Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost. The connection between this algorithm and some existing strategies is discussed and its efficiency is illustrated on a linear inverse problem of image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio

How to Apply ICA on Actual Data? Example of Mars Hyperspectral Image Analysis

International audienceAs any estimation method, results provided by ICA are dependent of a model — usually a linear mixture and separation model — and of a criterion — usually independence. In many actual problems, the model is a coarse approximation of the system physics and independence can be more or less satisfied, and consequently results are not reliable. Moreover, with many actual data, there is a lack of reliable knowledge on the sources to be extracted, and the interpretation of the independent components (IC) must be done very carefully, using partial prior information and with interactive discussions with experts. In this talk, we explain how such a scientific method can take place on the example of analysis of Mars hyperspectral images

Source Separation in Chemical Analysis : Recent Achievements and Perspectives

International audienceSource separation is one of the most relevant estimation problems found in chemistry. Indeed, dealing with mixtures is paramount in different kinds of chemical analysis. For instance, there are some cases where the analyte is a chemical mixture of different components, e.g., in the analysis of rocks and heterogeneous materials through spectroscopy. Moreover, a mixing process can also take place even when the components are not chemically mixed. For instance, in ionic analysis of liquid samples, the ions are not chemically connected, but, due to the lack of selectivity of the chemical sensors, the acquired responses may be influenced by ions that are not the desired ones. Finally, there are some situations where the pure components cannot be isolated chemically since they appear only in the presence of other components. In this case, BSS may provide these components that cannot be retrieved otherwise. In this paper, our aim is to shed some light on the use of BSS in chemical analysis. In this context, we firstly provide a brief overview on source separation (Section II), with particular attention to the classes of linear and nonlinear mixing models (Sections III and IV, respectively). Then, (in Section V), we will give some conclusions and focus on challenging aspects that are found in chemical analysis. Although dealing with a relatively new field of applications, this article is not an exhaustive survey of source separation methods and algorithms, since there are solutions originated in closely related domains (e.g. remote sensing and hyperspectral imaging) that suit well several problems found in chemical analysis. Moreover, we do not discuss the supervised source separation methods, which are basically multivariate regression techniques, that one can find in chemometrics

Blind unmixing of linear mixtures using a hierarchical Bayesian model. Application to spectroscopic signal analysis

International audienceThis paper addresses the problem of spectral unmixing when positivity and additivity constraints are imposed on the mixing coefficients. A hierarchical Bayesian model is introduced to satisfy these two constraints. A Gibbs sampler is then proposed to generate samples distributed according to the posterior distribution of the unknown parameters associated to this Bayesian model. Simulation results conducted with synthetic data illustrate the performance of the proposed algorithm. The accuracy of this approach is also illustrated by unmixing spectra resulting from a multicomponent chemical mixture analysis by infrared spectroscopy

Algorithme primal-dual de points intérieurs pour l'estimation pénalisée des cartes d'abondances en imagerie hyperspectrale

National audienceL'estimation des cartes d'abondances en imagerie hyperspectrale nécessite de résoudre un problème d'optimisation sous des contraintes de positivité et d'additivité. Nous nous plaçons dans le cadre où les spectres des composants présents au sein de l'image ont été préalablement estimés par un algorithme d'extraction des pôles de mélange. Afin de réduire le temps de calcul, nous proposons un algorithme rapide de points intérieurs de type primal-dual pour l'estimation de ces cartes. En comparaison avec la méthode de référence FCLS, l'algorithme proposé présente l'avantage d'un coût de calcul réduit. Un second avantage est de pouvoir traiter le cas d'un critère pénalisé favorisant la régularité spatiale des cartes d'abondances. Des exemples sur des données synthétiques et réelles illustrent les performances de cet algorithme

Reconstruction d'un spectre RMN 2D par maximum d'entropie

International audienceLa résonance magnétique nucléaire (RMN) est une méthode moderne de spectroscopie utilisée pour l'analyse de la composition de produits biologiques. Nous nous intéressons dans cet article à l'estimation d'un spectre corrélation T1-T2 à partir de mesures RMN. Les difficultés de l'estimation sont liées au caractère mal-posé de ce problème inverse et à la taille importante des données à traiter. La méthode d'estimation est fondée sur le Maximum d'Entropie et nous proposons deux algorithmes de reconstruction itérative ; le premier est fondé sur l'algorithme de Bryan et Skilling et le second utilise le gradient conjugué non-linéaire. Par ailleurs, la structure du modèle d'observation est avantageusement exploitée pour alléger le coût de calcul sans employer les approximations proposées récemment par Vankataramanan et al. De plus, nous proposons le rajout d'une étape de recherche de pas adaptée à la fonction entropique afin d'assurer une décroissance du critère. Les algorithmes sont évalués sur un exemple synthétique et leur applicabilité est illustrée sur des données réelle

Primal-dual interior point optimization for a regularized reconstruction of NMR relaxation time distributions

International audienceThis paper deals with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) spectroscopy. This large scale and ill-posed inverse problem is solved by the iterative minimization of a regularized objective function allowing to encode some prior assumptions on the sought distribution. The numerical optimization of the criterion is performed using a primal-dual interior point algorithm allowing to handle the non-negativity constraint. The performances of the proposed approach are illustrated through the processing of real data from a two-dimensional NMR experiment

Spatially regularized multi-exponential transverse relaxation times estimation from magnitude MRI images under Rician noise

International audienceSynopsis This work aims at improving the estimation of multi-exponential transverse relaxation times from noisy magnitude MRI images. A spatially regularized Maximum-Likelihood estimator accounting for the Rician distribution of the noise was introduced. This approach is compared to a Rician corrected least-square criterion with the introduction of spatial regularization. To deal with the large-scale optimization problem, a majoration-minimization approach was used, allowing the implementation of both the maximum-likelihood estimator and the spatial regularization. The importance of the regularization alongside the rician noise incorporation is shown both visually and numerically on magnitude MRI images acquired on fruit samples. Purpose Multi-exponential relaxation times and their associated amplitudes in an MRI image provide very useful information for assessing the constituents of the imaged sample. Typical examples are the detection of water compartments of plant tissues and the quanti cation of myelin water fraction for multiple sclerosis disease diagnosis. The estimation of the multi-exponential signal model from magnitude MRI images faces the problem of a relatively low signal to noise ratio (SNR), with a Rician distributed noise and a large-scale optimization problem when dealing with the entire image. Actually, maps are composed of coherent regions with smooth variations between neighboring voxels. This study proposes an e cient reconstruction method of values and amplitudes from magnitude images by incorporating this information in order to reduce the noise e ect. The main feature of the method is to use a regularized maximum likelihood estimator derived from a Rician likelihood and a Majorization-Minimization approach coupled with the Levenberg-Marquardt algorithm to solve the large-scale optimization problem. Tests were conducted on apples and the numerical results are given to illustrate the relevance of this method and to discuss its performances. Methods For each voxel of the MRI image, the measured signal at echo time is represented by a multi-exponential model: with The data are subject to an additive Gaussian noise in the complex domain and therefore magnitude MRI data follows a Rician distribution : is the rst kind modi ed Bessel function of order 0 and is the standard deviation of the noise which is usually estimated from the image background. For an MRI image with voxels, the model parameters are usually estimated by minimizing the least-squares (LS) criterion under the assumption of a Gaussian noise using nonlinear LS solvers such as Levenberg-Marquardt (LM). However, this approach does not yield satisfying results when applied to magnitude data. Several solutions to overcome this issue are proposed by adding a correction term to the LS criterion. In this study, the retained correction uses the expectation value of data model under the hypothesis of Rician distribution since it outperforms the other correction strategies: stands for the sum of squares. We refer to this method as Rician corrected LS (RCLS). A more direct way for solving this estimation problem is to use a maximum likelihood (ML) estimator which comes down to minimize: To solve this optimization problem when dealing with the entire image, a majorization-minimization (MM) technique was adopted. The resulting MM-ML algorithm is summarized in gure 1, the LM algorithm used in this method minimizes a set of LS criteria derived from the quadratic majorization strategy. A spatial regularization term based on a cost function was also added to both criteria (and) to ensure spatial smoothness of the estimated maps. In order to reduce the numerical complexity by maintaining variable separability between each voxel and it's neighboring voxels , the function is majorized by : where stands for the iteration number of the iterative optimization algorithm

Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery

This paper studies a fully Bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery. Each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model. The estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical Bayesian model. This model assumes conjugate prior distributions for these parameters, accounts for non-negativity and full-additivity constraints, and exploits the fact that the endmember proportions lie on a lower dimensional simplex. A Gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution. This sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples. The accuracy of the joint Bayesian estimator is illustrated by simulations conducted on synthetic and real AVIRIS images