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

Nearly degenerate heavy sterile neutrinos in cascade decay: mixing and oscillations

Some extensions beyond the Standard Model propose the existence of nearly degenerate heavy sterile neutrinos. If kinematically allowed these can be resonantly produced and decay in a cascade to common final states. The common decay channels lead to mixing of the heavy sterile neutrino states and interference effects. We implement non-perturbative methods to study the dynamics of the cascade decay to common final states, which features similarities but also noteworthy differences with the case of neutral meson mixing. We show that mixing and oscillations among the nearly degenerate sterile neutrinos can be detected as \emph{quantum beats} in the distribution of final states produced from their decay. These oscillations would be a telltale signal of mixing between heavy sterile neutrinos. We study in detail the case of two nearly degenerate sterile neutrinos produced in the decay of pseudoscalar mesons and decaying into a purely leptonic "visible" channel: $\nu_h \rightarrow e^+ e^- \nu_a$. Possible cosmological implications for the effective number of neutrinos $N_{eff}$ are discussed.Comment: updated references, more comments, same results, published version. arXiv admin note: text overlap with arXiv:1406.573

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

Linear inverse problems with noise: primal and primal-dual splitting

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian, Poisson). On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution, inpainting and denoising with some comparison to prior methods are also reported

Deconvolution of confocal microscopy images using proximal iteration and sparse representations

We propose a deconvolution algorithm for images blurred and degraded by a Poisson noise. The algorithm uses a fast proximal backward-forward splitting iteration. This iteration minimizes an energy which combines a \textit{non-linear} data fidelity term, adapted to Poisson noise, and a non-smooth sparsity-promoting regularization (e.g $\ell_1$-norm) over the image representation coefficients in some dictionary of transforms (e.g. wavelets, curvelets). Our results on simulated microscopy images of neurons and cells are confronted to some state-of-the-art algorithms. They show that our approach is very competitive, and as expected, the importance of the non-linearity due to Poisson noise is more salient at low and medium intensities. Finally an experiment on real fluorescent confocal microscopy data is reported