We propose a new method to better estimate and subtract the contribution of detected compact sources to the microwave sky. These bright compact source emissions contaminate the full-sky data over a significant fraction of the sky, and should therefore be accurately removed if a high resolution and full-sky estimate of the components is sought after. However the point source spectral variability hampers accurate blind source separation, even with state-of-the-art localized source separation techniques. In this work, we rather propose to estimate the flux of the brightest compact sources using a morphological separation approach, relying on a more sophisticated model for the background than in standard approaches. Essentially, this amounts to separate point sources with known support and shape from a background assumed sparse in the spherical harmonic domain. This approach is compared to standard local χ<sup>2</sup> minimization modeling the background as a low order polynomial on WMAP realistic simulations. If in noisy situations estimating more than a few parameter does not improve flux recovery, in the first WMAP channels the proposed method leads to lower biases (typically by factors of 2) and increased robustness.
In fluorescence microscopy, one can distinguish two kinds of imaging approaches, wide field and raster scan
microscopy, differing by their excitation and detection scheme. In both imaging modalities the acquisition is
independent of the information content of the image. Rather, the number of acquisitions N, is imposed by
the Nyquist-Shannon theorem. However, in practice, many biological images are compressible (or, equivalently
here, sparse), meaning that they depend on a number of degrees of freedom K that is smaller that their size N.
Recently, the mathematical theory of compressed sensing (CS) has shown how the sensing modality could take
advantage of the image sparsity to reconstruct images with no loss of information while largely reducing the number M of acquisition. Here we present a novel fluorescence microscope designed along the principles of CS. It uses a spatial light modulator (DMD) to create structured wide field excitation patterns and a sensitive point detector to measure the emitted fluorescence. On sparse fluorescent samples, we could achieve compression ratio N/M of up to 64, meaning that an image can be reconstructed with a number of measurements of only 1.5 % of its pixel number. Furthemore, we extend our CS acquisition scheme to an hyperspectral imaging system.
Recent advances in signal processing have focused on the use of sparse representations in various applications.
A new field of interest based on sparsity has recently emerged: compressed sensing. This theory
is a new sampling framework that provides an alternative to the well-known Shannon sampling theory.
In this paper we investigate how compressed sensing (CS) can provide new insights into astronomical
data compression. In a previous study1 we gave new insights into the use of Compressed Sensing (CS)
in the scope of astronomical data analysis. More specifically, we showed how CS is flexible enough to
account for particular observational strategies such as raster scans. This kind of CS data fusion concept
led to an elegant and effective way to solve the problem ESA is faced with, for the transmission to the
earth of the data collected by PACS, one of the instruments onboard the Herschel spacecraft which will
launched in late 2008/early 2009.
In this paper, we extend this work by showing how CS can be effectively used to jointly decode multiple
observations at the level of map making. This allows us to directly estimate large areas of the sky
from one or several raster scans. Beyond the particular but important Herschel example, we strongly
believe that CS can be applied to a wider range of applications such as in earth science and remote
sensing where dealing with multiple redundant observations is common place. Simple but illustrative
examples are given that show the effectiveness of CS when decoding is made from multiple redundant
The recent development of multi-channel sensors has motivated interest in devising new methods for the
coherent processing of multivariate data. An extensive work has already been dedicated to multivariate
data processing ranging from blind source separation (BSS) to multi/hyper-spectral data restoration.
Previous work has emphasized on the fundamental role played by sparsity and morphological diversity
to enhance multichannel signal processing.
GMCA is a recent algorithm for multichannel data analysis which was used successfully in a variety of
applications including multichannel sparse decomposition, blind source separation (BSS), color image
restoration and inpainting. Inspired by GMCA, a recently introduced algorithm coined HypGMCA
is described for BSS applications in hyperspectral data processing. It assumes the collected data is a
linear instantaneous mixture of components exhibiting sparse spectral signatures as well as sparse spatial
morphologies, each in specified dictionaries of spectral and spatial waveforms. We report on numerical
experiments with synthetic data and application to real observations which demonstrate the validity of
the proposed method.
Over the last few years, the development of multi-channel sensors motivated interest in methods for the
coherent processing of multivariate data. From blind source separation (BSS) to multi/hyper-spectral
data restoration, an extensive work has already been dedicated to multivariate data processing. Previous
work has emphasized on the fundamental role played by sparsity and morphological diversity to
enhance multichannel signal processing.
Morphological diversity has been first introduced in the mono-channel case to deal with contour/texture
extraction. The morphological diversity concept states that the data are the linear combination of several
so-called morphological components which are sparse in different incoherent representations. In
that setting, piecewise smooth features (contours) and oscillating components (textures) are separated
based on their morphological differences assuming that contours (respectively textures) are sparse in the
Curvelet representation (respectively Local Discrete Cosine representation).
In the present paper, we define a multichannel-based framework for sparse multivariate data representation.
We introduce an extension of morphological diversity to the multichannel case which boils down
to assuming that each multichannel morphological component is diversely sparse spectrally and/or spatially.
We propose the Generalized Morphological Component Analysis algorithm (GMCA) which aims
at recovering the so-called multichannel morphological components. Hereafter, we apply the GMCA
framework to two distinct multivariate inverse problems : blind source separation (BSS) and multichannel
data restoration. In the two aforementioned applications, we show that GMCA provides new and
essential insights into the use of morphological diversity and sparsity for multivariate data processing.
Further details and numerical results in multivariate image and signal processing will be given illustrating
the good performance of GMCA in those distinct applications.
The statistics of the temperature anisotropies in the primordial Cosmic Microwave Background radiation field
provide a wealth of information for cosmology and the estimation of cosmological parameters. An even more
acute inference should stem from the study of maps of the polarization state of the CMB radiation. Measuring
the latter extremely weak CMB polarization signal requires very sensitive instruments. The full-sky maps of
both temperature and polarization anisotropies of the CMB to be delivered by the upcoming Planck Surveyor
satellite experiment are hence awaited with excitement. Still, analyzing CMB data requires tackling a number
of practical difficulties, notably that several other astrophysical sources emit radiation in the frequency range
of CMB observations. Separating the different astrophysical foreground components and the CMB proper from
available multichannel data is a problem that has drawn much attention in the community. Nevertheless, 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 to fill in the gaps, based on an iterative thresholding scheme in a sparse
representation of the data. This algorithm relies on the variety of recently developed transforms on the sphere
among which several multiscale transforms which we will review. We also contribute to enlarging the set of
available transforms for polarized data on the sphere. We describe new multiscale decompositions namely the
isotropic undecimated wavelet and curvelet transforms for polarized data on the sphere. The proposed transforms
are invertible and so allow for applications in image restoration and denoising.
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.