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

Scale-discretised ridgelet transform on the sphere

We revisit the spherical Radon transform, also called the Funk-Radon transform, viewing it as an axisymmetric convolution on the sphere. Viewing the spherical Radon transform in this manner leads to a straightforward derivation of its spherical harmonic representation, from which we show the spherical Radon transform can be inverted exactly for signals exhibiting antipodal symmetry. We then construct a spherical ridgelet transform by composing the spherical Radon and scale-discretised wavelet transforms on the sphere. The resulting spherical ridgelet transform also admits exact inversion for antipodal signals. The restriction to antipodal signals is expected since the spherical Radon and ridgelet transforms themselves result in signals that exhibit antipodal symmetry. Our ridgelet transform is defined natively on the sphere, probes signal content globally along great circles, does not exhibit blocking artefacts, supports spin signals and exhibits an exact and explicit inverse transform. No alternative ridgelet construction on the sphere satisfies all of these properties. Our implementation of the spherical Radon and ridgelet transforms is made publicly available. Finally, we illustrate the effectiveness of spherical ridgelets for diffusion magnetic resonance imaging of white matter fibers in the brain.Comment: 5 pages, 4 figures, matches version accepted by EUSIPCO, code available at http://www.s2let.or

A novel sampling theorem on the rotation group

We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for publication. Code available at http://www.sothree.or

Sparse image reconstruction on the sphere: analysis and synthesis

We develop techniques to solve ill-posed inverse problems on the sphere by sparse regularisation, exploiting sparsity in both axisymmetric and directional scale-discretised wavelet space. Denoising, inpainting, 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 regularisation 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.Comment: 11 pages, 6 Figure

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

Directional spin wavelets on the sphere

We construct a directional spin wavelet framework on the sphere by generalising the scalar scale-discretised wavelet transform to signals of arbitrary spin. The resulting framework is the only wavelet framework defined natively on the sphere that is able to probe the directional intensity of spin signals. Furthermore, directional spin scale-discretised wavelets support the exact synthesis of a signal on the sphere from its wavelet coefficients and satisfy excellent localisation and uncorrelation properties. Consequently, directional spin scale-discretised wavelets are likely to be of use in a wide range of applications and in particular for the analysis of the polarisation of the cosmic microwave background (CMB). We develop new algorithms to compute (scalar and spin) forward and inverse wavelet transforms exactly and efficiently for very large data-sets containing tens of millions of samples on the sphere. By leveraging a novel sampling theorem on the rotation group developed in a companion article, only half as many wavelet coefficients as alternative approaches need be computed, while still capturing the full information content of the signal under analysis. Our implementation of these algorithms is made publicly available

All-sky radiative transfer and characterisation for cosmic structures

This thesis focuses on providing a solid theoretical foundation and the associated methodologies for the studies of cosmic magnetism and cosmological reionisation. It develops covariant formalisms of cosmological radiative transport of (i) polarised continuum radiation, and (ii) 21-cm line of neutral hydrogen that calculate, from first principles, the polarisation arising from the emergence and evolution of cosmic magnetic fields and the tomographic 21-cm line signals associated with cosmological reionisation, respectively. The two formalisms, namely the cosmological polarised radiative transfer (CPRT) and the cosmological 21-cm line radiative transfer (C21LRT), self-consistently account for the relevant radiation processes, relativistic and cosmological effects along a ray transported in an expanding, evolving Universe. Their all-sky algorithms adopt a ray-tracing method and a post-processing approach by which complex physical models, such as those obtained from cosmological simulations, can be accounted for in the radiative transfer calculations. The power of the CPRT calculations to compute unambiguous point-to-point polarisation of large-scale structures, such as a 3D simulated galaxy cluster and a modelled magnetised universe, is demonstrated. The ability of the C21LRT formulation to calculate the 21-cm line spectra across cosmic time, with full accounts of the essential cosmological radiative transfer effects, is verified. Furthermore, a new spherical curvelet transform for efficient extraction of directional, elongated features within spherical data is constructed. It is particularly useful for the studies in wide-field astronomical research, such as analyses of the data of continuum polarisation and the structured 21-cm line from all-sky surveys or the CPRT and C21LRT calculations. The formulations, methodologies and techniques developed in this work together establish a solid framework within which reliable theoretical predictions and robust data characterisation can be made, ultimately laying a foundation for the meaningful physical interpretation of observations and studying the structural evolution of the magnetic ionised Universe