Welcome to the healpy documentation

healpy is a Python package to handle pixelated data on the sphere. It is based on the Hierarchical Equal Area isoLatitude Pixelization (HEALPix) scheme and bundles the HEALPix C++ library.

HEALPix was developed to efficiently process Cosmic Microwave Background data from Cosmology experiments like BOOMERANG and WMAP but it is now used in other branches of Astrophysics to store data from all-sky surveys. The target audience used to be primarily the Cosmology scientific community but currently anyone interested in handling pixelated data on the sphere is very welcome to propose new features.

healpy provides utilities to:

  • convert between sky coordinates and pixel indices in HEALPix nested and ring schemes

  • find pixels within a disk, a polygon or a strip in the sky

  • apply coordinate transformations between Galactic, Ecliptic and Equatorial reference frames

  • apply custom rotations either to vectors or full maps

  • read and write HEALPix maps to disk in FITS format

  • upgrade and downgrade the resolution of existing HEALPix maps

  • visualize maps in Mollweide, Gnomonic and Cartographic projections

  • transform maps to Spherical Harmonics space and back using multi-threaded C++ routines

  • compute Auto and Cross Power Spectra from maps and create map realizations from spectra

Tutorial

healpy tutorial

See the Jupyter Notebook version of this tutorial at https://github.com/healpy/healpy/blob/master/doc/healpy_tutorial.ipynb

See a executed version of the notebook with embedded plots at https://gist.github.com/zonca/9c114608e0903a3b8ea0bfe41c96f255

Choose the inline backend of maptlotlib to display the plots inside the Jupyter Notebook

import matplotlib.pyplot as plt

%matplotlib inline

import numpy as np
import healpy as hp

NSIDE and ordering

Maps are simply numpy arrays, where each array element refers to a location in the sky as defined by the Healpix pixelization schemes (see the healpix website).

Note: Running the code below in a regular Python session will not display the maps; it’s recommended to use an IPython shell or a Jupyter notebook.

The resolution of the map is defined by the NSIDE parameter, which is generally a power of 2.

NSIDE = 32
print(
    "Approximate resolution at NSIDE {} is {:.2} deg".format(
        NSIDE, hp.nside2resol(NSIDE, arcmin=True) / 60
    )
)

The function healpy.pixelfunc.nside2npix gives the number of pixels NPIX of the map:

NPIX = hp.nside2npix(NSIDE)
print(NPIX)

The same pixels in the map can be ordered in 2 ways, either RING, where they are numbered in the array in horizontal rings starting from the North pole:

m = np.arange(NPIX)
hp.mollview(m, title="Mollview image RING")
hp.graticule()

The standard coordinates are the colatitude \(\theta\), \(0\) at the North Pole, \(\pi/2\) at the equator and \(\pi\) at the South Pole and the longitude \(\phi\) between \(0\) and \(2\pi\) eastward, in a Mollview projection, \(\phi=0\) is at the center and increases eastward toward the left of the map.

We can also use vectors to represent coordinates, for example vec is the normalized vector that points to \(\theta=\pi/2, \phi=3/4\pi\):

vec = hp.ang2vec(np.pi / 2, np.pi * 3 / 4)
print(vec)

We can find the indices of all the pixels within \(10\) degrees of that point and then change the value of the map at those indices:

ipix_disc = hp.query_disc(nside=32, vec=vec, radius=np.radians(10))

m = np.arange(NPIX)
m[ipix_disc] = m.max()
hp.mollview(m, title="Mollview image RING")

We can retrieve colatitude and longitude of each pixel using pix2ang, in this case we notice that the first 4 pixels cover the North Pole with pixel centers just ~\(1.5\) degrees South of the Pole all at the same latitude. The fifth pixel is already part of another ring of pixels.

theta, phi = np.degrees(hp.pix2ang(nside=32, ipix=[0, 1, 2, 3, 4]))

theta

phi

The RING ordering is necessary for the Spherical Harmonics transforms, the other option is NESTED ordering which is very efficient for map domain operations because scaling up and down maps is achieved just multiplying and rounding pixel indices. See below how pixel are ordered in the NESTED scheme, notice the structure of the 12 HEALPix base pixels (NSIDE 1):

m = np.arange(NPIX)
hp.mollview(m, nest=True, title="Mollview image NESTED")

All healpy routines assume RING ordering, in fact as soon as you read a map with read_map, even if it was stored as NESTED, it is transformed to RING. However, you can work in NESTED ordering passing the nest=True argument to most healpy routines.

Reading and writing maps to file

For the following section, it is required to download larger maps by executing from the terminal the bash script healpy_get_wmap_maps.sh which should be available in your path.

This will download the higher resolution WMAP data into the current directory.

!healpy_get_wmap_maps.sh

wmap_map_I = hp.read_map("wmap_band_iqumap_r9_7yr_W_v4.fits")

By default, input maps are converted to RING ordering, if they are in NESTED ordering. You can otherwise specify nest=True to retrieve a map is NESTED ordering, or nest=None to keep the ordering unchanged.

By default, read_map loads the first column, for reading other columns you can specify the field keyword.

write_map writes a map to disk in FITS format, if the input map is a list of 3 maps, they are written to a single file as I,Q,U polarization components:

hp.write_map("my_map.fits", wmap_map_I, overwrite=True)

Visualization

As shown above, mollweide projection with mollview is the most common visualization tool for HEALPIX maps. It also supports coordinate transformation, coord does Galactic to ecliptic coordinate transformation, norm='hist' sets a histogram equalized color scale and xsize increases the size of the image. graticule adds meridians and parallels.

hp.mollview(
    wmap_map_I,
    coord=["G", "E"],
    title="Histogram equalized Ecliptic",
    unit="mK",
    norm="hist",
    min=-1,
    max=1,
)
hp.graticule()

gnomview instead provides gnomonic projection around a position specified by rot, for example you can plot a projection of the galactic center, xsize and ysize change the dimension of the sky patch.

hp.gnomview(wmap_map_I, rot=[0, 0.3], title="GnomView", unit="mK", format="%.2g")

mollzoom is a powerful tool for interactive inspection of a map, it provides a mollweide projection where you can click to set the center of the adjacent gnomview panel. ## Masked map, partial maps

By convention, HEALPIX uses \(-1.6375 * 10^{30}\) to mark invalid or unseen pixels. This is stored in healpy as the constant UNSEEN.

All healpy functions automatically deal with maps with UNSEEN pixels, for example mollview marks in grey those sections of a map.

There is an alternative way of dealing with UNSEEN pixel based on the numpyMaskedArray class, hp.ma loads a map as a masked array, by convention the mask is 0 where the data are masked, while numpy defines data masked when the mask is True, so it is necessary to flip the mask.

mask = hp.read_map("wmap_temperature_analysis_mask_r9_7yr_v4.fits").astype(np.bool)
wmap_map_I_masked = hp.ma(wmap_map_I)
wmap_map_I_masked.mask = np.logical_not(mask)

Filling a masked array fills in the UNSEEN value and return a standard array that can be used by mollview. compressed() instead removes all the masked pixels and returns a standard array that can be used for examples by the matplotlib hist() function:

hp.mollview(wmap_map_I_masked.filled())

plt.hist(wmap_map_I_masked.compressed(), bins=1000);

Spherical Harmonics transforms

healpy provides bindings to the C++ HEALPIX library for performing spherical harmonic transforms. hp.anafast computes the angular power spectrum of a map:

LMAX = 1024
cl = hp.anafast(wmap_map_I_masked.filled(), lmax=LMAX)
ell = np.arange(len(cl))

therefore we can plot a normalized CMB spectrum and write it to disk:

plt.figure(figsize=(10, 5))
plt.plot(ell, ell * (ell + 1) * cl)
plt.xlabel("$\ell$")
plt.ylabel("$\ell(\ell+1)C_{\ell}$")
plt.grid()
hp.write_cl("cl.fits", cl, overwrite=True)

Gaussian beam map smoothing is provided by hp.smoothing:

wmap_map_I_smoothed = hp.smoothing(wmap_map_I, fwhm=np.radians(1.))
hp.mollview(wmap_map_I_smoothed, min=-1, max=1, title="Map smoothed 1 deg")

For more information see the HEALPix primer

Installation

Installation procedure for Healpy

Requirements

Healpy depends on the HEALPix C++ and cfitsio C libraries. Source code for both is included with Healpy and is built automatically, so you do not need to install them yourself. Only Linux and MAC OS X are supported, not Windows.

Source installation with Pip

It is possible to build the latest healpy with pip

pip install --user healpy

If you have installed with pip, you can keep your installation up to date by upgrading from time to time:

pip install --user --upgrade healpy

On Linux with newer compilers many users reported compilation errors like configure: error: cannot run C compiled programs, the solution is to specifiy the flags for the C and CXX compiler:

CC=gcc CXX=g++ CFLAGS=’-fPIC’ CXXFLAGS=’-fPIC’ pip install –user healpy

Compilation issues with Mac OS

Currently most people report they cannot install healpy on Mac OS either via pip or building from source, due to the impossibility of compiling the HEALPix based extension. The only options right now are using conda-forge or Macports.

Installation on Mac OS with MacPorts

If you are using a Mac and have the MacPorts package manager, it’s even easer to install Healpy with:

sudo port install py27-healpy

Binary apt-get style packages are also available in the development versions of Debian (sid) and Ubuntu (utopic).

Almost-as-quick installation from official source release

Healpy is also available in the Python Package Index (PyPI). You can download it with:

curl -O https://pypi.python.org/packages/source/h/healpy/healpy-1.7.4.tar.gz

and build it with:

tar -xzf healpy-1.7.4.tar.gz
pushd healpy-1.7.4
python setup.py install --user
popd

If everything goes fine, you can test it:

python

>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> import healpy as hp
>>> hp.mollview(np.arange(12))
>>> plt.show()

or run the test suite with nose:

cd healpy-1.7.4 && python setup.py test

Building against external Healpix and cfitsio

Healpy uses pkg-config to detect the presence of the Healpix and cfitsio libraries. pkg-config is available on most systems. If you do not have pkg-config installed, then Healpy will download and use (but not install) a Python clone called pykg-config.

If you want to provide your own external builds of Healpix and cfitsio, then download the following packages:

If you are going to install the packages in a nonstandard location (say, --prefix=/path/to/local), then you should set the environment variable PKG_CONFIG_PATH=/path/to/local/lib/pkgconfig when building. No other environment variable settings are necessary, and you do not need to set PKG_CONFIG_PATH to use Healpy after you have built it.

Then, unpack each of the above packages and build them with the usual configure; make; make install recipe.

Development install

Developers building from a snapshot of the github repository need:

  • autoconf and libtool (in Debian or Ubuntu: sudo apt-get install autoconf automake libtool pkg-config)

  • cython > 0.16

  • run git submodule init and git submodule update to get the bundled HEALPix sources

the best way to install healpy if you plan to develop is to build the C++ extensions in place with:

python setup.py build_ext --inplace

then add the healpy repository folder to your PYTHONPATH (e.g. if you cloned this repository to $REPOS such that $REPOS/healpy/INSTALL.rst exists, then add $REPOS/healpy to your PYTHONPATH).

In case of compilation errors, see the note above in the pip section.

Clean

When you run “python setup.py”, temporary build products are placed in the “build” directory. If you want to clean out and remove the build directory, then run:

python setup.py clean --all

Reference

sphtfunc – Spherical harmonic transforms

From map to spherical harmonics

anafast(map1[, map2, nspec, lmax, mmax, …])

Computes the power spectrum of a Healpix map, or the cross-spectrum between two maps if map2 is given.

map2alm(maps[, lmax, mmax, iter, pol, …])

Computes the alm of a Healpix map.

From spherical harmonics to map

synfast(cls, nside[, lmax, mmax, alm, pol, …])

Create a map(s) from cl(s).

alm2map(alms, nside[, lmax, mmax, pixwin, …])

Computes a Healpix map given the alm.

alm2map_der1(alm, nside[, lmax, mmax])

Computes a Healpix map and its first derivatives given the alm.

Spherical harmonic transform tools

smoothing(map_in[, fwhm, sigma, …])

Smooth a map with a Gaussian symmetric beam.

smoothalm(alms[, fwhm, sigma, beam_window, …])

Smooth alm with a Gaussian symmetric beam function.

alm2cl(alms1[, alms2, lmax, mmax, lmax_out, …])

Computes (cross-)spectra from alm(s).

synalm(cls[, lmax, mmax, new, verbose])

Generate a set of alm given cl.

almxfl(alm, fl[, mmax, inplace])

Multiply alm by a function of l.

pixwin(nside[, pol, lmax])

Return the pixel window function for the given nside.

Alm()

This class provides some static methods for alm index computation.

Other tools

gauss_beam(fwhm[, lmax, pol])

Gaussian beam window function

beam2bl(beam, theta, lmax)

Computes a transfer (or window) function b(l) in spherical harmonic space from its circular beam profile b(theta) in real space.

bl2beam(bl, theta)

Computes a circular beam profile b(theta) in real space from its transfer (or window) function b(l) in spherical harmonic space.

visufunc – Visualisation

Map projections

mollview([map, fig, rot, coord, unit, …])

Plot a healpix map (given as an array) in Mollweide projection.

gnomview([map, fig, rot, coord, unit, …])

Plot a healpix map (given as an array) in Gnomonic projection.

cartview([map, fig, rot, zat, coord, unit, …])

Plot a healpix map (given as an array) in Cartesian projection.

orthview([map, fig, rot, coord, unit, …])

Plot a healpix map (given as an array) in Orthographic projection.

Graticules

graticule([dpar, dmer, coord, local])

Draw a graticule on the current Axes.

delgraticules()

Delete all graticules previously created on the Axes.

Tracing lines or points

projplot(\*args, \*\*kwds)

projplot is a wrapper around matplotlib.Axes.plot() to take into account the spherical projection.

projscatter(\*args, \*\*kwds)

Projscatter is a wrapper around matplotlib.Axes.scatter() to take into account the spherical projection.

projtext(\*args, \*\*kwds)

Projtext is a wrapper around matplotlib.Axes.text() to take into account the spherical projection.

Pixel querying routines

query_disc()

Returns pixels whose centers lie within the disk defined by vec and radius (in radians) (if inclusive is False), or which overlap with this disk (if inclusive is True).

query_polygon()

Returns the pixels whose centers lie within the convex polygon defined by the vertices array (if inclusive is False), or which overlap with this polygon (if inclusive is True).

query_strip()

Returns pixels whose centers lie within the colatitude range defined by theta1 and theta2 (if inclusive is False), or which overlap with this region (if inclusive is True).

boundaries()

Returns an array containing vectors to the boundary of the nominated pixel.

rotator – Rotation and geometry functions

Rotation

Rotator([rot, coord, inv, deg, eulertype])

Rotation operator, including astronomical coordinate systems.

rotateVector(rotmat, vec[, vy, vz, do_rot])

Rotate a vector (or a list of vectors) using the rotation matrix given as first argument.

rotateDirection(rotmat, theta[, phi, …])

Rotate the vector described by angles theta,phi using the rotation matrix given as first argument.

Geometrical helpers

vec2dir(vec[, vy, vz, lonlat])

Transform a vector to angle given by theta,phi.

dir2vec(theta[, phi, lonlat])

Transform a direction theta,phi to a unit vector.

angdist(dir1, dir2[, lonlat])

Returns the angular distance between dir1 and dir2.

projector – Spherical projections

Basic classes

SphericalProj([rot, coord, flipconv])

This class defines functions for spherical projection.

GnomonicProj([rot, coord, xsize, ysize, reso])

This class provides class methods for Gnomonic projection.

MollweideProj([rot, coord, xsize])

This class provides class methods for Mollweide projection.

CartesianProj([rot, coord, xsize, ysize, …])

This class provides class methods for Cartesian projection.

zoomtool – Interactive visualisation

Interactive map visualization

mollzoom([map, fig, rot, coord, unit, …])

Interactive mollweide plot with zoomed gnomview.

Indices and tables

License

Licenses

Healpy License

Healpy is licensed under the GNU General Public License.

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Version 2, June 1991

Copyright (C) 1989, 1991 Free Software Foundation, Inc.

51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

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  1. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING

WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

END OF TERMS AND CONDITIONS

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest

possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest

to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

<one line to give the program’s name and a brief idea of what it does.> Copyright (C) <year> <name of author>

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

Also add information on how to contact you by electronic and paper mail.

If the program is interactive, make it output a short notice like this when it starts in an interactive mode:

Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w’. This is free software, and you are welcome to redistribute it under certain conditions; type `show c’ for details.

The hypothetical commands `show w’ and `show c’ should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w’ and `show c’; they could even be mouse-clicks or menu items–whatever suits your program.

You should also get your employer (if you work as a programmer) or your school, if any, to sign a “copyright disclaimer” for the program, if necessary. Here is a sample; alter the names:

Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision’ (which makes passes at compilers) written by James Hacker.

<signature of Ty Coon>, 1 April 1989 Ty Coon, President of Vice

This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.