Australia Telescope National Facility

National Radio Astronomy Observatory

The original FITS papers provided a simplified, non-specific method
for describing the physical coordinate values of image pixels. In the
intervening 20 years, there have been a number of implementations of
coordinates in astronomical software packages and several attempts to
obtain agreement on their representation in FITS. A general consensus
on the important details has been obtained and is described here.

The initial paper describing the Flexible Image Transport System, or
FITS format (Wells, Greisen, & Harten 1981) proposed keywords to
describe the physical coordinates of image data. The original authors
chose to defer discussion of the technical details of coordinate
specification until the basic FITS format was accepted generally and
until a deeper understanding of image coordinate specification and
computation could be obtained. While participating in the development
of the AIPS software package of the National Radio Astronomy
Observatory, Greisen (1983) developed FITS-like syntax and semantics
to define both velocity and celestial coordinates. The latter have
been widely used for interchanging imagery from a number of
instruments at widely differing spectral domains and form the basis
of the present proposal. That proposal is presented in three papers
which are available from our WWW home
pages:
^{3}

1. | Representations of world coordinates in FITS |

by Greisen and Calabretta (general conventions, units) | |

2. | Representations of celestial coordinates in FITS |

by Calabretta and Greisen (celestial projections) | |

3. | Representations of spectral coordinates in FITS |

by Greisen (spectral and velocity axes) |

The computation of the coordinates associated with each pixel is
broken into three steps, a correction for arbitrary small image
distortions, a linear transformation (including rotation, scale and
skew), and transformation from intermediate world coordinates to the
required world coordinate system via a predefined algorithm. These
steps and the FITS header keywords used to define them are illustrated in
Figure 1. The pixel regularization correction is
encoded in an image extension and accounts for small distortions which
can't easily be described by other means. It will often be omitted by
FITS writers and may be ignored when present by many FITS readers.
The linear transformation is done with a translation specified by the
familiar ` CRPIX`* i* followed by a matrix multiplication
specified by ` CD j_ i` keywords. The resulting
``intermediate world coordinates'' are in physical units (e.g., deg,
m/s), but require conversion via a predefined linear or non-linear
algorithm into the final world coordinate system. The algorithm and
final coordinates are selected by the two halves of the

In some cases, an image may be described by more than one set of
coordinates. For example, spectral axes may be described by
frequency, wavelength, and velocity (only one of which might be
linear) and spatial axes may be described, for example, by position in
the telescope focal plane in meters as well as in degrees on the sky.
To permit up to 26 secondary coordinate descriptions, a version code
may be appended to each of the standard header keywords (e.g., `
CRVAL27Q`). These codes are the letters ` A` through ` Z`,
with the blank character being reserved for the primary version. If
an alternate coordinate description is specified, all coordinate
keywords for that version must be given even if they do not differ
from those of the primary version. Note that this convention (and the
` CD j_ i` keyword) reduce the maximum axis number from
999 to a more reasonable 99.

Paper I contains a detailed proposal for a new IAU standard for the representation of units when they must appear in plain character form. This proposal is a merger and clarification of the systems devised by George & Angelini (1995) and Ochsenbein et al. (1996) and is a supplement to the IAU Style Manual (1988) which defines units as they would appear in a published document.

Image data are represented in FITS not only in the primary array and
in an ` IMAGE` extension, but also in a multi-dimensional vector
in a single element of a FITS binary table and a tabulated list of
pixel coordinates in a FITS ASCII or binary table. Keywords to
specify the coordinates of such data are given in appendices to Papers
I, II, and III and, for the primary coordinate description, match
those already in use by some software systems.

Paper II applies the general formalism of Paper I to the problem of
representing celestial coordinate systems. With detailed mathematical
formulæ, it covers all (24) spherical map projections likely to be
of interest in astronomy including zenithal, cylindrical,
pseudo-cylindrical, conic, polyconic, pseudo-conic, and quad-cube
projections. The familiar gnomonic or ``` -TAN`'' projection is
extended to allow up to 80 parameters to describe a full astrometric
solution to the distortions in a photographic or CCD image. The
projections support terrestrial and planetary mapping as well as
celestial mapping. Several examples of header interpretation are
provided and header construction is exemplified by cases taken from
full-sky imagery, the Digitized Sky Survey, a long-slit spectrograph
image, and an image of the Moon. The major celestial coordinate
systems are supported including generalized longitude-latitude pairs
and a new keyword ` RADECSYS` is defined to specify the coordinate
system (i.e. ` 'FK4'`, ` 'FK5'`).

In the computation of celestial coordinates, ``intermediate world
coordinates'' are identified with Cartesian coordinates in the plane
of projection. The computation of world coordinates then requires
first de-projection of the Cartesian coordinates to obtain ``native
spherical coordinates'' and then spherical coordinate transformation
with rotation angles defined by ` CRVAL`* j* and new `
LONPOLE` and ` LATPOLE` keywords. The native coordinate system is the
one in which the projection is most conveniently defined and the
reference values in the desired system must correspond to the
``reference point'' of the projection. This is fixed for each class
of projection. Oblique projections are generated naturally in this
fashion as illustrated in Figure 2.

The coordinates of Paper II are supported by the ` WCSLIB` software
package (Calabretta 1999) available without charge under GNU
licensing.

Paper III extends the methods of Papers I and II to represent frequency, wavelength, and velocity including the radio, optical and redshift conventional velocities. Conversion formulæ for data measured in any one of these and expressed in any one of these are given. Provision is made for higher order pixel corrections. Keywords are defined to specify velocity coordinate reference frames and any parameters needed to switch between them. Methods to describe the dependence of spectral coordinates on celestial coordinates are given for velocity reference frames and for instruments similar to Fabry-Perot interferometers. It is noted that objective prism data, in which one spectral and two spatial axes are mingled on a two-dimensional image, cannot be described by the methods of Papers I and III. When these data are reduced to tabular form, they are easily described by the conventions presented in all three papers for such data.

Calabretta, M. 1999, ` WCSLIB`, Version 2.4, located at

` ftp://ftp.atnf.csiro.au/pub/software/wcslib/wcslib.tar.gz`.

George, I. M. &Angelini, L. 1995, Specification
of Physical Units within OGIP FITS files, OGIP Memo OGIP/93-001,
NASA Goddard Space Flight Center, Greenbelt, Maryland,

` ftp://fits.cv.nrao.edu/fits/wcs/OGIP93_001.ps`

Greisen, E. W. 1983, AIPS Memo No. 27, National Radio Astronomy Observatory, Charlottesville, Virginia

IAU Style Manual 1983, * IAU Inf. Bull.*, No. 49, 14.

Ochsenbein, F., Paul, N., & Kuin, M. 1996,
Standards for Astronomical Catalogues, Version 1.5,
` http://vizier.u-strasbg.fr/doc/catstd.htx`

Wells, D. C., Greisen, E. W., & Harten, R. H. 1981, A&AS, 44, 363

- ... Facility
^{1} - The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
- ... Observatory
^{2} - The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
- ... Observatory
^{3} - http://www.atnf.csiro.au/people/mcalabre/ or http://www.cv.nrao.edu/~egreisen

© Copyright 2000 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

adass@cfht.hawaii.edu