Jason D. McEwen

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.

Detection of the integrated Sachs–Wolfe effect and corresponding dark energy constraints made with directional spherical wavelets

Using a directional spherical wavelet analysis we detect the integrated Sachs-Wolfe (ISW) effect, indicated by a positive correlation between the Wilkinson Microwave Anisotropy Probe (WMAP) and NRAO VLA Sky Survey (NVSS) data, at the 3.9 sigma level. In a flat universe the detection of the ISW effect provides direct and independent evidence for dark energy. Moreover, we use our detection to constrain the dark energy density Omega_Lambda. We obtain estimates for Omega_Lambda consistent with other analysis techniques and data sets and rule out a zero cosmological constant at greater than 99% significance.

Fast Directional Continuous Spherical Wavelet Transform Algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Goacuterski, provides a savings of O(radicNpix) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 106 pixel map on a personal computer

Cosmological Applications of a Wavelet Analysis on the Sphere

The cosmic microwave background (CMB) is a relic radiation of the Big Bang and as such it contains a wealth of cosmological information. Statistical analyses of the CMB, in conjunction with other cosmological observables, represent some of the most powerful techniques available to cosmologists for placing strong constraints on the cosmological parameters that describe the origin, content and evolution of the Universe. The last decade has witnessed the introduction of wavelet analyses in cosmology and, in particular, their application to the CMB. We review here spherical wavelet analyses of the CMB that test the standard cosmological concordance model. The assumption that the temperature anisotropies of the CMB are a realisation of a statistically isotropic Gaussian random field on the sphere is questioned. Deviations from both statistical isotropy and Gaussianity are detected in the reviewed works, suggesting more exotic cosmological models may be required to explain our Universe. We also review spherical wavelet analyses that independently provide evidence for dark energy, an exotic component of our Universe of which we know very little currently. The effectiveness of accounting correctly for the geometry of the sphere in the wavelet analysis of full-sky CMB data is demonstrated by the highly significant detections of physical processes and effects that are made in these reviewed works.

Compressed sensing for wide-field radio interferometric imaging

For the next generation of radio interferometric telescopes it is of paramount importance to incorporate wide field-of-view (WFOV) considerations in interferometric imaging, otherwise the fidelity of reconstructed images will suffer greatly. We extend compressed sensing techniques for interferometric imaging to a WFOV and recover images in the spherical coordinate space in which they naturally live, eliminating any distorting projection. The effectiveness of the spread spectrum phenomenon, highlighted recently by one of the authors, is enhanced when going to a WFOV, while sparsity is promoted by recovering images directly on the sphere. Both of these properties act to improve the quality of reconstructed interferometric images. We quantify the performance of compressed sensing reconstruction techniques through simulations, highlighting the superior reconstruction quality achieved by recovering interferometric images directly on the sphere rather than the plane.

Markov chain Monte Carlo analysis of Bianchi VIIh models

We have extended the previous analyses of Jaffe et al. to a complete Markov chain Monte Carlo (MCMC) parameter space study of the Bianchi-type VII h models including a dark energy density, using Wilkinson Microwave Anisotropy Probe (WMAP) cosmic microwave background (CMB) data from the 1- and 3-yr releases. Since we perform the analysis in a Bayesian framework our entire inference is contained in the multidimensional posterior distribution from which we can extract marginalized parameter constraints and the comparative Bayesian evidence. Treating the left-handed Bianchi CMB anisotropy as a template centred upon the coldspot in the Southern hemisphere, the parameter estimates derived for the total energy density, tightness and vorticity from 3-yr data are found to be: Ω tot =0.43± 0.04, h = 0.32 +0.02 ¯0.13ω =9.7 +1.6 ¯1.5 x 10 -10 with orientation y =337○ +17 ¯23○ ). This template is preferred by a factor of roughly unity in log-evidence over a concordance cosmology alone. A Bianchi-type template is supported by the data only if its position on the sky is heavily restricted. All other Bianchi VII h templates including all right-handed models, are disfavoured. The low total energy density of the preferred template, implies a geometry that is incompatible with cosmologies inferred from recent CMB observations. Jaffe et al. found that extending the Bianchi model to include a term in ΩΔ creates a degeneracy in the Ω m¯ ΩΔ plane. We explore this region fully by MCMC and find that the degenerate likelihood contours do not intersect areas of parameter space that 1- or 3-yr WMAP data would prefer at any significance above 2σ. Thus we can confirm the conclusion that a physical Bianchi VII h model is not responsible for this signature, which we have treated in our analysis as merely a template.

Complex Data Processing: Fast Wavelet Analysis on the Sphere

In the general context of complex data processing, this article reviews a recent practical approach to the continuous wavelet formalism on the sphere. This formalism notably yields a correspondence principle which relates wavelets on the plane and on the sphere. Two fast algorithms are also presented for the analysis of signals on the sphere with steerable wavelets.

Non‐Gaussianity detections in the Bianchi VIIh corrected WMAP one‐year data made with directional spherical wavelets

Many of the current anomalies reported in the WMAP text (WMAP) one-year data disappear after ‘correcting’ for the best-fitting embedded Bianchi type VIIh component, albeit assuming no dark energy component. We investigate the effect of this Bianchi correction on the detections of non-Gaussianity in the WMAP data that we previously made using directional spherical wavelets. We confirm that the deviations from Gaussianity in the kurtosis of spherical Mexican hat wavelet coefficients are eliminated once the data are corrected for the Bianchi component, as previously discovered by Jaffe et al. This is due to the reduction of the cold spot at Galactic coordinates (l, b) = (209 ◦ , −57 ◦ ), which Cruz et al. claimed to be the sole source of nonGaussianity introduced in the kurtosis. Our previous detections of non-Gaussianity observed in the skewness of spherical wavelet coefficients are not reduced by the Bianchi correction. Indeed, the most significant detection of non-Gaussianity made with the spherical real Morlet wavelet at a significant level of 98.4 per cent remains (using a very conservative method to estimate the significance). Furthermore, we perform preliminary tests to determine if foregrounds or systematics are the source of this non-Gaussian signal, concluding that it is unlikely that these factors are responsible. We make our code to simulate Bianchi-induced temperature fluctuations publicly available.

Detection of the ISW effect and corresponding dark energy constraints made with directional spherical wavelets

Using a directional spherical wavelet analysis we detect the integrated Sachs-Wolfe (ISW) effect, indicated by a positive correlation between the Wilkinson Microwave Anisotropy Probe (WMAP) and NRAO VLA Sky Survey (NVSS) data, at the 3.9 sigma level. In a flat universe the detection of the ISW effect provides direct and independent evidence for dark energy. Moreover, we use our detection to constrain the dark energy density Omega_Lambda. We obtain estimates for Omega_Lambda consistent with other analysis techniques and data sets and rule out a zero cosmological constant at greater than 99% significance.

Bianchi VIIh models and the cold spot texture

We have returned to our previous Bianchi VII h analysis in light of the Cruz et al. suggestion that the cold spot observed near the Southern Galactic Pole may be a remnant temperature perturbation of a cosmic texture. Bridges et al. used two favoured left-handed Bianchi VII h templates with restricted prior probabilities so that the template was centred close to the cold spot. Using Wilkinson Microwave Anisotropy Probe (WMAP) data corrected for the texture fit we have now re-examined both models to assess any changes to these conclusions. We find that both models are left almost entirely unconstrained by the data and consequently exhibit significantly reduced Bayesian evidences. Both models are now disfavoured by the data. This result reinforces our previous assertion that the cold spot was significant in the detection of a Bianchi VII h component.