Fast Convolution on the Sphere

Abstract

We propose fast, exact and efficient algorithms for the convolution of two arbitrary functions on the sphere which speed up computations by a factor \order{\sqrt{N}} compared to present methods where $N$ is the number of pixels. No simplifying assumptions are made other than bandlimitation. This reduces typical computation times for convolving the full sky with the asymmetric beam pattern of a megapixel Cosmic Microwave Background (CMB) mission from months to minutes. Our methods enable realistic simulation and careful analysis of data from such missions, taking into account the effects of asymmetric ``point spread functions'' and far side lobes of the physical beam. While motivated by CMB studies, our methods are general and hence applicable to the convolution or filtering of any scalar field on the sphere with an arbitrary, asymmetric kernel. We show in an appendix that the same ideas can be applied to the inverse problems of map-making and beam reconstruction by similarly accelerating the {\em transpose convolution} which is needed for the iterative solution of the normal equations.

7 pages, 1 figure; results unchanged, minor changes to text. Submitted to Physical Review D

References in corpus (2)

• Asymmetric Beams in Cosmic Microwave Background Anisotropy Experiments
• All-sky convolution for polarimetry experiments

Cited by in corpus (72)

• Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis
• Cosmic Microwave Background Anisotropies
• Evidence of vorticity and shear at large angular scales in the WMAP data: a violation of cosmological isotropy?
• Myths and Truths Concerning Estimation of Power Spectra
• Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Profiles, Data Processing, Radiometer Characterization and Systematic Error Limits
• Xspect, estimation of the angular power spectrum by computing cross-power spectra with analytical error bars
• A high-significance detection of non-Gaussianity in the WMAP 1-year data using directional spherical wavelets
• Asymmetric Beams and CMB Statistical Anisotropy
• Planck 2013 results. VII. HFI time response and beams
• Exact reconstruction with directional wavelets on the sphere
• Detection of the ISW effect and corresponding dark energy constraints made with directional spherical wavelets
• Detection of the ISW effect and corresponding dark energy constraints
• Mapping the gravitational wave background
• Bianchi Type VII_h Models and the WMAP 3-year Data
• Libsharp - spherical harmonic transforms revisited
• A high-significance detection of non-Gaussianity in the WMAP 5-year data using directional spherical wavelets
• Fast and Efficient Template Fitting of Deterministic Anisotropic Cosmological Models Applied to WMAP Data
• A simulation pipeline for the Planck mission
• Fast directional continuous spherical wavelet transform algorithms
• Fast Pixel Space Convolution for CMB Surveys with Asymmetric Beams and Complex Scan Strategies: FEBeCoP
• BICEP2 III: Instrumental Systematics
• Making CMB temperature and polarization maps with Madam
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Signal Contamination from Sidelobe Pickup
• Hi intensity mapping with MeerKAT: Primary beam effects on foreground cleaning
• Continuous phase-space representations for finite-dimensional quantum states and their tomography
• Fast directional correlation on the sphere with steerable filters
• Destriping CMB temperature and polarization maps
• Fast and Exact Spin-s Spherical Harmonic Transforms
• Fast, exact CMB power spectrum estimation for a certain class of observational strategies
• Detecting Sunyaev-Zel'dovich clusters with PLANCK: II. Foreground components and optimised filtering schemes
• Comparison of map-making algorithms for CMB experiments
• Complex data processing: fast wavelet analysis on the sphere
• Efficient analysis and representation of geophysical processes using localized spherical basis functions
• Analysis of cosmic microwave background data on an incomplete sky
• Elliptical Beams in CMB Temperature and Polarization Anisotropy Experiments: An Analytic Approach
• BeyondPlanck III. Commander3
• Making Maps from Planck LFI 30GHz Data
• Deconvolution Map-Making for Cosmic Microwave Background Observations
• Constraining topology in harmonic space
• Algorithm for the evaluation of reduced Wigner matrices
• Optimal filters for detecting cosmic bubble collisions
• On the computation of directional scale-discretized wavelet transforms on the sphere
• BeyondPlanck I. Global Bayesian analysis of the Planck Low Frequency Instrument data
• Harmonic analysis of cosmic microwave background data I: ring reductions and point-source catalogue
• CMB power spectrum estimation using noncircular beams
• Harmonic analysis of cosmic microwave background data II: from ring-sets to the sky
• The Atacama Cosmology Telescope: Measurement and Analysis of 1D Beams for DR4
• The effect of asymmetric beams in the Wilkinson Microwave Anisotropy Probe experiment
• Probing frequency-dependent half-wave plate systematics for CMB experiments with full-sky beam convolution simulations
• From BeyondPlanck to Cosmoglobe: Preliminary $WMAP$ $Q$-band analysis
• Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics
• PReBeaM for Planck: A Polarized Regularized Beam Deconvolution Map-Making Method
• Asymfast, a method for convolving maps with asymmetric main beams
• Fast directional spatially localized spherical harmonic transform
• Full-sky beam convolution for cosmic microwave background applications
• Removing beam asymmetry bias in precision CMB temperature and polarisation experiments
• Sifting Convolution on the Sphere
• Filamentary Dust Polarization and the Morphology of Neutral Hydrogen Structures
• Spherical harmonic analysis of anisotropies in polarized stochastic gravitational wave background with interferometry experiments
• Non-Circular beam correction to the CMB power spectrum
• Full-sky correlation functions for CMB experiments with asymmetric window functions
• BeyondPlanck VIII. Efficient Sidelobe Convolution and Correction through Spin Harmonics
• Euclid preparation. LIX. Angular power spectra from discrete observations
• A search for concentric rings with unusual variance in the 7-year WMAP temperature maps using a fast convolution approach
• Estimating SI violation in CMB due to non-circular beam and complex scan in minutes
• ARKCoS: Artifact-Suppressed Accelerated Radial Kernel Convolution on the Sphere
• Pixel space convolution for cosmic microwave background experiments
• Spatial-Slepian Transform on the Sphere
• Efficient Generalized Spherical CNNs
• Compressed convolution
• Convolutions on the Sphere: Commutation with Differential Operators
• A phase-sensitive method for filtering on the sphere