David Brie

Separation of Non-Negative Mixture of Non-Negative Sources Using a Bayesian Approach and MCMC Sampling

This paper addresses blind-source separation in the case where both the source signals and the mixing coefficients are non-negative. The problem is referred to as non-negative source separation and the main application concerns the analysis of spectrometric data sets. The separation is performed in a Bayesian framework by encoding non-negativity through the assignment of Gamma priors on the distributions of both the source signals and the mixing coefficients. A Markov chain Monte Carlo (MCMC) sampling procedure is proposed to simulate the resulting joint posterior density from which marginal posterior mean estimates of the source signals and mixing coefficients are obtained. Results obtained with synthetic and experimental spectra are used to discuss the problem of non-negative source separation and to illustrate the effectiveness of the proposed method

Automated Force Volume Image Processing for Biological Samples

Atomic force microscopy (AFM) has now become a powerful technique for investigating on a molecular level, surface forces, nanomechanical properties of deformable particles, biomolecular interactions, kinetics, and dynamic processes. This paper specifically focuses on the analysis of AFM force curves collected on biological systems, in particular, bacteria. The goal is to provide fully automated tools to achieve theoretical interpretation of force curves on the basis of adequate, available physical models. In this respect, we propose two algorithms, one for the processing of approach force curves and another for the quantitative analysis of retraction force curves. In the former, electrostatic interactions prior to contact between AFM probe and bacterium are accounted for and mechanical interactions operating after contact are described in terms of Hertz-Hooke formalism. Retraction force curves are analyzed on the basis of the Freely Jointed Chain model. For both algorithms, the quantitative reconstruction of force curves is based on the robust detection of critical points (jumps, changes of slope or changes of curvature) which mark the transitions between the various relevant interactions taking place between the AFM tip and the studied sample during approach and retraction. Once the key regions of separation distance and indentation are detected, the physical parameters describing the relevant interactions operating in these regions are extracted making use of regression procedure for fitting experiments to theory. The flexibility, accuracy and strength of the algorithms are illustrated with the processing of two force-volume images, which collect a large set of approach and retraction curves measured on a single biological surface. For each force-volume image, several maps are generated, representing the spatial distribution of the searched physical parameters as estimated for each pixel of the force-volume image.

From Bernoulli–Gaussian Deconvolution to Sparse Signal Restoration

Formulated as a least square problem under an l0 constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, matching pursuit (MP), orthogonal matching pursuit (OMP), orthogonal least squares (OLS), stepwise regression algorithms and the exhaustive search. We revisit the single most likely replacement (SMLR) algorithm, developed in the mid-1980s for Bernoulli-Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli-Gaussian signal restoration leads to an l0-penalized least square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called single best replacement (SBR), can be interpreted as a forward-backward extension of OLS sharing similarities with stepwise regression algorithms. Some structural properties of SBR are put forward. A fast and stable implementation is proposed. The approach is illustrated on two inverse problems involving highly correlated dictionaries. We show that SBR is very competitive with popular sparse algorithms in terms of tradeoff between accuracy and computation time.

A CANDECOMP/PARAFAC Perspective on Uniqueness of DOA Estimation Using a Vector Sensor Array

We address the uniqueness problem in estimating the directions-of-arrival (DOAs) of multiple narrowband and fully polarized signals impinging on a passive sensor array composed of identical vector sensors. The data recorded on such an array present the so-called “multiple invariances,” which can be linked to the CANDECOMP/PARAFAC (CP) model. CP refers to a family of low-rank decompositions of three-way or higher way (mutidimensional) data arrays, where each dimension is termed as a “mode.” A sufficient condition is derived for uniqueness of the CP decomposition of a three-way (three mode) array in the particular case where one of the three loading matrices, each associated to one mode, involved in the decomposition has full column rank. Based on this, upper bounds on the maximal number of identifiable DOAs are deduced for the two typical cases, i.e., the general case of uncorrelated or partially correlated sources and the case where the sources are coherent.

Non-negative source separation: range of admissible solutions and conditions for the uniqueness of the solution

A main issue in source separation is to deal with the indeterminacies. Well known are the ordering and scale ambiguities, but other types of indeterminacies may also occur. In this paper we address these indeterminacies in the case of non-negative sources and non-negative mixing coefficients. On the one hand, we fully develop the case of two sources. On the other hand, in the general case we formulate necessary conditions for the uniqueness of the solution (up to ordering and scale ambiguities).

Uni-mode and Partial Uniqueness Conditions for CANDECOMP/PARAFAC of Three-Way Arrays with Linearly Dependent Loadings

In this paper, three sufficient conditions are derived for the three-way CANDECOMP/PARAFAC (CP) model, which ensure uniqueness in one of the three modes (“uni-mode-uniqueness”). Based on these conditions, a partial uniqueness condition is proposed which allows collinear loadings in only one mode. We prove that if there is uniqueness in one mode, then the initial CP model can be uniquely decomposed in a sum of lower-rank tensors for which identifiability can be independently assessed. This condition is simpler and easier to check than other similar conditions existing in the specialized literature. These theoretical results are illustrated by numerical examples.

Regularization aspects in continuous-time model identification

This paper presents an analysis of some regularization aspects in continuous-time model identification. The study particulary focuses on linear filter methods and shows that filtering the data before estimating their derivatives corresponds to a regularized signal derivative estimation by minimizing a compound criterion whose expression is given explicitly. A new structure based on a null phase filter corresponding to a true regularization filter is proposed and allows to discuss the filter phase effects on parameter estimation by comparing its performances with those of the Poisson filter-based methods. Based on this analysis, a formulation of continuous-time model identification as a joint system input-output signal and model parameter estimation is suggested. In this framework, two linear filter methods are interpreted and a compound criterion is proposed in which the regularization is ensured by a model fitting measure, resulting in a new regularization filter structure for signal estimation.

Sparse multidimensional modal analysis using a multigrid dictionary refinement

We address the problem of multidimensional modal estimation using sparse estimation techniques coupled with an efficient multigrid approach. Modal dictionaries are obtained by discretizing modal functions (damped complex exponentials). To get a good resolution, it is necessary to choose a fine discretization grid resulting in intractable computational problems due to the huge size of the dictionaries. The idea behind the multigrid approach amounts to refine the dictionary over several levels of resolution. The algorithm starts from a coarse grid and adaptively improves the resolution in dependence of the active set provided by sparse approximation methods. The proposed method is quite general in the sense that it allows one to process in the same way mono-and multidimensional signals. We show through simulations that, as compared to high-resolution modal estimation methods, the proposed sparse modal method can greatly enhance the estimation accuracy for noisy signals and shows good robustness with respect to the choice of the number of components.

Approximate joint diagonalization by nonorthogonal nonparametric Jacobi transformations

We propose a novel algorithm for the problem of nonorthogonal joint diagonalization of a set of structured matrices based on successive Jacobi-like transformations. Though the elementary transformation matrices we use are not optimal in the sense of the global criterion, they are ensured to be nonsingular, and can be computed in closed form. The algorithm is efficient in virtue of its low computational complexity and fast convergence. The performance of the new algorithm is compared in simulations to the similar algorithms of the recent literature.

Simulation of postive normal variables using several proposal distributions

In this paper, we propose a new methodology to generate random variables distributed according to a Gaussian with positive support. We narrow the study to the univariate case. The method consists in an accept-reject algorithm in which a previous step is added consisting in choosing among several proposal distributions the one which gives the highest average probability of acceptance for given parameters of the target distribution. This results in a very fast method since it generates low reject