Search

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S 2 and rotation group SO (3) , i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d -functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT ▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity. • Differentiable and accelerated spherical harmonic and Wigner transforms. • Novel Wigner d-function recursion which is stable and scalable to high resolution. • Highly efficient gradient propagation scheme for harmonic analysis. • Open sourced and professionally developed JAX software package. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Next-generation radio interferometers, such as the Square Kilometre Array, will revolutionize our understanding of the Universe through their unprecedented sensitivity and resolution. However, to realize these goals significant challenges in image and data processing need to be overcome. The standard methods in radio interferometry for reconstructing images, such as CLEAN, have served the community well over the last few decades and have survived largely because they are pragmatic. However, they produce reconstructed interferometric images that are limited in quality and scalability for big data. In this work, we apply and evaluate alternative interferometric reconstruction methods that make use of state-of-the-art sparse image reconstruction algorithms motivated by compressive sensing, which have been implemented in the PURIFY software package. In particular, we implement and apply the proximal alternating direction method of multipliers algorithm presented in a recent article. First, we assess the impact of the interpolation kernel used to perform gridding and degridding on sparse image reconstruction. We find that the Kaiser-Bessel interpolation kernel performs as well as prolate spheroidal wave functions while providing a computational saving and an analytic form. Secondly, we apply PURIFY to real interferometric observations from the Very Large Array and the Australia Telescope Compact Array and find that images recovered by PURIFY are of higher quality than those recovered by CLEAN. Thirdly, we discuss how PURIFY reconstructions exhibit additional advantages over those recovered by CLEAN. The latest version of PURIFY, with developments presented in this work, is made publicly available. [ABSTRACT FROM AUTHOR]

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 polarization 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. [ABSTRACT FROM AUTHOR]

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 polarization 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. [ABSTRACT FROM AUTHOR]

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 polarization 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. [ABSTRACT FROM AUTHOR]

We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitedness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and Fourier-Laguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating-point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example. [ABSTRACT FROM PUBLISHER]

We develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension. The fundamental property of any sampling theorem is the number of samples required to represent a band-limited signal. To represent exactly a signal on the sphere band-limited at L, all sampling theorems on the sphere require \cal O(L^2) samples. However, our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical, but smaller, number of samples than the Gauss-Legendre sampling theorem. The complexity of our algorithms scale as \cal O(L^3), however, the continual use of fast Fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably, resulting in algorithms that are fast. Furthermore, we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational complexity or computation time. We make our implementation of these algorithms available publicly and perform numerical experiments demonstrating their speed and accuracy up to very high band-limits. Finally, we highlight the advantages of our sampling theorem in the context of potential applications, notably in the field of compressive sampling. [ABSTRACT FROM AUTHOR]

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 et al. 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 Górski, provides a savings of O(√Npix) 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. [ABSTRACT FROM AUTHOR]

The 21-cm hyperfine line of neutral hydrogen is a useful tool to probe the conditions of the Universe during the Dark Ages, Cosmic Dawn, and the Epoch of Reionization. In most of the current calculations, the 21-cm line signals at given frequencies are computed, using an integrated line-of-sight line opacity, with the correction for cosmological expansion. These calculations have not fully captured the line and continuum interactions in the radiative transfer, in response to evolution of the radiation field and the variations of thermal and dynamic properties of the line-of-sight medium. We construct a covariant formulation for the radiative transfer of the 21-cm line and derive the cosmological 21-cm line radiative transfer (C21LRT) equation. The formulation properly accounts for local emission and absorption processes and the interaction between the line and continuum when the radiation propagates across the expanding Universe to the present observer. Our C21LRT calculations show that methods simply summing the line optical depth could lead to error of 5  per cent in the 21-cm signals for redshift z ∼ 12–35 and of |$\gt 10~{{\ \rm per\ cent}}$| for redshift z ≲ 8. Proper covariant radiative transfer is therefore necessary for producing correct theoretical templates for extracting information of the structural evolution of the Universe through the Epoch of Reionization from the 21-cm tomographic data. [ABSTRACT FROM AUTHOR]

The emerging generation of radio interferometric (RI) telescopes, such as the Square Kilometre Array (SKA), will acquire massive volumes of data and transition radio astronomy to a big-data era. The ill-posed inverse problem of imaging the raw visibilities acquired by RI telescopes will become significantly more computationally challenging, particularly in terms of data storage and computational cost. Current RI imaging methods, such as CLEAN, its variants, and compressive sensing approaches (i.e. sparse regularization), have yielded excellent reconstruction fidelity. However, scaling these methods to big-data remains difficult if not impossible in some cases. All state-of-the-art methods in RI imaging lack the ability to process data streams as they are acquired during the data observation stage. Such approaches are referred to as online processing methods. We present an online sparse regularization methodology for RI imaging. Image reconstruction is performed simultaneously with data acquisition, where observed visibilities are assimilated into the reconstructed image as they arrive and then discarded. Since visibilities are processed online, good reconstructions are recovered much faster than standard (offline) methods that cannot start until the data acquisition stage completes. Moreover, the online method provides additional computational savings and, most importantly, dramatically reduces data storage requirements. Theoretically, the reconstructed images are of the same fidelity as those recovered by the equivalent offline approach and, in practice, very similar reconstruction fidelity is achieved. We anticipate that online imaging techniques, as proposed here, will be critical in scaling RI imaging to the emerging big-data era of radio astronomy. [ABSTRACT FROM AUTHOR]

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, for massive data sizes, like those anticipated from the Square Kilometre Array, it will be difficult if not impossible to apply any MCMC technique due to its inherent computational cost. We formulate Bayesian inference problems with sparsity-promoting priors (motivated by compressive sensing), for which we recover maximum a posteriori (MAP) point estimators of radio interferometric images by convex optimization. Exploiting recent developments in the theory of probability concentration, we quantify uncertainties by post-processing the recovered MAP estimate. Three strategies to quantify uncertainties are developed: (i) highest posterior density credible regions, (ii) local credible intervals (cf. error bars) for individual pixels and superpixels, and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. Our MAP-based methods are approximately 105 times faster computationally than state-of-the-art MCMC methods and, in addition, support highly distributed and parallelized algorithmic structures. For the first time, our MAP-based techniques provide a means of quantifying uncertainties for radio interferometric imaging for realistic data volumes and practical use, and scale to the emerging big data era of radio astronomy. [ABSTRACT FROM AUTHOR]

Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Since radio interferometric imaging requires solving a high-dimensional, ill-posed inverse problem, uncertainty quantification is difficult but also critical to the accurate scientific interpretation of radio observations. Statistical sampling approaches to perform Bayesian inference, like Markov chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, traditional high-dimensional sampling methods are generally limited to smooth (e.g. Gaussian) priors and cannot be used with sparsity-promoting priors. Sparse priors, motivated by the theory of compressive sensing, have been shown to be highly effective for radio interferometric imaging. In this article proximal MCMC methods are developed for radio interferometric imaging, leveraging proximal calculus to support non-differential priors, such as sparse priors, in a Bayesian framework. Furthermore, three strategies to quantify uncertainties using the recovered posterior distribution are developed: (i) local (pixel-wise) credible intervals to provide error bars for each individual pixel; (ii) highest posterior density credible regions; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. [ABSTRACT FROM AUTHOR]

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, 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 \ell 1 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. [ABSTRACT FROM PUBLISHER]

We formulate and solve the analog of Slepian spatial-spectral concentration problem on the three-dimensional ball. Both the standard Fourier–Bessel and also the Fourier–Laguerre spectral domains are considered since the latter exhibits a number of practical advantages such as spectral decoupling and exact computation. The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the space-bandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used to represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeed permits a sparse representation of concentrated signals. [ABSTRACT FROM AUTHOR]

We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at L using only L^2 samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The number of samples required in our scheme is a factor of two or four fewer than existing techniques, which require either 2L^2 or 4L^2 samples. We note, however, that we do not recover a sampling theorem on the sphere, where spherical harmonic transforms are theoretically exact. Nevertheless, we achieve high accuracy even for very large band-limits. For our optimal-dimensionality sampling scheme, we develop a fast and accurate algorithm to compute the spherical harmonic transform (and inverse), with computational complexity comparable with existing schemes in practice. We conduct numerical experiments to study in detail the stability, accuracy and computational complexity of the proposed transforms. We also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications. [ABSTRACT FROM PUBLISHER]

Despite the technological advancements in Virtual Reality (VR), users are constantly combating feelings of nausea and disorientation, the so-called cybersickness. Cybersickness symptoms cause severe discomfort and hinder the immersive VR experience. Here we investigated cybersickness in 360-degree head-mounted display VR. In traditional 360-degree VR experiences, translational movement in the real world is not reflected in the virtual world, and therefore self-motion information is not corroborated by matching visual and vestibular cues, which may trigger symptoms of cybersickness. We evaluated whether a new Artificial Intelligence (AI) software designed to supplement the 360-degree VR experience with artificial six-degrees-of-freedom motion may reduce cybersickness. Explicit (simulator sickness questionnaire and Fast Motion Sickness (FMS) rating) and implicit (heart rate) measurements were used to evaluate cybersickness symptoms during and after 360-degree VR exposure. Simulator sickness scores showed a significant reduction in feelings of nausea during the AI-supplemented six-degrees-of-freedom motion VR compared to traditional 360-degree VR. However, six-degrees-of-freedom motion VR did not reduce oculomotor or disorientation measures of sickness. No changes were observed in FMS and heart rate measures. Improving the congruency between visual and vestibular cues in 360-degree VR, as provided by the AI-supplemented six-degrees-of-freedom motion system considered, is essential for a more engaging, immersive and safe VR experience, which is critical for educational, cultural and entertainment applications. [ABSTRACT FROM AUTHOR]

Polarization of radiation is a powerful tool to study cosmic magnetism and analysis of polarization can be used as a diagnostic tool for large-scale structures. In this work, we present a solid theoretical foundation for using polarized light to investigate large-scale magnetic field structures: the cosmological polarized radiative transfer (CPRT) formulation. The CPRT formulation is fully covariant. It accounts for cosmological and relativistic effects in a self-consistent manner and explicitly treats Faraday rotation, as well as Faraday conversion, emission, and absorption processes. The formulation is derived from the first principles of conservation of phase–space volume and photon number. Without loss of generality, we consider a flat Friedmann–Robertson–Walker (FRW) space–time metric and construct the corresponding polarized radiative transfer equations. We propose an all-sky CPRT calculation algorithm, based on a ray-tracing method, which allows cosmological simulation results to be incorporated and, thereby, model templates of polarization maps to be constructed. Such maps will be crucial in our interpretation of polarized data, such as those to be collected by the Square Kilometer Array (SKA). We describe several tests which are used for verifying the code and demonstrate applications in the study of the polarization signatures in different distributions of electron number density and magnetic fields. We present a pencil-beam CPRT calculation and an all-sky calculation, using a simulated galaxy cluster or a model magnetized universe obtained from GCMHD+ simulations as the respective input structures. The implications on large-scale magnetic field studies are discussed; remarks on the standard methods using rotation measure are highlighted. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the commonly used limiting cases, or approximations, for two-point cosmic-shear statistics. We discuss the most prominent assumptions in this statistic: the flat-sky (small angle limit), the Limber (Bessel-to-delta function limit) and the Hankel transform (large ℓ-mode limit) approximations; that the vast majority of cosmic-shear results to date have used simultaneously. We find that the combined effect of these approximations can suppress power by ≳ 1 per cent on scales of ℓ ≲ 40. A fully non-approximated cosmic-shear study should use a spherical-sky, non-Limber-approximated power spectrum analysis and a transform involving Wigner small-d matrices in place of the Hankel transform. These effects, unaccounted for, would constitute at least 11 per cent of the total budget for systematic effects for a power spectrum analysis of a Euclid-like experiment; but they are unnecessary. [ABSTRACT FROM AUTHOR]

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 \mathcal O(L^5) to \mathcal O(L^3\log 2{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. [ABSTRACT FROM AUTHOR]

We present Spin-SILC, a new foreground component separation method that accurately extracts the cosmic microwave background (CMB) polarization E and B modes from raw multifrequency Stokes Q and U measurements of the microwave sky. Spin-SILC is an internal linear combination method that uses spin wavelets to analyse the spin-2 polarization signal P = Q + iU. The wavelets are additionally directional (non-axisymmetric). This allows different morphologies of signals to be separated and therefore the cleaning algorithm is localized using an additional domain of information. The advantage of spin wavelets over standard scalar wavelets is to simultaneously and self-consistently probe scales and directions in the polarization signal P = Q + iU and in the underlying E and B modes, therefore providing the ability to perform component separation and E-B decomposition concurrently for the first time. We test Spin-SILC on full-mission Planck simulations and data and show the capacity to correctly recover the underlying cosmological E and B modes. We also demonstrate a strong consistency of our CMB maps with those derived from existing component separation methods. Spin-SILC can be combined with the pseudo- and pure E-B spin wavelet estimators presented in a companion paper to reliably extract the cosmological signal in the presence of complicated sky cuts and noise. Therefore, it will provide a computationally efficient method to accurately extract the CMB E and B modes for future polarization experiments. [ABSTRACT FROM AUTHOR]

In the context of next-generation radio telescopes, like the Square Kilometre Array (SKA), the efficient processing of large-scale data sets is extremely important. Convex optimization tasks under the compressive sensing framework have recently emerged and provide both enhanced image reconstruction quality and scalability to increasingly larger data sets. We focus herein mainly on scalability and propose two new convex optimization algorithmic structures able to solve the convex optimization tasks arising in radio-interferometric imaging. They rely on proximal splitting and forward-backward iterations and can be seen, by analogy, with the clean major-minor cycle, as running sophisticated clean-like iterations in parallel in multiple data, prior, and image spaces. Both methods support any convex regularization function, in particular, the well-studied ℓ1 priors promoting image sparsity in an adequate domain. Tailored for big-data, they employ parallel and distributed computations to achieve scalability, in terms of memory and computational requirements. One of them also exploits randomization, over data blocks at each iteration, offering further flexibility. We present simulation results showing the feasibility of the proposed methods as well as their advantages compared to state-of-the-art algorithmic solvers. Our matlab code is available online on GitHub. [ABSTRACT FROM AUTHOR]

We present a new framework for testing the isotropy of the Universe using cosmic microwave background data, building on the nested-sampling ANICOSMO code. Uniquely, we are able to constrain the scalar, vector and tensor degrees of freedom alike; previous studies only considered the vector mode (linked to vorticity). We employ Bianchi type VIIh cosmologies to model the anisotropic Universe, from which other types may be obtained by taking suitable limits. In a separate development, we improve the statistical analysis by including the effect of Bianchi power in the high-ℓ, as well as the low-ℓ, likelihood. To understand the effect of all these changes, we apply our new techniques to Wilkinson Microwave Anisotropy Probe data. We find no evidence for anisotropy, constraining shear in the vector mode to (σV/H)0 < 1.7 × 10-10 (95 per cent confidence level). For the first time, we place limits on the tensor mode; unlike other modes, the tensor shear can grow from a near-isotropic early Universe. The limit on this type of shear is (σT,reg/H)0 < 2.4 × 10-7 (95 per cent confidence level). [ABSTRACT FROM AUTHOR]

We present new clean maps of the cosmic microwave background (CMB) temperature anisotropies (as measured by Planck) constructed with a novel internal linear combination (ILC) algorithm using directional, scale-discretized wavelets - scale-discretized, directional wavelet ILC or Scale-discretised, directional wavelet Internal Linear Combination (SILC). Directional wavelets, when convolved with signals on the sphere, can separate the anisotropic filamentary structures which are characteristic of both the CMB and foregrounds. Extending previous component separation methods, which use the frequency, spatial and harmonic signatures of foregrounds to separate them from the cosmological background signal, SILC can additionally use morphological information in the foregrounds and CMB to better localize the cleaning algorithm. We test the method on Planck data and simulations, demonstrating consistency with existing component separation algorithms, and discuss how to optimize the use of morphological information by varying the number of directional wavelets as a function of spatial scale. We find that combining the use of directional and axisymmetric wavelets depending on scale could yield higher quality CMB temperature maps. Our results set the stage for the application of SILC to polarization anisotropies through an extension to spin wavelets. [ABSTRACT FROM AUTHOR]

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented. [ABSTRACT FROM AUTHOR]

• Proposed a segmentation framework for images on the sphere, for the first time. • It is compatible with any type of spherical wavelets. • It is efficient and flexible for any type of spherical images. • Devised a new hybrid directional-curvelet wavelet construction on the sphere. • Numerical results with a series of applications are presented. Segmentation, a useful/powerful technique in pattern recognition, is the process of identifying object outlines within images. There are a number of efficient algorithms for segmentation in Euclidean space that depend on the variational approach and partial differential equation modelling. Wavelets have been used successfully in various problems in image processing, including segmentation, inpainting, noise removal, super-resolution image restoration, and many others. Wavelets on the sphere have been developed to solve such problems for data defined on the sphere, which arise in numerous fields such as cosmology and geophysics. In this work, we propose a wavelet-based method to segment images on the sphere, accounting for the underlying geometry of spherical data. Our method is a direct extension of the tight-frame based segmentation method used to automatically identify tube-like structures such as blood vessels in medical imaging. It is compatible with any arbitrary type of wavelet frame defined on the sphere, such as axisymmetric wavelets, directional wavelets, curvelets, and hybrid wavelet constructions. Such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process. In particular, directional wavelets and curvelets, which were designed to efficiently capture directional signal content, provide additional advantages in segmenting images containing prominent directional and curvilinear features. We present several numerical experiments, applying our wavelet-based segmentation method, as well as the common K-means method, on real-world spherical images, including an Earth topographic map, a light probe image, solar data-sets, and spherical retina images. These experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images, including those with prominent directional features. Moreover, our algorithm is efficient with convergence usually within a few iterations. [ABSTRACT FROM AUTHOR]

A fundamental assumption in the standard model of cosmology is that the Universe is isotropic on large scales. Breaking this assumption leads to a set of solutions to Einstein's field equations, known as Bianchi cosmologies, only a subset of which have ever been tested against data. For the first time, we consider all degrees of freedom in these solutions to conduct a general test of isotropy using cosmic microwave background temperature and polarization data from Planck. For the vector mode (associated with vorticity), we obtain a limit on the anisotropic expansion of (σV/H)0<4.7×10−11 (95% C.L.), which is an order of magnitude tighter than previous Planck results that used cosmic microwave background temperature only. We also place upper limits on other modes of anisotropic expansion, with the weakest limit arising from the regular tensor mode, (σT,reg/H)0<1.0×10−6 (95% C.L.). Including all degrees of freedom simultaneously for the first time, anisotropic expansion of the Universe is strongly disfavored, with odds of 121 000:1 against. [ABSTRACT FROM AUTHOR]