Detection/estimation of the modulus of a vector. Application to point-source detection in polarization data

Given a set of images, whose pixel values can be considered as the components of a vector, it is interesting to estimate the modulus of such a vector in some localized areas corresponding to a compact signal. For instance, the detection/estimation of a polarized signal in compact sources immersed in a background is relevant in some fields like astrophysics. We develop two different techniques, one based on the Neyman–Pearson lemma, the Neyman–Pearson filter (NPF), and another based on pre-filtering before fusion, the filtered fusion (FF), to deal with the problem of detection of the source and estimation of the polarization given two or three images corresponding to the different components of polarization (two for linear polarization, three including circular polarization). For the case of linear polarization, we have performed numerical simulations on two-dimensional patches to test these filters following two different approaches (a blind and a non-blind detection), considering extragalactic point sources immersed in cosmic microwave background (CMB) and non-stationary noise with the conditions of the 70 GHz Planck channel. The FF outperforms the NPF, especially for low fluxes. We can detect with the FF extragalactic sources in a high noise zone with fluxes      Jy for (blind/non-blind) detection and in a low noise zone with fluxes      Jy for (blind/non-blind) detection with low errors in the estimated flux and position.     

Harmonic analysis of cosmic microwave background data – I. Ring reductions and point-source catalogue

We present a harmonic model for the data analysis of an all-sky cosmic microwave background survey, such as Planck , where the survey is obtained through ring-scans of the sky. In this model, resampling and pixelization of the data are avoided. The spherical transforms of the sky at each frequency, in total intensity and polarization, as well as the bright-point-source catalogue, are derived directly from the data reduced on to the rings. Formal errors and the most significant correlation coefficients for the spherical transforms of the frequency maps are preserved. A clean and transparent path from the original samplings in the time domain to the final scientific products is thus obtained. The data analysis is largely based on Fourier analysis of rings; the positional stability of the instrument's spin axis during these scans is a requirement for the data model and is investigated here for the Planck satellite. Brighter point sources are recognized and extracted as part of the ring reductions and, on the basis of accumulated data, used to build the bright-point-source catalogue. The analysis of the rings is performed in an iterative loop, involving a range of geometric and detector response calibrations. The geometric calibrations are used to reconstruct the paths of the detectors over the sky during a scan and the phase offsets between scans of different detectors; the response calibrations eliminate short- and long-term variations in detector response. Point-source information may allow the reconstruction of the beam profile. The reconstructed spherical transforms of the sky in each frequency channel form the input to the subsequent analysis stages. Although the methods in this paper were developed with the data processing for the Planck satellite in mind, there are many aspects which have wider implementation possibilities, including the construction of real-space pixelized maps.     

The observability of topological defects with ground-based interferometers

Topological defect theories lead to non-Gaussian features on maps of fluctuations of the cosmic microwave background radiation (CMBR), which enable us to distinguish them from maps predicted by standard inflationary models. We have recently presented a maximum entropy method (MEM) which simultaneously deconvolves interferometer maps of CMBR fluctuations, and separates out foreground contaminants. By applying this method to simulated observations using a realistic ground-based interferometer, we demonstrate that it is possible to recover the prominent hotspots in the CMBR maps which delineate individual defects, even in the presence of a significant Galactic foreground.     

Wavelet analysis and the detection of non-Gaussianity in the cosmic microwave background

We investigate the use of wavelet transforms in detecting and characterizing non-Gaussian structure in maps of the cosmic microwave background (CMB). We apply the method to simulated maps of the KaiserStebbins effect resulting from cosmic strings, on to which Gaussian signals of varying amplitudes are superposed. We find that the method significantly outperforms standard techniques based on measuring the moments of the pixel temperature distribution. We also compare the results with those obtained using techniques based on Minkowski functionals, and we again find the wavelet method to be superior. In particular, using the wavelet technique, we find that it is possible to detect non-Gaussianity even in the presence of a superposed Gaussian signal with 3 times the rms amplitude of the original cosmic string map. We also find that the wavelet technique is useful in characterizing the angular scales at which the non-Gaussian signal occurs.     

Foreground separation methods for satellite observations of the cosmic microwave background

A maximum entropy method (MEM) is presented for separating the emission resulting from different foreground components from simulated satellite observations of the cosmic microwave background radiation (CMBR). In particular, the method is applied to simulated observations by the proposed Planck Surveyor satellite. The simulations, performed by Bouchet &38; Gispert, include emission from the CMBR and the kinetic and thermal Sunyaev–Zel'dovich (SZ) effects from galaxy clusters, as well as Galactic dust, free–free and synchrotron emission. We find that the MEM technique performs well and produces faithful reconstructions of the main input components. The method is also compared with traditional Wiener filtering and is shown to produce consistently better results, particularly in the recovery of the thermal SZ effect.     

Wavelet domain Bayesian denoising of string signal in the cosmic microwave background

An algorithm is proposed for denoising the signal induced by cosmic strings in the cosmic microwave background. A Bayesian approach is taken, based on modelling the string signal in the wavelet domain with generalized Gaussian distributions. Good performance of the algorithm is demonstrated by simulated experiments at arcminute resolution under noise conditions including primary and secondary cosmic microwave background anisotropies, as well as instrumental noise.     

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of a previously developed wavelet formalism developed by Antoine & Vandergheynst and Wiaux et al. The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation are first identified. A scale-discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background data, such as the imprint of topological defects, in particular, cosmic strings, and for their reconstruction after separation from the other signal components.     

Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach

We present the first tests of a new method, the correlated component analysis (CCA) based on second-order statistics, to estimate the mixing matrix, a key ingredient to separate astrophysical foregrounds superimposed to the Cosmic Microwave Background (CMB). In the present application, the mixing matrix is parametrized in terms of the spectral indices of Galactic synchrotron and thermal dust emissions, while the free–free spectral index is prescribed by basic physics, and is thus assumed to be known. We consider simulated observations of the microwave sky with angular resolution and white stationary noise at the nominal levels for the Planck satellite, and realistic foreground emissions, with a position-dependent synchrotron spectral index. We work with two sets of Planck frequency channels: the low-frequency set, from 30 to 143 GHz, complemented with the Haslam 408 MHz map, and the high-frequency set, from 217 to 545 GHz. The concentration of intense free–free emission on the Galactic plane introduces a steep dependence of the spectral index of the global Galactic emission with Galactic latitude, close to the Galactic equator. This feature makes difficult for the CCA to recover the synchrotron spectral index in this region, given the limited angular resolution of Planck , especially at low frequencies. A cut of a narrow strip around the Galactic equator  (| b | < 3°)  , however, allows us to overcome this problem. We show that, once this strip is removed, the CCA allows an effective foreground subtraction, with residual uncertainties inducing a minor contribution to errors on the recovered CMB power spectrum.