximate knowledge of the PSFs, the accuracy of the reconstruction depends on the difference of magnitude between the ring and the central binary.

Application of the Richardson-Lucy algorithm to the deconvolution of two-fold probability density functions

The authors use the Richardson-Lucy algorithm to deconvolve a set of images which are grey level representations of slices of two-fold probability density functions (PDF). These PDFs are computed from the one-dimensional signal obtained with the ESO slit-scanning infrared specklograph. In these conditions, it is shown that the PDF of the true signal is blurred by the PDF of the noise. The deconvolution is first performed on simulated data, for two levels of additive noise, i.e. for two different widths of the blurring function. These images are linked to one another, and they check the goodness of the deconvolution procedure by verifying that the properties of the image power spectrum (a quantity that can be derived from the whole set of PDF) are well conserved during the deconvolution. They discuss the quality of the result, which depends on the number of iterations. An application is made to real physical data.

Image quality and seeing measurements for long horizontal overwater propagation

The effect of atmospheric turbulence on the imaging of scenes, for horizontal propagation of the light over a distance of 20 km, 15 m above the sea surface, was analysed at visible wavelengths using a 20 cm telescope. Point-source images were recorded during the night, and the Fried parameter was derived using several methods, leading to values ranging from 2 to 4 cm depending on observing conditions. A very high level of scintillation was observed. Studies of correlations between close-by sources lead to a very small domain of isoplanatism. Daytime observations of an extended source are also performed; an image motion of small spatial coherent length seems to be drawn by a horizontal wind producing a wave-like distortion of horizontal lines and a boiling-like distortion of vertical ones.

First results of the PML monitor of atmospheric turbulence profile with high vertical resolution

Aims. Future extremely large telescopes will certainly be equipped with wide-field adaptive optics systems. The optimization of the performances of these techniques requires a precise specification of the different components of these AO systems. Most of these technical specifications are related to the atmospheric turbulence parameters, particularly the profile of the refractive index structure constantC 2 (h). A new monitor called Profiler of Moon Limb (PML) for the extraction of theC 2 N (h) profile with high vertical resolution and its first results are presented. Methods. The PML instrument uses an optical method based on the observation of the Moon limb through two subapertures. The use of the lunar limb leads to a continuum of double stars allowing a scan of the whole atmosphere with high resolution in altitude. Results. The first prototype of the PML has been installed at Dome C in Antarctica and the first results of the PML are presented and compared to radio-sounding balloon profiles. In addition to the C 2 (h) profile obtained with high vertical resolution, PML is also able to provide other atmospheric turbulence parameters such as the outer scale profile, the total seeing, and the isoplanatic and isopistonic angles.

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.