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Analysis Issues for Large CMB Data Sets

K. M. Gorski, E. Hivon, B. D. Wandelt

TL;DR

The paper addresses the challenge of analyzing very large, multi-frequency full-sky CMB data sets from MAP and Planck, where each map may contain $N_{\text{pix}} \sim \text{a few} \times 1.5 \times 10^6$ pixels. It introduces HEALPIX, a Hierarchical Equal Area iso-Latitude Pixelisation, which partitions the sphere into $N_{\text{pix}} = 12 N_{\text{side}}^2$ equal-area pixels with iso-latitude rings and two numbering schemes to support fast transforms and neighbor queries, including explicit boundary relations like $\cos \theta = a + b \phi$ (equator) and $\cos \theta = a + b / \phi^2$ (poles). The work documents a public software package and demonstrates HEALPIX's adoption by MAP and Planck for simulations and map analyses, highlighting its suitability for efficient spherical harmonic transforms and wavelet analyses. A brief non-Gaussianity digression contextualizes CMB statistics, illustrating the need for scalable, robust analysis methods in next-generation data sets.

Abstract

Multi-frequency, high resolution, full sky measurements of the anisotropy in both temperature and polarisation of the cosmic microwave background radiation are the goals of the satellite missions MAP (NASA) and Planck (ESA). The ultimate data products of these missions - multiple microwave sky maps, each of which will have to comprise more than 10^6 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. Some considerations of the relevant aspects of the mathematical structure of future CMB data sets are presented in this contribution. >>> for better on-screen rendition of the figures see <<< http://www.tac.dk/~healpix or http://www.mpa-garching.mpg.de/~cosmo/contributions.html

Analysis Issues for Large CMB Data Sets

TL;DR

The paper addresses the challenge of analyzing very large, multi-frequency full-sky CMB data sets from MAP and Planck, where each map may contain pixels. It introduces HEALPIX, a Hierarchical Equal Area iso-Latitude Pixelisation, which partitions the sphere into equal-area pixels with iso-latitude rings and two numbering schemes to support fast transforms and neighbor queries, including explicit boundary relations like (equator) and (poles). The work documents a public software package and demonstrates HEALPIX's adoption by MAP and Planck for simulations and map analyses, highlighting its suitability for efficient spherical harmonic transforms and wavelet analyses. A brief non-Gaussianity digression contextualizes CMB statistics, illustrating the need for scalable, robust analysis methods in next-generation data sets.

Abstract

Multi-frequency, high resolution, full sky measurements of the anisotropy in both temperature and polarisation of the cosmic microwave background radiation are the goals of the satellite missions MAP (NASA) and Planck (ESA). The ultimate data products of these missions - multiple microwave sky maps, each of which will have to comprise more than 10^6 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. Some considerations of the relevant aspects of the mathematical structure of future CMB data sets are presented in this contribution. >>> for better on-screen rendition of the figures see <<< http://www.tac.dk/~healpix or http://www.mpa-garching.mpg.de/~cosmo/contributions.html

Paper Structure

This paper contains 3 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Discretisation of Functions on a Sphere with HEALPIX for High Resolution Applications
  3. Digression: Non-Gaussianity in the DMR data?

Figures (2)

  • Figure 1: Orthographic view of HEALPIX division of a sphere. Overplot of equator and meridians illustrates octahedral symmetry of the HEALPIX construction. Light-gray shading shows one of eight (four north, and four south) identical polar base-resolution pixels. Dark-gray shading shows one of four identical equatorial base-resolution pixels. Moving clockwise from the upper left panel the grid is hierarchically subdivided with the grid resolution parameter equal to $N_{side} = \,1,\,2,\,4,\,8$, and the total number of pixels equal to $N_{pix} = 12 \times N_{side}^2 = \,12,\,48,\,192,\,768$. All pixel centers are located on $N_{ring} = 4 \times N_{side} - 1$ rings of constant latitude. Within each panel the areas of all pixels are identical.
  • Figure 2: Cylindrical projection of the HEALPIX division of a sphere and two natural pixel numbering schemes (ring and nested) allowed by HEALPIX. Both numbering schemes map the two dimensional distribution of discrete area elements on a sphere into the one dimensional, integer pixel number array, which is essential for computations involving data sets with very large total pixel numbers. >From top to bottom: Panel one (resolution parameter $N_{side} = 2$) and panel two ($N_{side} = 4$) show the ring scheme for pixel numbering, with the pixel number winding down from north to south pole through the consecutive isolatitude rings. Panel three (resolution parameter $N_{side} = 2$) and panel four ($N_{side} = 4$) show the nested scheme for pixel numbering within which the pixel number grows with consecutive hierarchical subdivisions on a tree structure seeded by the twelve base-resolution pixels.