HEALPix

Splitting a flat surface into discrete pixels (picture elements) is fairly easy. You can just use a grid. Pixelating a curved surface (in the case of astronomy, a sphere) is a bit more complicated.

Some desirable features might be fairly obvious; having equal area pixels tends to make things easier (for example, this property means that the signal to noise of each pixel is the same). Other desirable features only make sense in the context in which these maps will be used; cosmologists need to calculate Legendgre polynomials¹ and for computational reasons want the pixels to be at as few distinct latitudes as possible.

The current preferred method for pixelating a sphere is HEALPix (Hierarchical Equal Area isoLatitude pixelization) (Górski et al. 2005). As the name suggests, each pixel in this scheme has equal area and the centers lie only on a few latitudes. Hierarchical means that pixels that are nearby in space live nearby in the data structure – this makes it easy to pull out and analyze subregions.

HEALPix divides the sphere into 12 base regions. Each of these can then be subdivided, usually by a power of 2 to allow for a simple hierarchy, into the final pixels. These pixels are ordered either by the “ring” or “nest” scheme; which option is better depends on what the data will be used for. The latest maps of the CMB from Planck use 2048 subdivisions, resulting in maps with roughly 50 million pixels. To get a sense of the organization and shapes of the pixels, here are some examples with slightly fewer subdivisions than that.

The shapes, particularly of the pixels around the poles can be seen more easily on a globe.

Note that the red borders are not quite right when there are few subdivisions. Getting the interpolation right is a bit tricky!

References