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.Comment: Accepted for publication in A&A. Manuscript with all figures can be downloaded at http://jstarck.free.fr/aa_sphere05.pd

On Preferred Axes in WMAP Cosmic Microwave Background Data after Subtraction of the Integrated Sachs-Wolfe Effect

There is currently a debate over the existence of claimed statistical anomalies in the cosmic microwave background (CMB), recently confirmed in Planck data. Recent work has focussed on methods for measuring statistical significance, on masks and on secondary anisotropies as potential causes of the anomalies. We investigate simultaneously the method for accounting for masked regions and the foreground integrated Sachs-Wolfe (ISW) signal. We search for trends in different years of WMAP CMB data with different mask treatments. We reconstruct the ISW field due to the 2 Micron All-Sky Survey (2MASS) and the NRAO VLA Sky Survey (NVSS) up to l=5, and we focus on the Axis of Evil (AoE) statistic and even/odd mirror parity, both of which search for preferred axes in the Universe. We find that removing the ISW reduces the significance of these anomalies in WMAP data, though this does not exclude the possibility of exotic physics. In the spirit of reproducible research, all reconstructed maps and codes will be made available for download at http://www.cosmostat.org/anomaliesCMB.html.Comment: Figure 1-2 and Tables 1, D.1, D.2 updated. Main conclusions unchanged. Accepted for publication in A&A. In the spirit of reproducible research, all statistical and sparse inpainting codes as well as resulting products which constitute main results of this paper will be made public here: http://www.cosmostat.org/anomaliesCMB.htm

3D galaxy clustering with future wide-field surveys: Advantages of a spherical Fourier-Bessel analysis

Upcoming spectroscopic galaxy surveys are extremely promising to help in addressing the major challenges of cosmology, in particular in understanding the nature of the dark universe. The strength of these surveys comes from their unprecedented depth and width. Optimal extraction of their three-dimensional information is of utmost importance to best constrain the properties of the dark universe. Although there is theoretical motivation and novel tools to explore these surveys using the 3D spherical Fourier-Bessel (SFB) power spectrum of galaxy number counts $C_\ell(k,k^\prime)$, most survey optimisations and forecasts are based on the tomographic spherical harmonics power spectrum $C^{(ij)}_\ell$. We performed a new investigation of the information that can be extracted from the tomographic and 3D SFB techniques by comparing the forecast cosmological parameter constraints obtained from a Fisher analysis in the context of planned stage IV wide-field galaxy surveys. The comparison was made possible by careful and coherent treatment of non-linear scales in the two analyses. Nuisance parameters related to a scale- and redshift-dependent galaxy bias were also included for the first time in the computation of both the 3D SFB and tomographic power spectra. Tomographic and 3D SFB methods can recover similar constraints in the absence of systematics. However, constraints from the 3D SFB analysis are less sensitive to unavoidable systematics stemming from a redshift- and scale-dependent galaxy bias. Even for surveys that are optimised with tomography in mind, a 3D SFB analysis is more powerful. In addition, for survey optimisation, the figure of merit for the 3D SFB method increases more rapidly with redshift, especially at higher redshifts, suggesting that the 3D SFB method should be preferred for designing and analysing future wide-field spectroscopic surveys.Comment: 12 pages, 6 Figures. Python package for cosmological forecasts available at https://cosmicpy.github.io . Updated figures. Matches published versio

Polarized wavelets and curvelets on the sphere

The statistics of the temperature anisotropies in the primordial cosmic microwave background radiation field provide a wealth of information for cosmology and for estimating cosmological parameters. An even more acute inference should stem from the study of maps of the polarization state of the CMB radiation. Measuring the 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 being awaited with excitement. Multiscale methods, such as isotropic wavelets, steerable wavelets, or curvelets, have been proposed in the past to analyze the CMB temperature map. In this paper, we contribute to enlarging the set of available transforms for polarized data on the sphere. We describe a set of new multiscale decompositions for polarized data on the sphere, including decimated and undecimated Q-U or E-B wavelet transforms and Q-U or E-B curvelets. The proposed transforms are invertible and so allow for applications in data restoration and denoising.Comment: Accepted. Full paper will figures available at http://jstarck.free.fr/aa08_pola.pd

Low-l CMB Analysis and Inpainting

Reconstruction of the CMB in the Galactic plane is extremely difficult due to the dominant foreground emissions such as Dust, Free-Free or Synchrotron. For cosmological studies, the standard approach consists in masking this area where the reconstruction is not good enough. This leads to difficulties for the statistical analysis of the CMB map, especially at very large scales (to study for e.g., the low quadrupole, ISW, axis of evil, etc). We investigate in this paper how well some inpainting techniques can recover the low-$\ell$ spherical harmonic coefficients. We introduce three new inpainting techniques based on three different kinds of priors: sparsity, energy and isotropy, and we compare them. We show that two of them, sparsity and energy priors, can lead to extremely high quality reconstruction, within 1% of the cosmic variance for a mask with Fsky larger than 80%.Comment: Submitte

True CMB Power Spectrum Estimation

The cosmic microwave background (CMB) power spectrum is a powerful cosmological probe as it entails almost all the statistical information of the CMB perturbations. Having access to only one sky, the CMB power spectrum measured by our experiments is only a realization of the true underlying angular power spectrum. In this paper we aim to recover the true underlying CMB power spectrum from the one realization that we have without a need to know the cosmological parameters. The sparsity of the CMB power spectrum is first investigated in two dictionaries; Discrete Cosine Transform (DCT) and Wavelet Transform (WT). The CMB power spectrum can be recovered with only a few percentage of the coefficients in both of these dictionaries and hence is very compressible in these dictionaries. We study the performance of these dictionaries in smoothing a set of simulated power spectra. Based on this, we develop a technique that estimates the true underlying CMB power spectrum from data, i.e. without a need to know the cosmological parameters. This smooth estimated spectrum can be used to simulate CMB maps with similar properties to the true CMB simulations with the correct cosmological parameters. This allows us to make Monte Carlo simulations in a given project, without having to know the cosmological parameters. The developed IDL code, TOUSI, for Theoretical pOwer spectrUm using Sparse estImation, will be released with the next version of ISAP

The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement

Sparsity and morphological diversity for hyperspectral data analysis

Recently morphological diversity and sparsity have emerged as new and effective sources of diversity for Blind Source Separation. Based on these new concepts, novelmethods such as Generalized Morphological Component Analysis have been put forward. The latter takes advantage of the very sparse representation of structured data in large overcomplete dictionaries, to separate sources based on their morphology. Building on GMCA, the purpose of this contribution is to describe a new algorithm for hyperspectral data processing. Large-scale hyperspectral data refers to collected data that exhibit sparse spectral signatures in addition to sparse spatial morphologies, in specified dictionaries of spectral and spatial waveforms. Numerical experiments are reported which demonstrate the validity of the proposed extension for solving source separation problems involving hyperspectral data