Sifting convolution on the sphere

A novel spherical convolution is defined through the sifting property of the Dirac delta on the sphere. The so-called sifting convolution is defined by the inner product of one function with a translated version of another, but with the adoption of an alternative translation operator on the sphere. This translation operator follows by analogy with the Euclidean translation when viewed in harmonic space. The sifting convolution satisfies a variety of desirable properties that are lacking in alternate definitions, namely: it supports directional kernels; it has an output which remains on the sphere; and is efficient to compute. An illustration of the sifting convolution on a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering, while its output remains on the sphere

Sparse Image Reconstruction on the Sphere: Analysis and Synthesis

We develop techniques to solve ill-posed inverse problems on the sphere by sparse regularization, exploiting sparsity in both axisymmetric and directional scale-discretized wavelet space. Denoising, in painting, and deconvolution problems and combinations thereof, are considered as examples. Inverse problems are solved in both the analysis and synthesis settings, with a number of different sampling schemes. The most effective approach is that with the most restricted solution-space, which depends on the interplay between the adopted sampling scheme, the selection of the analysis/synthesis problem, and any weighting of the ℓ1 norm appearing in the regularization problem. More efficient sampling schemes on the sphere improve reconstruction fidelity by restricting the solution-space and also by improving sparsity in wavelet space. We apply the technique to denoise Planck 353-GHz observations, improving the ability to extract the structure of Galactic dust emission, which is important for studying Galactic magnetism

Localisation of directional scale-discretised wavelets on the sphere

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets

Novel perspectives gained from new reconstruction algorithms

Since the 1970s, much of traditional interferometric imaging has been built around variations of the CLEAN algorithm, in both terminology, methodology, and algorithm development. Recent developments in applying new algorithms from convex optimization to interferometry has allowed old concepts to be viewed from a new perspective, ranging from image restoration to the development of computationally distributed algorithms. We present how this has ultimately led the authors to new perspectives in wide-field imaging, allowing for the first full individual non-coplanar corrections applied during imaging over extremely wide-fields of view for the Murchison Widefield Array (MWA) telescope. Furthermore, this same mathematical framework has provided a novel understanding of wide-band polarimetry at low frequencies, where instrumental channel depolarization can be corrected through the new δ λ2 -projection algorithm. This is a demonstration that new algorithm development outside of traditional radio astronomy is valuable for the new theoretical and practical perspectives gained. These perspectives are timely with the next generation of radio telescopes coming online

Wavelet reconstruction of E and B modes for CMB polarisation and cosmic shear analyses

We present new methods for mapping the curl-free (E-mode) and divergence-free (B-mode) components of spin 2 signals using spin directional wavelets. Our methods are equally applicable to measurements of the polarisation of the cosmic microwave background (CMB) and the shear of galaxy shapes due to weak gravitational lensing. We derive pseudo and pure wavelet estimators, where E-B mixing arising due to incomplete sky coverage is suppressed in wavelet space using scale- and orientation-dependent masking and weighting schemes. In the case of the pure estimator, ambiguous modes (which have vanishing curl and divergence simultaneously on the incomplete sky) are also cancelled. On simulations, we demonstrate the improvement (i.e., reduction in leakage) provided by our wavelet space estimators over standard harmonic space approaches. Our new methods can be directly interfaced in a coherent and computationally-efficient manner with component separation or feature extraction techniques that also exploit wavelets

Second-Generation Curvelets on the Sphere

Curvelets are efficient to represent highly anisotropic signal content, such as a local linear and curvilinear structure. First-generation curvelets on the sphere, however, suffered from blocking artefacts. We present a new second-generation curvelet transform, where scale-discretized curvelets are constructed directly on the sphere. Scale-discretized curvelets exhibit a parabolic scaling relation, are well localized in both spatial and harmonic domains, support the exact analysis and synthesis of both scalar and spin signals, and are free of blocking artefacts. We present fast algorithms to compute the exact curvelet transform, reducing computational complexity from O(L5) to O(L3 log2 L) for signals band limited at L. The implementation of these algorithms is made publicly available. Finally, we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve-like features in natural spherical images

Sparse Bayesian mass-mapping with uncertainties: Full sky observations on the celestial sphere

To date weak gravitational lensing surveys have typically been restricted to small fields of view, such that the flat-sky approximation has been sufficiently satisfied. However, with Stage IV surveys (e.g. LSST and Euclid) imminent, extending mass-mapping techniques to the sphere is a fundamental necessity. As such, we extend the sparse hierarchical Bayesian massmapping formalism presented in previous work to the spherical sky. For the first time, this allows us to construct maximum a posteriori spherical weak lensing dark-matter mass-maps, with principled Bayesian uncertainties, without imposing or assuming Gaussianty. We solve the spherical mass-mapping inverse problem in the analysis setting adopting a sparsity promoting Laplacetype wavelet prior, though this theoretical framework supports all log-concave posteriors. Our spherical mass-mapping formalism facilitates principled statistical interpretation of reconstructions. We apply our framework to convergence reconstruction on high resolution N-body simulations with pseudo-Euclid masking, polluted with a variety of realistic noise levels, and show a significant increase in reconstruction fidelity compared to standard approaches. Furthermore, we perform the largest joint reconstruction to date of the majority of publicly available shear observational data sets (combining DESY1, KiDS450, and CFHTLens) and find that our formalism recovers a convergence map with significantly enhanced small-scale detail. Within our Bayesian framework we validate, in a statistically rigorous manner, the community’s intuition regarding the need to smooth spherical Kaiser-Squires estimates to provide physically meaningful convergence maps. Such approaches cannot reveal the small-scale physical structures that we recover within our framework

Sparse Bayesian mass mapping with uncertainties: peak statistics and feature locations

Weak lensing convergence maps – upon which higher order statistics can be calculated – can be recovered from observations of the shear field by solving the lensing inverse problem. For typical surveys this inverse problem is ill-posed (often seriously) leading to substantial uncertainty on the recovered convergence maps. In this paper we propose novel methods for quantifying the Bayesian uncertainty in the location of recovered features and the uncertainty in the cumulative peak statistic – the peak count as a function of signal-to-noise ratio (SNR). We adopt the sparse hierarchical Bayesian mass-mapping framework developed in previous work, which provides robust reconstructions and principled statistical interpretation of reconstructed convergence maps without the need to assume or impose Gaussianity. We demonstrate our uncertainty quantification techniques on both Bolshoi N-body (cluster scale) and Buzzard V-1.6 (large-scale structure) N-body simulations. For the first time, this methodology allows one to recover approximate Bayesian upper and lower limits on the cumulative peak statistic at well-defined confidence levels

Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations

For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT

Iterative Residual Fitting for Spherical Harmonic Transform of Band-Limited Signals on the Sphere: Generalization and Analysis

We present the generalized iterative residual fitting (IRF) for the computation of the spherical harmonic transform (SHT) of band-limited signals on the sphere. The proposed method is based on the partitioning of the subspace of band-limited signals into orthogonal subspaces. There exist sampling schemes on the sphere which support accurate computation of SHT. However, there are applications where samples~(or measurements) are not taken over the predefined grid due to nature of the signal and/or acquisition set-up. To support such applications, the proposed IRF method enables accurate computation of SHTs of signals with randomly distributed sufficient number of samples. In order to improve the accuracy of the computation of the SHT, we also present the so-called multi-pass IRF which adds multiple iterative passes to the IRF. We analyse the multi-pass IRF for different sampling schemes and for different size partitions. Furthermore, we conduct numerical experiments to illustrate that the multi-pass IRF allows sufficiently accurate computation of SHTs