Yassir Moudden

Morphological Component Analysis: An Adaptive Thresholding Strategy

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the lscr1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.

Sparsity and Morphological Diversity in Blind Source Separation

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.

Wavelets, ridgelets and curvelets on the sphere

We present in this paper new multiscale transforms on the sphere, namely the isotropic undecimated wavelet transform, the pyramidal wavelet transform, the ridgelet transform and the curvelet transform. All of these transforms can be inverted i.e. we can exactly reconstruct the original data from its coefficients in either representation. Several applications are described. We show how these transforms can be used in denoising and especially in a Combined Filtering Method, which uses both the wavelet and the curvelet transforms, thus benefiting from the advantages of both transforms. An application to component separation from multichannel data mapped to the sphere is also described in which we take advantage of moving to a wavelet representation.

Image Decomposition and Separation Using Sparse Representations: An Overview

This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method-morphological component analysis (MCA)-based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation.

Morphological diversity and source separation

This letter describes a new method for blind source separation, adapted to the case of sources having different morphologies. We show that such morphological diversity leads to a new and very efficient separation method, even in the presence of noise. The algorithm, coined multichannel morphological component analysis (MMCA), is an extension of the morphological component analysis (MCA) method. The latter takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. MCA has been shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The proposed extension, MMCA, extends the above for multichannel data, achieving a better source separation in those circumstances. Furthermore, the new algorithm can efficiently achieve good separation in a noisy context where standard independent component analysis methods fail. The efficiency of the proposed scheme is confirmed in numerical experiments

Morphological component analysis

The Morphological Component Analysis (MCA) is a a new method which allows us to separate features contained in an image when these features present different morphological aspects. We show that MCA can be very useful for decomposing images into texture and piecewise smooth (cartoon) parts or for inpainting applications. We extend MCA to a multichannel MCA (MMCA) for analyzing multispectral data and present a range of examples which illustrates the results.

CMB data analysis and sparsity

The statistical analysis of the soon to come Planck satellite CMB data will help set tighter bounds on major cosmological parameters. On the way, a number of practical difficulties need to be tackled, notably that several other astrophysical sources emit radiation in the frequency range of CMB observations. Some level of residual contributions, most significantly in the galactic region and at the locations of strong radio point sources will unavoidably contaminate the estimated spherical CMB map. Masking out these regions is common practice but the gaps in the data need proper handling. In order to restore the stationarity of a partly incomplete CMB map and thus lower the impact of the gaps on non-local statistical tests, we developed an inpainting algorithm on the sphere based on a sparse representation of the data, to fill in and interpolate across the masked regions. c 2007 Elsevier B.V. All rights reserved.

SZ and CMB reconstruction using Generalized Morphological Component Analysis

Abstract In the last decade, the study of cosmic microwave background (CMB) data has become one of the most powerful tools for studying and understanding the Universe. More precisely, measuring the CMB power spectrum leads to the estimation of most cosmological parameters. Nevertheless, accessing such precious physical information requires extracting several different astrophysical components from the data. Recovering those astrophysical sources (CMB, Sunyaev–Zel’dovich clusters, galactic dust) thus amounts to a component separation problem which has already led to an intensive activity in the field of CMB studies. In this paper, we introduce a new sparsity-based component separation method coined Generalized Morphological Component Analysis (GMCA). The GMCA approach is formulated in a Bayesian maximum a posteriori (MAP) framework. Numerical results show that this new source recovery technique performs well compared to state-of-the-art component separation methods already applied to CMB data.

Blind component separation in wavelet space: application to CMB analysis

It is a recurrent issue in astronomical data analysis that observations are incomplete maps with missing patches or intentionally masked parts. In addition, many astrophysical emissions are nonstationary processes over the sky. All these effects impair data processing techniques which work in the Fourier domain. Spectral matching ICA (SMICA) is a source separation method based on spectral matching in Fourier space designed for the separation of diffuse astrophysical emissions in cosmic microwave background observations. This paper proposes an extension of SMICA to the wavelet domain and demonstrates the effectiveness of wavelet-based statistics for dealing with gaps in the data.

First evidence of a gravitational lensing-induced echo in gamma rays with Fermi LAT

Aims. This article shows the first evidence ever of gravitational lensing phenomena in high energy gamma-rays. This evidence comes from the observation of an echo in the light curve of the distant blazar PKS 1830-211 induced by a gravitational lens system. Methods. Traditional methods for estimating time delays in gravitational lensing systems rely on the cross-correlation of the light curves from individual images. We used the 300 MeV‐30 GeV photons detected by the Fermi-LAT instrument. It cannot separate the images of known lenses, so the observed light curve is the superposition of individual image light curves. The Fermi-LAT instrument has the advantage of providing long, evenly spaced, time series with very low photon noise. This allows us to use Fourier transform methods directly. Results. A time delay between the two compact images of PKS 1830-211 has been searched for by both the autocorrelation method and the “double power spectrum” method. The double power spectrum shows a 4.2σ proof of a time delay of 27.1±0.6 days, consistent with others’ results.