Localisation of directional scale-discretised wavelets on the sphere

Abstract

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.

28 pages, 8 figures, minor changes to match version accepted for publication by ACHA

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Cited by in corpus (28)

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• Spherical Bayesian mass-mapping with uncertainties: full sky observations on the celestial sphere
• Wavelet-Bayesian inference of cosmic strings embedded in the cosmic microwave background
• Non-Gaussian modelling and statistical denoising of Planck dust polarization full-sky maps using scattering transforms
• The Atacama Cosmology Telescope: Map-Based Noise Simulations for DR6
• SILC: a new Planck Internal Linear Combination CMB temperature map using directional wavelets
• Wavelet reconstruction of E and B modes for CMB polarisation and cosmic shear analyses
• Sparse image reconstruction on the sphere: analysis and synthesis
• Spin-SILC: CMB polarisation component separation with spin wavelets
• Differentiable and accelerated spherical harmonic and Wigner transforms
• A Novel CMB Component Separation Method: Hierarchical Generalized Morphological Component Analysis
• Generative models of astrophysical fields with scattering transforms on the sphere
• Wavelet-Based Segmentation on the Sphere
• Posterior sampling for inverse imaging problems on the sphere in seismology and cosmology
• Slepian Scale-Discretised Wavelets on the Sphere
• Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs
• Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows
• Sparse image reconstruction on the sphere: a general approach with uncertainty quantification
• Spherical framelets from spherical designs
• Spatial-Slepian Transform on the Sphere
• Efficient Generalized Spherical CNNs
• Directional polynomial wavelets on spheres
• Polar Wavelets in Space
• A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations
• Bayesian variational regularization on the ball
• Scale-discretised ridgelet transform on the sphere
• Edge detection with polynomial frames on the sphere