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The analysis of functions on domains with spherical topology occupies a central place in physical science and engineering disciplines. This is particularly apparent in the fields of astronomy, cosmology, geophysics, atomic and nuclear physics. In many cases the geometry is either dictated by the object under study or approximate spherical symmetry can be exploited to yield powerful perturbation methods. Practical limits for the purely analytical study of these problems create an urgent necessity for efficient and accurate numerical tools.

The simplicity of the spherical form belies the intricacy of global analysis on the sphere. There is no known point set which achieves the analogue of uniform sampling in Euclidean space and allows exact and invertible discrete spherical harmonic decompositions of arbitrary but band-limited functions. Any existing proposition of practical schemes for the discrete treatment of such functions on the sphere introduces some (hopefully tiny) systematic error dependent on the global properties of the point set. The goal is to minimise these errors and faithfully represent deterministic functions as well as realizations of random variates both in configuration and Fourier space while maintaining computational efficiency.

We illustrate these points using as an example the field which is particularly close to the authors' hearts, Cosmic Microwave Background (CMB) anisotropies. Here we are in the happy situation of expecting an explosion of available data within the next decade. The Microwave Anisotropy Probe (MAP) (NASA) and Planck Surveyor (ESA) missions are aiming to provide multi-frequency, high resolution, full sky measurements of the anisotropy in both temperature and polarization of the cosmic microwave background radiation. The ultimate data products of these missions -- multiple microwave sky maps, each of which will have to comprise more than $\sim $ 106 pixels in order to render the angular resolution of the instruments -- will present serious challenges to those involved in the analysis and scientific exploitation of the results of both surveys.

As we have learned while working with the COBE mission products, the digitised sky map is an essential intermediate stage in information processing between the entry point of data acquisition by the instruments -- very large time ordered data streams, and the final stage of astrophysical analysis -- typically producing a 'few' numerical values of physical parameters of interest. COBE-DMR sky maps (angular resolution of $7^\circ$ (FWHM) in three frequency bands, two channels each, 6144 pixels per map) were considered large at the time of their release.

As for future CMB maps, a whole sky CMB survey at the angular resolution of $\sim 10'$ (FWHM), discretised with a few pixels per resolution element (so that the discretisation effects on the signal are sub-dominant with respect to the effects of instrument's angular response), will require map sizes of at least $N_{pix} \sim $ a few $\times 1.5\, 10^6$ pixels. More pixels than that will be needed to represent the Planck-HFI higher resolution channels. This estimate, Npix, should be multiplied by the number of frequency bands (or, indeed, by the number of individual observing channels -- 74 in the case of Planck -- for the analysis work to be done before the final coadded maps are made for each frequency band) to render an approximate expected size of the already very compressed form of survey data which would be the input to the astrophysical analysis pipeline.

It appears to us that very careful attention ought to be given to devising high resolution CMB map structures which can maximally facilitate the forthcoming analyses of large size data sets, for the following reasons:

* It is clearly very easy to end up with an estimated size of many GBy for the data objects which would be directly involved in the science extraction part of the future CMB missions.
* Many essential scientific questions can only be answered by global studies of future data sets.

This document is an introduction to the properties of our proposed approach for a high resolution numerical representation of functions on the sphere -- the Hierarchical Equal Area and iso-Latitude Pixelization (HEALPix, see, and the associated multi-purpose computer software package.

Eric Hivon 2010-06-18
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