arXiv subjects

Jean-Luc Starck

Publications and source records attributed to Jean-Luc Starck.

At least 91 records · Page 5Linked to original sources

arXiv2013

GLIMPSE: Accurate 3D weak lensing reconstructions using sparsity

We present GLIMPSE - Gravitational Lensing Inversion and MaPping with Sparse Estimators - a new algorithm to generate density reconstructions in three dimensions from photometric weak lensing measurements. This is an extension of earlier work in one dimension aimed at applying compressive sensing theory to the inversion of gravitational lensing measurements to recover 3D density maps. Using the assumption that the density can be represented sparsely in our chosen basis - 2D transverse wavelets and 1D line of sight dirac functions - we show that clusters of galaxies can be identified and accurately localised and characterised using this method. Throughout, we use simulated data consistent with the quality currently attainable in large surveys. We present a thorough statistical analysis of the errors and biases in both the redshifts of detected structures and their amplitudes. The GLIMPSE method is able to produce reconstructions at significantly higher resolution than the input data; in this paper we show reconstructions with 6x finer redshift resolution than the shear data. Considering cluster simulations with 0.05 <= z <= 0.75 and 3e13/h Msun <= Mvir <= 1e15/h Msun, we show that the redshift extent of detected peaks is typically 1-2 pixels, or Dz <~ 0.07, and that we are able to recover an unbiased estimator of the redshift of a detected cluster by considering many realisations of the noise. We also recover an accurate estimator of the mass, that is largely unbiased when the redshift is known, and whose bias is constrained to <~ 5% in the majority of our simulations when the estimated redshift is taken to be the true redshift. This shows a substantial improvement over earlier 3D inversion methods, which showed redshift smearing with a typical standard deviation of 0.2-0.3, a significant damping of the amplitude of the peaks detected, and a bias in the detected redshift.

astro-ph.CO↗

arXiv2013

Astronomical Image Denoising Using Dictionary Learning

Astronomical images suffer a constant presence of multiple defects that are consequences of the intrinsic properties of the acquisition equipments, and atmospheric conditions. One of the most frequent defects in astronomical imaging is the presence of additive noise which makes a denoising step mandatory before processing data. During the last decade, a particular modeling scheme, based on sparse representations, has drawn the attention of an ever growing community of researchers. Sparse representations offer a promising framework to many image and signal processing tasks, especially denoising and restoration applications. At first, the harmonics, wavelets, and similar bases and overcomplete representations have been considered as candidate domains to seek the sparsest representation. A new generation of algorithms, based on data-driven dictionaries, evolved rapidly and compete now with the off-the-shelf fixed dictionaries. While designing a dictionary beforehand leans on a guess of the most appropriate representative elementary forms and functions, the dictionary learning framework offers to construct the dictionary upon the data themselves, which provides us with a more flexible setup to sparse modeling and allows to build more sophisticated dictionaries. In this paper, we introduce the Centered Dictionary Learning (CDL) method and we study its performances for astronomical image denoising. We show how CDL outperforms wavelet or classic dictionary learning denoising techniques on astronomical images, and we give a comparison of the effect of these different algorithms on the photometry of the denoised images.

astro-ph.IM↗

arXiv2012

Multichannel Poisson denoising and deconvolution on the sphere : Application to the Fermi Gamma Ray Space Telescope

A multiscale representation-based denoising method for spherical data contaminated with Poisson noise, the multiscale variance stabilizing transform on the sphere (MS-VSTS), has been previously proposed. This paper first extends this MS-VSTS to spherical two and one dimensions data (2D-1D), where the two first dimensions are longitude and latitude, and the third dimension is a meaningful physical index such as energy or time. We then introduce a novel multichannel deconvolution built upon the 2D-1D MS-VSTS, which allows us to get rid of both the noise and the blur introduced by the point spread function (PSF) in each energy (or time) band. The method is applied to simulated data from the Large Area Telescope (LAT), the main instrument of the Fermi Gamma-Ray Space Telescope, which detects high energy gamma-rays in a very wide energy range (from 20 MeV to more than 300 GeV), and whose PSF is strongly energy-dependent (from about 3.5° at 100 MeV to less than 0.1° at 10 GeV).

astro-ph.IM↗

arXiv2012

Fast Calculation of the Weak Lensing Aperture Mass Statistic

The aperture mass statistic is a common tool used in weak lensing studies. By convolving lensing maps with a filter function of a specific scale, chosen to be larger than the scale on which the noise is dominant, the lensing signal may be boosted with respect to the noise. This allows for detection of structures at increased fidelity. Furthermore, higher-order statistics of the aperture mass (such as its skewness or kurtosis), or counting of the peaks seen in the resulting aperture mass maps, provide a convenient and effective method to constrain the cosmological parameters. In this paper, we more fully explore the formalism underlying the aperture mass statistic. We demonstrate that the aperture mass statistic is formally identical to a wavelet transform at a specific scale. Further, we show that the filter functions most frequently used in aperture mass studies are not ideal, being non-local in both real and Fourier space. In contrast, the wavelet formalism offers a number of wavelet functions that are localized both in real and Fourier space, yet similar to the 'optimal' aperture mass filters commonly adopted. Additionally, for a number of wavelet functions, such as the starlet wavelet, very fast algorithms exist to compute the wavelet transform. This offers significant advantages over the usual aperture mass algorithm when it comes to image processing time, demonstrating speed-up factors of ~ 5 - 1200 for aperture radii in the range 2 to 64 pixels on an image of 1024 x 1024 pixels.

astro-ph.CO↗

arXiv2012

Cosmological constraints from the capture of non-Gaussianity in Weak Lensing data

Weak gravitational lensing has become a common tool to constrain the cosmological model. The majority of the methods to derive constraints on cosmological parameters use second-order statistics of the cosmic shear. Despite their success, second-order statistics are not optimal and degeneracies between some parameters remain. Tighter constraints can be obtained if second-order statistics are combined with a statistic that is efficient to capture non-Gaussianity. In this paper, we search for such a statistical tool and we show that there is additional information to be extracted from statistical analysis of the convergence maps beyond what can be obtained from statistical analysis of the shear field. For this purpose, we have carried out a large number of cosmological simulations along the σ8-Ωm degeneracy, and we have considered three different statistics commonly used for non-Gaussian features characterization: skewness, kurtosis and peak count. To be able to investigate non-Gaussianity directly in the shear field we have used the aperture mass definition of these three statistics for different scales. Then, the results have been compared with the results obtained with the same statistics estimated in the convergence maps at the same scales. First, we show that shear statistics give similar constraints to those given by convergence statistics, if the same scale is considered. In addition, we find that the peak count statistic is the best to capture non-Gaussianities in the weak lensing field and to break the σ8-Ωm degeneracy. We show that this statistical analysis should be conducted in the convergence maps: first, because there exist fast algorithms to compute the convergence map for different scales, and secondly because it offers the opportunity to denoise the reconstructed convergence map, which improves non-Gaussian features extraction.

astro-ph.CO↗

arXiv2011

A Compressed Sensing Approach to 3D Weak Lensing

(Abridged) Weak gravitational lensing is an ideal probe of the dark universe. In recent years, several linear methods have been developed to reconstruct the density distribution in the Universe in three dimensions, making use of photometric redshift information to determine the radial distribution of lensed sources. In this paper, we aim to address three key issues seen in these methods; namely, the bias in the redshifts of detected objects, the line of sight smearing seen in reconstructions, and the damping of the amplitude of the reconstruction relative to the underlying density. We consider the problem under the framework of compressed sensing (CS). Under the assumption that the data are sparse in an appropriate dictionary, we construct a robust estimator and employ state-of-the-art convex optimisation methods to reconstruct the density contrast. For simplicity in implementation, and as a proof of concept of our method, we reduce the problem to one-dimension, considering the reconstruction along each line of sight independently. Despite the loss of information this implies, we demonstrate that our method is able to accurately reproduce cluster haloes up to a redshift of z=1, deeper than state-of-the-art linear methods. We directly compare our method with these linear methods, and demonstrate minimal radial smearing and redshift bias in our reconstructions, as well as a reduced damping of the reconstruction amplitude as compared to the linear methods. In addition, the CS framework allows us to consider an underdetermined inverse problem, thereby allowing us to reconstruct the density contrast at finer resolution than the input data.

astro-ph.CO↗

arXiv2011

Uncertainty in 2-point correlation function estimators and BAO detection in SDSS DR7

We study the uncertainty in different two-point correlation function (2PCF) estimators in currently available galaxy surveys. This is motivated by the active subject of using the baryon acoustic oscillations (BAOs) feature in the correlation function as a tool to constrain cosmological parameters, which requires a fine analysis of the statistical significance. We discuss how estimators are affected by both the uncertainty in the mean density $\bar{n}$ and the integral constraint $\frac{1}{V^2}\int_{V^2} \hatξ (r) d^3r =0$ which necessarily causes a bias. We quantify both effects for currently available galaxy samples using simulated mock catalogues of the Sloan Digital Sky Survey (SDSS) following a lognormal model, with a Lambda-Cold Dark Matter ($Λ\text{CDM}$) correlation function and similar properties as the samples (number density, mean redshift for the $Λ\text{CDM}$ correlation function, survey geometry, mass-luminosity bias). Because we need extensive simulations to quantify small statistical effects, we cannot use realistic N-body simulations and some physical effects are neglected. Our simulations still enable a comparison of the different estimators by looking at their biases and variances. We also test the reliability of the BAO detection in the SDSS samples and study the compatibility of the data results with our $Λ\text{CDM}$ simulations.

astro-ph.CO↗

arXiv2011

Linear inverse problems with noise: primal and primal-dual splitting

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian, Poisson). On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution, inpainting and denoising with some comparison to prior methods are also reported.

stat.AP↗

arXiv2011

Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by Poisson noise. A proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution and comparison to prior methods are also reported.

stat.AP↗

arXiv2011

Data augmentation for galaxy density map reconstruction

The matter density is an important knowledge for today cosmology as many phenomena are linked to matter fluctuations. However, this density is not directly available, but estimated through lensing maps or galaxy surveys. In this article, we focus on galaxy surveys which are incomplete and noisy observations of the galaxy density. Incomplete, as part of the sky is unobserved or unreliable. Noisy as they are count maps degraded by Poisson noise. Using a data augmentation method, we propose a two-step method for recovering the density map, one step for inferring missing data and one for estimating of the density. The results show that the missing areas are efficiently inferred and the statistical properties of the maps are very well preserved.

stat.AP↗

arXiv2011

Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets.

stat.AP↗

arXiv2010

Feasibility and performances of compressed-sensing and sparse map-making with Herschel/PACS data

The Herschel Space Observatory of ESA was launched in May 2009 and is in operation since. From its distant orbit around L2 it needs to transmit a huge quantity of information through a very limited bandwidth. This is especially true for the PACS imaging camera which needs to compress its data far more than what can be achieved with lossless compression. This is currently solved by including lossy averaging and rounding steps on board. Recently, a new theory called compressed-sensing emerged from the statistics community. This theory makes use of the sparsity of natural (or astrophysical) images to optimize the acquisition scheme of the data needed to estimate those images. Thus, it can lead to high compression factors. A previous article by Bobin et al. (2008) showed how the new theory could be applied to simulated Herschel/PACS data to solve the compression requirement of the instrument. In this article, we show that compressed-sensing theory can indeed be successfully applied to actual Herschel/PACS data and give significant improvements over the standard pipeline. In order to fully use the redundancy present in the data, we perform full sky map estimation and decompression at the same time, which cannot be done in most other compression methods. We also demonstrate that the various artifacts affecting the data (pink noise, glitches, whose behavior is a priori not well compatible with compressed-sensing) can be handled as well in this new framework. Finally, we make a comparison between the methods from the compressed-sensing scheme and data acquired with the standard compression scheme. We discuss improvements that can be made on ground for the creation of sky maps from the data.

astro-ph.IM↗

arXiv2009

Scale-Based Gaussian Coverings: Combining Intra and Inter Mixture Models in Image Segmentation

By a "covering" we mean a Gaussian mixture model fit to observed data. Approximations of the Bayes factor can be availed of to judge model fit to the data within a given Gaussian mixture model. Between families of Gaussian mixture models, we propose the Rényi quadratic entropy as an excellent and tractable model comparison framework. We exemplify this using the segmentation of an MRI image volume, based (1) on a direct Gaussian mixture model applied to the marginal distribution function, and (2) Gaussian model fit through k-means applied to the 4D multivalued image volume furnished by the wavelet transform. Visual preference for one model over another is not immediate. The Rényi quadratic entropy allows us to show clearly that one of these modelings is superior to the other.

cs.CV↗

arXiv2009

Astronomical Data Analysis and Sparsity: from Wavelets to Compressed Sensing

Wavelets have been used extensively for several years now in astronomy for many purposes, ranging from data filtering and deconvolution, to star and galaxy detection or cosmic ray removal. More recent sparse representations such ridgelets or curvelets have also been proposed for the detection of anisotropic features such cosmic strings in the cosmic microwave background. We review in this paper a range of methods based on sparsity that have been proposed for astronomical data analysis. We also discuss what is the impact of Compressed Sensing, the new sampling theory, in astronomy for collecting the data, transferring them to the earth or reconstructing an image from incomplete measurements.

astro-ph.IM↗

arXiv2009

Full-Sky Weak Lensing Simulation with 70 Billion Particles

We have performed a 70 billion dark-matter particles N-body simulation in a 2 $h^{-1}$ Gpc periodic box, using the concordance, cosmological model as favored by the latest WMAP3 results. We have computed a full-sky convergence map with a resolution of $Δθ\simeq 0.74$ arcmin$^{2}$, spanning 4 orders of magnitude in angular dynamical range. Using various high-order statistics on a realistic cut sky, we have characterized the transition from the linear to the nonlinear regime at $\ell \simeq 1000$ and shown that realistic galactic masking affects high-order moments only below $\ell < 200$. Each domain (Gaussian and non-Gaussian) spans 2 decades in angular scale. This map is therefore an ideal tool for testing map-making algorithms on the sphere. As a first step in addressing the full map reconstruction problem, we have benchmarked in this paper two denoising methods: 1) Wiener filtering applied to the Spherical Harmonics decomposition of the map and 2) a new method, called MRLens, based on the modification of the Maximum Entropy Method on a Wavelet decomposition. While the latter is optimal on large spatial scales, where the signal is Gaussian, MRLens outperforms the Wiener method on small spatial scales, where the signal is highly non-Gaussian. The simulated full-sky convergence map is freely available to the community to help the development of new map-making algorithms dedicated to the next generation of weak-lensing surveys.

astro-ph↗

arXiv2008

Stein Block Thresholding For Image Denoising

In this paper, we investigate the minimax properties of Stein block thresholding in any dimension $d$ with a particular emphasis on $d=2$. Towards this goal, we consider a frame coefficient space over which minimaxity is proved. The choice of this space is inspired by the characterization provided in \cite{BorupNielsen} of family of smoothness spaces on $\mathbb{R}^d$, a subclass of so-called decomposition spaces \cite{Feichtinger}. These smoothness spaces cover the classical case of Besov spaces, as well as smoothness spaces corresponding to curvelet-type constructions. Our main theoretical result investigates the minimax rates over these decomposition spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax (up to a $\log$ factor) in the least favorable situation. Another contribution is that the minimax rates given here are stated for a general noise sequence model in the transform coefficient domain beyond the usual i.i.d. Gaussian case. The choice of the threshold parameter is theoretically discussed and its optimal value is stated for some noise models such as the (non-necessarily i.i.d.) Gaussian case. We provide a simple, fast and a practical procedure. We also report a comprehensive simulation study to support our theoretical findings. The practical performance of our Stein block denoising compares very favorably to the BLS-GSM state-of-the art denoising algorithm on a large set of test images. A toolbox is made available for download on the Internet to reproduce the results discussed in this paper.

math.ST↗

arXiv2008

Wavelet and Curvelet Moments for Image Classification: Application to Aggregate Mixture Grading

We show the potential for classifying images of mixtures of aggregate, based themselves on varying, albeit well-defined, sizes and shapes, in order to provide a far more effective approach compared to the classification of individual sizes and shapes. While a dominant (additive, stationary) Gaussian noise component in image data will ensure that wavelet coefficients are of Gaussian distribution, long tailed distributions (symptomatic, for example, of extreme values) may well hold in practice for wavelet coefficients. Energy (2nd order moment) has often been used for image characterization for image content-based retrieval, and higher order moments may be important also, not least for capturing long tailed distributional behavior. In this work, we assess 2nd, 3rd and 4th order moments of multiresolution transform -- wavelet and curvelet transform -- coefficients as features. As analysis methodology, taking account of image types, multiresolution transforms, and moments of coefficients in the scales or bands, we use correspondence analysis as well as k-nearest neighbors supervised classification.

cs.CV↗

arXiv2007

Dark matter maps reveal cosmic scaffolding

Ordinary baryonic particles (such as protons and neutrons) account for only one-sixth of the total matter in the Universe. The remainder is a mysterious "dark matter" component, which does not interact via electromagnetism and thus neither emits nor reflects light. As dark matter cannot be seen directly using traditional observations, very little is currently known about its properties. It does interact via gravity, and is most effectively probed through gravitational lensing: the deflection of light from distant galaxies by the gravitational attraction of foreground mass concentrations. This is a purely geometrical effect that is free of astrophysical assumptions and sensitive to all matter -- whether baryonic or dark. Here we show high fidelity maps of the large-scale distribution of dark matter, resolved in both angle and depth. We find a loose network of filaments, growing over time, which intersect in massive structures at the locations of clusters of galaxies. Our results are consistent with predictions of gravitationally induced structure formation, in which the initial, smooth distribution of dark matter collapses into filaments then into clusters, forming a gravitational scaffold into which gas can accumulate, and stars can be built.

astro-ph↗