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Register a new userFilter Design for the Detection/Estimation of the Modulus of a Vector. Application to Polarization Data
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We consider a set of M images, whose pixel intensities at a common point can be treated as the components of a M-dimensional vector. We are interested in the estimation of the modulus of such a vector associated to a compact source. For instance, the detection/estimation of the polarized signal of compact sources immersed in a noisy background is relevant in some fields like Astrophysics. We develop two different techniques, one based on the Maximum Likelihood Estimator (MLE) applied to the modulus distribution, the modulus filter (ModF) and other based on prefiltering the components before fusion, the filtered fusion (FF), to deal with this problem. We present both methods in the general case of M images and apply them to the particular case of three images (linear plus circular polarization). Numerical simulations have been performed to test these filters considering polarized compact sources immersed in stationary noise. The FF performs better than the ModF in terms of errors in the estimated amplitude and position of the source, especially in the low signal-to-noise case. We also compare both methods with the direct application of a matched filter (MF) on the polarization data. This last technique is clearly outperformed by the new methods.
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We investigate the performance of a simple Bayesian fitting approach to correct the cosmic microwave background (CMB) B-mode polarization for gravitational lensing effects in the recovered probability distribution of the tensor-to-scalar ratio. We perform a two-dimensional power spectrum fit of the amplitude of the primordial B-modes (tensor-to-scalar ratio, $r$) and the amplitude of the lensing B-modes (parameter $A_{lens}$), jointly with the estimation of the astrophysical foregrounds including both synchrotron and thermal dust emissions. Using this Bayesian framework, we forecast the ability of the proposed CMB space mission LiteBIRD to constrain $r$ in the presence of realistic lensing and foreground contributions. We compute the joint posterior distribution of $r$ and $A_{lens}$, which we improve by adopting a prior on $A_{lens}$ taken from the South Pole Telescope (SPT) measurement. As it applies to the power spectrum, this approach cannot mitigate the uncertainty on $r$ that is due to E-mode cosmic variance transferred to B-modes by lensing, unlike standard delensing techniques that are performed on maps. However, the method allows to correct for the bias on $r$ induced by lensing, at the expense of a larger uncertainty due to the increased volume of the parameter space. We quantify, for different values of the tensor-to-scalar ratio, the trade-off between bias correction and increase of uncertainty on $r$. For LiteBIRD simulations, which include foregrounds and lensing contamination, we find that correcting the foreground-cleaned CMB B-mode power spectrum for the lensing bias, not the lensing cosmic variance, still guarantees a $3sigma$ detection of $r=5times 10^{-3}$. The significance of the detection is increased to $6sigma$ when the current SPT prior on $A_{lens}$ is adopted.
The estimation of the polarization $P$ of extragalactic compact sources in Cosmic Microwave Background images is a very important task in order to clean these images for cosmological purposes -- as, for example, to constrain the tensor-to-scalar ratio of primordial fluctuations during inflation -- and also to obtain relevant astrophysical information about the compact sources themselves in a frequency range, $ u sim 10$--$200$ GHz, where observations have only very recently started to be available. In this paper we propose a Bayesian maximum a posteriori (MAP) approach estimation scheme which incorporates prior information about the distribution of the polarization fraction of extragalactic compact sources between 1 and 100 GHz. We apply this Bayesian scheme to white noise simulations and to more realistic simulations that include CMB intensity, Galactic foregrounds and instrumental noise with the characteristics of the QUIJOTE experiment Wide Survey at 11 GHz. Using these simulations, we also compare our Bayesian method with the frequentist Filtered Fusion method that has been already used in WMAP data and in the emph{Planck} mission. We find that the Bayesian method allows us to decrease the threshold for a feasible estimation of $P$ to levels below $sim 100$ mJy (as compared to $sim 500$ mJy that was the equivalent threshold for the frequentist Filtered Fusion). We compare the bias introduced by the Bayesian method and find it to be small in absolute terms. Finally, we test the robustness of the Bayesian estimator against uncertainties in the prior and in the flux density of the sources. We find that the Bayesian estimator is robust against moderate changes in the parameters of the prior and almost insensitive to realistic errors in the estimated photometry of the sources.
We present results from an end-to-end simulation pipeline interferometric observations of cosmic microwave background polarization. We use both maximum-likelihood and Gibbs sampling techniques to estimate the power spectrum. In addition, we use Gibbs sampling for image reconstruction from interferometric visibilities. The results indicate the level to which various systematic errors (e.g., pointing errors, gain errors, beam shape errors, cross- polarization) must be controlled in order to successfully detect and characterize primordial B modes as well as other scientific goals. In addition, we show that Gibbs sampling is an effective method of image reconstruction for interferometric data in other astrophysical contexts.
The calculation of the characteristic function of the signal fluctuations due to clustered astrophysical sources is performed in this paper. For the typical case of power-law differential number counts and two-point angular correlation function, we present an extension of Zolotarevs theorem that allows us to compute the cumulants of the logarithm of the absolute value of the intensity. As a test, simulations based on recent observations of radio galaxies are then carried out, showing that these cumulants can be very useful for determining the fundamental parameters defining the number counts and the correlation. If the angular correlation scale of the observed source population is known, the method presented here is able to obtain estimators of the amplitude and slope of the power-law number counts with mean absolute errors that are one order of magnitude better than previous techniques, that did not take into account the correlation. Even if the scale of correlation is not well known, the method is able to estimate it and still performs much better than if the effect of correlations is not considered.
The matched filter (MF) is one of the most popular and reliable techniques to the detect signals of known structure and amplitude smaller than the level of the contaminating noise. Under the assumption of stationary Gaussian noise, MF maximizes the probability of detection subject to a constant probability of false detection or false alarm (PFA). This property relies upon a priori knowledge of the position of the searched signals, which is usually not available. Recently, it has been shown that when applied in its standard form, MF may severely underestimate the PFA. As a consequence the statistical significance of features that belong to noise is overestimated and the resulting detections are actually spurious. For this reason, an alternative method of computing the PFA has been proposed that is based on the probability density function (PDF) of the peaks of an isotropic Gaussian random field. In this paper we further develop this method. In particular, we discuss the statistical meaning of the PFA and show that, although useful as a preliminary step in a detection procedure, it is not able to quantify the actual reliability of a specific detection. For this reason, a new quantity is introduced called the specific probability of false alarm (SPFA), which is able to carry out this computation. We show how this method works in targeted simulations and apply it to a few interferometric maps taken with the Atacama Large Millimeter/submillimeter Array (ALMA) and the Australia Telescope Compact Array (ATCA). We select a few potential new point sources and assign an accurate detection reliability to these sources.
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