Sébastien Bourguignon

A Sparsity-Based Method for the Estimation of Spectral Lines From Irregularly Sampled Data

We address the problem of estimating spectral lines from irregularly sampled data within the framework of sparse representations. Spectral analysis is formulated as a linear inverse problem, which is solved by minimizing an l1-norm penalized cost function. This approach can be viewed as a basis pursuit de-noising (BPDN) problem using a dictionary of cisoids with high frequency resolution. In the studied case, however, usual BPDN characterizations of uniqueness and sparsity do not apply. This paper deals with the l1-norm penalization of complex-valued variables, that brings satisfactory prior modeling for the estimation of spectral lines. An analytical characterization of the minimizer of the criterion is given and geometrical properties are derived about the uniqueness and the sparsity of the solution. An efficient optimization strategy is proposed. Convergence properties of the iterative coordinate descent (ICD) and iterative reweighted least-squares (IRLS) algorithms are first examined. Then, both strategies are merged in a convergent procedure, that takes advantage of the specificities of ICD and IRLS, considerably improving the convergence speed. The computation of the resulting spectrum estimator can be implemented efficiently for any sampling scheme. Algorithm performance and estimation quality are illustrated throughout the paper using an artificial data set, typical of some astrophysical problems, where sampling irregularities are caused by day/night alternation. We show that accurate frequency location is achieved with high resolution. In particular, compared with sequential Matching Pursuit methods, the proposed approach is shown to achieve more robustness regarding sampling artifacts.

Bernoulli-Gaussian spectral analysis of unevenly spaced astrophysical data

We address the problem of line spectra detection and estimation from astrophysical data. As observations generally suffer sampling irregularities, false peaks may appear in the Fourier spectrum. We propose a linear spectral model with an arbitrarily large number of fixed frequencies and search for a sparse solution by modelling the spectrum as a Bernoulli-Gaussian process. The use of Markov chain Monte Carlo methods to compute the posterior mean estimate is discussed in the unsupervised framework. The original work by Cheng et al. (1996) is modified to account for specificities of the spectral analysis problem. Simulations reveal the efficiency of the method and its relevance to the astrophysical frequency detection context is emphasized. Finally, an application to astrophysical data is presented

Modeling the spatial PSF at the VLT focal plane for MUSE WFM data analysis purpose

MUSE is the Multi Unit Spectroscopic Explorer, an AO-assisted integral field spectrograph for visible and near-IR wavelengths which is planned to be commissioned at the UT4 of the Very Large Telescope in 2012.1 We present the status on the modeling of the spatial PSF at the UT focus and its Field-of-View (FoV) and spectral variations. Modeling these variations and studying their implications is a cornerstone for some MUSE data analysis and processing problems such as fusion, source extraction and deconvolution of MUSE datacubes. In Wide Field Mode (WFM, 1 square arc-minute FoV, 0.2 arcsec spatial sampling), MUSE can operate without Adaptive Optics (AO) correction or with a Ground Layer Adaptive Optics facility aimed at providing an almost uniform correction over a large field of view. In Narrow Field Mode (7.5 square arcseconds FoV, 0.025 arcsec spatial sampling) MUSE will make use of a Laser Tomography Adaptive Optics reconstruction, implying stronger spatial variations. By using the adaptive optics simulation tool PAOLA, we simulate in WFM the spatial PSF as a function of atmospheric turbulence parameters, observed wavelengths, AO mode and position in the field of view. We then develop a mathematical model fitting the generated data which allows, with a small number of parameters, to approximate the PSF at any spatial and spectral position of MUSE datacube. Finally, we evaluate the possibility to estimate the model parameters directly from the (future) MUSE data themselves.

Joint Reconstruction Strategy for Structured Illumination Microscopy With Unknown Illuminations

The blind structured illumination microscopy strategy proposed by Mudry et al. is fully re-founded in this paper, unveiling the central role of the sparsity of the illumination patterns in the mechanism that drives super-resolution in the method. A numerical analysis shows that the resolving power of the method can be further enhanced with optimized one-photon or two-photon speckle illuminations. A much improved numerical implementation is provided for the reconstruction problem under the image positivity constraint. This algorithm rests on a new preconditioned proximal iteration faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

Resolution enhancement of ultrasonic signals by up-sampled sparse deconvolution

This paper deals with the estimation of the arrival times of overlapping ultrasonic echoes. We focus on approaches based on discrete sparse deconvolution. Such methods are limited by the time resolution imposed by the model discretization, which is usually considered at the data sampling rate. In order to get closer to the continuous-time model, we propose to increase the time precision by introducing an up-sampling factor in the discrete model. The problem is then recast as a Multiple Input Single Output (MISO) deconvolution problem. Then, we propose to revisit standard sparse deconvolution algorithms for MISO systems. Specific and efficient algorithmic implementation is derived in such setting. Algorithms are evaluated on synthetic data, showing improvements in robustness toward discretization errors and competitive computational time compared to the standard approaches.

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.

Including frequency-dependent attenuation for the deconvolution of ultrasonic signals

Ultrasonic non-destructive testing (NDT) is a standard process for detecting flaws or discontinuities in industrial parts. A pulse is emitted by an ultrasonic transducer through a material, and a reflected wave is produced at each impedance change. In many cases, echoes can overlap in the received signal and deconvolution can be applied to perform echo separation and to enhance the resolution. Common deconvolution techniques assume that the shape of the echoes is invariant to the propagation distance. This can cause poor performances with materials such as plastics or composites, in particular because acoustic propagation suffers from frequency-dependent attenuation. In geophysics, biomedical imaging or NDT, various frequency-dependent attenuation models have been proposed under different formulations. This communication compares the related possible constructions in order to account for attenuation in deconvolution methods. Especially, we introduce a discrete model for the data, that includes an attenuat...

Fluorescence blind structured illumination microscopy: A new reconstruction strategy

In this communication, a fast reconstruction algorithm is proposed for fluorescence blind structured illumination microscopy (SIM) under the sample positivity constraint. This new algorithm is by far simpler and faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

Sampling schemes and parameter estimation for nonlinear Bernoulli-Gaussian sparse models

We address the sparse approximation problem in the case where the data are approximated by the linear combination of a small number of elementary signals, each of these signals depending non-linearly on additional parameters. Sparsity is explicitly expressed through a Bernoulli-Gaussian hierarchical model in a Bayesian framework. Posterior mean estimates are computed using Markov Chain Monte-Carlo algorithms. We generalize the partially marginalized Gibbs sampler proposed in the linear case in [1], and build an hybrid Hastings-within-Gibbs algorithm in order to account for the nonlinear parameters. All model parameters are then estimated in an unsupervised procedure. The resulting method is evaluated on a sparse spectral analysis problem. It is shown to converge more efficiently than the classical joint estimation procedure, with only a slight increase of the computational cost per iteration, consequently reducing the global cost of the estimation procedure.

Reconstruction of 3-D Microwave Images based on a Block-BiCGStab Algorithm

In 3-D microwave imaging, gradient-based optimization algorithms usually make use of the so-called stabilized version of the biconjugate gradient iterative method (BiCGStab) in order to solve multiple linear systems. We propose to use a block version of BiCGStab to jointly solve the mutiple right-hand side linear systems. Illuminations are partitioned in subgroups, which makes the method more efficient. The reconstruction process is studied on realistic simulated data and illustrates the efficiency of the method compared to BiCGStab.