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Estimating the cosmological microwave background is of utmost importance for cosmology. However, its estimation from full-sky surveys such as WMAP or more recently Planck is challenging: CMB maps are generally estimated via the application of some source separation techniques which never prevent the final map from being contaminated with noise and foreground residuals. These spurious contaminations whether noise or foreground residuals are well known to be a plague for most cosmologically relevant tests or evaluations; this includes CMB lensing reconstruction or non-Gaussian signatures search. Noise reduction is generally performed by applying a simple Wiener filter in spherical harmonics; however, this does not account for the non-stationarity of the noise. Foreground contamination is usually tackled by masking the most intense residuals detected in the map, which makes CMB evaluation harder to perform. In this paper, we introduce a novel noise reduction framework coined LIW-Filtering for Linear Iterative Wavelet Filtering which is able to account for the noise spatial variability thanks to a wavelet-based modeling while keeping the highly desired linearity of the Wiener filter. We further show that the same filtering technique can effectively perform foreground contamination reduction thus providing a globally cleaner CMB map. Numerical results on simulated Planck data are provided.

In mid-2009, European Spatial Agency (ESA) put in orbit the latest space observatory Planck to investigate the Cosmological Microwave Background (CMB). These data are of particular scientific importance as it will provide more insight into the understanding of the birth of our universe. Most cosmological parameters can be derived from the study of these CMB data. After a series of successful CMB experiments (Archeops, Boomerang, Maxima, COBE, and WMAP [

The very specific scanning pattern in full-sky CMB experiments leads to an instrumental noise closely Gaussian that exhibits significant spatial variation: Planck noise is far from being homogeneous. More precisely, the statistics of the noise vary spatially; in the field of statistics, it is said that the noise is nonstationary. As an illustration, Figure

Noise can be a significant limitation for most cosmological tests or reconstructions. High noise level can hamper non-Gaussianity tests as the noisy map is likely to be “more Gaussian” than the noise-free map. The classical approach in the field of cosmology generally consists in reducing the noise of the CMB via a Wiener filter in spherical harmonics. If

Another less traditional but commonly encountered noise reducing technique is the local Wiener filtering. It boils down to apply the Wiener filter in the Fourier space on patches in the pixel domain. The main drawback of this method is that while large patches should be favored to capture the nonstationarity of noise, they are enable to capture correlations at scales larger than the patch size.

The first evaluation of already sophisticated CMB-dedicated source separation methods has been performed in [

The contribution of this paper is twofold:

It introduces a novel noise reduction framework that, in opposition to the classical spherical-harmonics based Wiener filter, accounts for the nonstationarity of the noise. This new method preserves the linearity of the filtering. In fact, linearity is crucial in this field as it makes the study of error propagation via Monte-Carlo simulations much more convenient to carry out. In this framework, we make use of a simple, but efficient, wavelet-based modeling of the noise that allows for the modeling of nonstationarities at different scales. This will be discussed in Section

We discuss an extension of this method to further estimate an approximate contribution of the foreground residuals per wavelet scales by means of RMS maps. Thereby contamination reduction could be tackled within the same filtering framework.

In a very general context and more precisely in the context of Planck, it is customary to assume that the input noise data, denoted by

More precisely, the CMB

Written differently,

As said clearly in the previous paragraph, computing the MAP solution requires inverting a system of equations which turns to be intractable due to the large scale of Planck data. The most straightforward numerical solver is the well-known conjugate gradient. However, while this solver is known to be a fast and accurate numerical method for solving linear systems of equations, it lacks the flexibility of more modern optimization techniques. For this reason we rather opt for the more flexible optimization framework of proximal calculus (see [

The problem in (

In the problem of interest in this paper,

(1)

(2)

(a) Forward step of the FPS:

(b) Backward step of the FPS:

(3)

The computational cost of this iterative Wiener algorithm is particularly low. The first step (2-a) only requires entrywise multiplications and additions of vectors. It is therefore of the order of

The proposed iterative filtering technique preserves the linearity of the overall CMB map processing. In fact, in every step of the proposed algorithm, each output depends linearly on their inputs. This property is essential as it allows for a simple way to study the propagation of errors via Monte-Carlo simulations; each simulation then has to undergo exactly the same processing step described previously. It is worthwhile to outline that the solution to the proposed iterative Wiener filter is exactly the solution of the Wiener filter described in (

Assuming a perfect calibration of the data, the nonstationarities of instrumental noise mainly originate from the nonuniform scanning of the sky. The lower the number of scanning time in one area is, the higher the noise level is and vice versa. Figure

The way the sky is scanned by Planck makes the noise correlated along the scanning direction. This means that the noise is correlated along elongated patterns in the pixel domain; for numerical reasons, this can hardly be accounted for in the pixelwise covariance matrix

Most modern CMB-dedicated source separation methods, Needlet ILC [

It then seems natural to model non-stationary-correlated noise: (i) in the wavelet domain to capture its correlation and (ii) locally to capture its nonstationarity.

Wavelets are well known to be the tool of choice to analyze nonstationary signals [

As emphasized previously, one elegant way to model the instrumental noise is to consider the distribution of its expansion coefficients in the spherical wavelet domain. The basic idea consists in assuming that the wavelet coefficients of noise are approximately decorrelated. This idea takes its roots in the field of multiscale statistical modeling: it has been shown that wavelets exhibit (almost)-decorrelating properties for some classes of nonstationary stochastic processes. To give only one example, this is the case for fractional Brownian motion [

In the following, numerical experiments will be performed on simulated Planck data described in [

The noise that contaminates the raw CMB map at the output of most component separation techniques is nonstationary and correlated. The CMB map estimates being estimated from several observations at different resolutions, the output noise is obviously correlated. Source separation techniques like L-GMCA [

In this section, we particularly focus on the noise reduction aspect of the proposed method. The reference denoising method in the field of astrophysics and more specifically CMB data analysis is the

As detailed in the previous section, the modeling of the noise contribution is performed in the wavelet domain: the noise is assumed to be nonstationary but decorrelated at each wavelet scale

The global Wiener solution is computed by filtering the original CMB map

Iterative Wiener has been applied to the original noisy CMB map

Figure

It is well known that the Wiener filter provides a CMB map that has a biased power spectrum. Thereby the Wiener filter—in red in Figure

Figure

A more complete measure of discrepancy between the original map

To better illustrate the differences between the

As a brief conclusion for these experimental results, it appears clearly that the nonstationarity of the noise has an impact on the CMB estimate. The most classical noise reducing technique in the field, that is, the

As stated in the introduction in Section

Unlike noise, the foreground residual are generally non-Gaussian, with the exception of the CIB (see [

In this equation, the correlation of foreground pixels implies that

So as to capture the correlation of foreground pixels, a natural and simple strategy boils down to adopting the wavelet-based statistical modeling used to model correlated noise in the previous Section

In the rest of this paper, we will generally use the term contamination for the contribution of both the foreground residuals and the instrumental noise. In the following, as suggested previously, we also opt for exactly the same model to capture the pixel dependencies of the instrumental noise. Therefore, we will now define locally at pixel

From the estimation of the local variance

Let

where

The dataset used in this section is exactly the same that we used to study noise reduction. It is described in Section

As emphasized in Section

The proposed Wiener-based iterative method has been applied on the aforementioned Planck simulated data. Figure

Figure

Figure

Figure

(a)

Assuming that the Planck instrumental noise is Gaussian is widely considered as a reasonable assumption. This is obviously not the case for foreground residuals the presence of which is likely to largely distort the search for non-Gaussian features in the CMB. Thereby, reducing the amount of contaminant should help preventing non-Gaussianity test from the non-Gaussian impact of foreground residuals. In the field of CMB non-Gaussianity evaluation, a classical technique boils down to measuring higher-order statistics in the wavelet domain [

The red dashed line in Figure

Figure

Important cosmological tests and evaluations are performed on estimated CMB maps; this is particularly the case for CMB lensing reconstruction [

As shown in the previous experiments, the proposed filtering clearly limits the impact of noise by taking into account its nonstationary behavior. Furthermore, results are presented that clearly show that modeling the contribution of foreground residuals make contamination reduction effective even on the galactic plane. This has two major consequences: (i) it makes it possible the use of a much smaller mask prior to any analysis of the estimated CMB map and (ii) it helps reducing the non-Gaussian features that originate from the presence of foreground residuals.

Whether noise or foreground residual is at stake, the central assumption that is at the heart of the modeling of contamination is the decorrelation hypothesis we made in Section

The contamination modeling used so far in this paper makes profit of the decorrelation assumption; it particularly helps simplifying the filtering process by only requiring the handling of diagonal matrices (i.e., root mean squared maps). Departing from the decorrelation assumption will largely increase the complexity of the proposed filtering techniques. However, a straightforward way of extending the proposed methods is to choose multiscale signal representations which are better adapted to the morphology or structure of the signal to be modeled.

It is important to wonder whether the contamination reduction filtering introduced in Section

Cosmological microwave background maps estimated from full-sky surveys such as WMAP or more recently Planck generally suffers from various sources of contaminations: (i) instrumental noise is generally nonstationary which may generate non-Gaussian signatures and (ii) foreground residuals generally remain even after the application of state-of-the-art source separation methods. In this context, the most classical denoising technique, aka.

Future work will focus on refining the modeling of noise and foreground residuals. Without losing the simplicity of this approach, we will more specifically focus on studying more adapted signal representation to better model the noise/foreground contamination spatial behavior.

The developed IDL code will be released with the next version of ISAP (Interactive Sparse astronomical data Analysis Packages) via the following web site:

This work was supported by the French National Agency for Research (ANR-08-EMER-009-01) and the European Research Council Grant SparseAstro (ERC-228261).