We review scale-discretized wavelets on the sphere, which are directional and allow one to probe oriented structure in data defined on the sphere. Furthermore, scale-discretized wavelets allow in practice the exact synthesis of a signal from its wavelet coefficients. We present exact and efficient algorithms to compute the scale-discretized wavelet transform of band-limited signals on the sphere. These algorithms are implemented in the publicly available S2DW code. We release a new version of S2DW that is parallelized and contains additional code optimizations. Note that scale-discretized wavelets can be viewed as a directional generalization of needlets. Finally, we outline future improvements to the algorithms presented, which can be achieved by exploiting a new sampling theorem on the sphere developed recently by some of the authors.
Pressing questions in cosmology such as the nature of dark matter and dark energy can be addressed using large galaxy surveys, which measure the positions, properties and redshifts of galaxies in order to map the large-scale structure of the Universe. We review the Fourier-Laguerre transform, a novel transform in 3D spherical coordinates which is based on spherical harmonics combined with damped Laguerre polynomials and appropriate for analysing galaxy surveys. We also recall the construction of aglets, 3D wavelets obtained through a tiling of the Fourier-Laguerre space, which can be used to extract scale-dependent, spatially localised features on the ball. We exploit a sampling theorem to obtain exact Fourier-Laguerre and aglet transforms, such that band-limited signals can analysed and reconstructed at oating point accuracy on a nite number of voxels on the ball. We present a potential application of the aglet transform for nding voids in galaxy surveys and studying the large-scale structure of the Universe.
KEYWORDS: Spherical lenses, Optical spheres, Wavelets, Computing systems, Space operations, Signal processing, Physics, Astronomy, Signal analyzers, Current controlled current source
This paper considers the construction of Reproducing Kernel Hilbert Spaces (RKHS) on the sphere as an alternative to the conventional Hilbert space using the inner product that yields the L2(S2) function space of finite energy signals. In comparison with wavelet representations, which have multi-resolution properties on L2(S2), the representations that arise from the RKHS approach, which uses different inner products, have an overall smoothness constraint, which may offer advantages and simplifications in certain contexts. The key contribution of this paper is to construct classes of closed-form kernels, such as one based on the von Mises-Fisher distribution, which permits efficient inner product computation using kernel evaluations. Three classes of RKHS are defined: isotropic kernels and non-isotropic kernels both with spherical harmonic eigenfunctions, and general anisotropic kernels.
We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association
between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem
requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as
the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent
a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the
dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on
the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.
Dark energy dominates the energy density of our Universe, yet we know very little about its nature and origin.
Although strong evidence in support of dark energy is provided by the cosmic microwave background, the relic
radiation of the Big Bang, in conjunction with either observations of supernovae or of the large scale structure of
the Universe, the verification of dark energy by independent physical phenomena is of considerable interest. We
review works that, through a wavelet analysis on the sphere, independently verify the existence of dark energy by
detecting the integrated Sachs-Wolfe effect. The effectiveness of a wavelet analysis on the sphere is demonstrated
by the highly statistically significant detections of dark energy that are made. Moreover, the detection is used
to constrain properties of dark energy. A coherent picture of dark energy is obtained, adding further support to
the now well established cosmological concordance model that describes our Universe.
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