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- Shared Python Interface for World Coordinate Systems

# Shared Python Interface for World Coordinate Systems

## Background

The `WCS` class implements what is considered the most common ‘standard’ for representing world coordinate systems in FITS files, but it cannot represent arbitrarily complex transformations and there is no agreement on how to use the standard beyond FITS files. Therefore, other world coordinate system transformation approaches exist, such as the gwcs package being developed for the James Webb Space Telescope (which is also applicable to other data).

Since one of the goals of the Astropy Project is to improve interoperability between packages, we have collaboratively defined a standardized application programming interface (API) for world coordinate system objects to be used in Python. This API is described in the Astropy Proposal for Enhancements (APE) 14: A shared Python interface for World Coordinate Systems.

The core astropy package provides base classes that define the low- and high- level APIs described in APE 14 in the `astropy.wcs.wcsapi` module, and these are listed in the Reference/API section below.

## Overview

While the full details and motivation for the API are detailed in APE 14, this documentation summarizes the elements that are implemented directly in the astropy core package. The high-level interface is likely of most interest to the average user. In particular, the most important methods are the `pixel_to_world()` and `world_to_pixel()` methods. These provide the essential elements of WCS: mapping to and from world coordinates. The remainder generally provide information about the kind of world coordinates or similar information about the structure of the WCS.

In a bit more detail, the key classes implemented here are a high-level that provides the main user interface (`BaseHighLevelWCS` and subclasses), and a lower-level interface (`BaseLowLevelWCS` and subclasses). These can be distinct objects or the same one. For FITS-WCS, the `WCS` object meant for FITS-WCS follows both interfaces, allowing immediate use of this API with files that already contain FITS-WCS. More concrete examples are outlined below.

## Basic usage

Let’s start off by looking at the shared Python interface for WCS by using a simple image with two celestial axes (Right Ascension and Declination):

    >>> from astropy.wcs import WCS
    >>> from astropy.utils.data import get_pkg_data_filename
    >>> from astropy.io import fits
    >>> filename = get_pkg_data_filename('galactic_center/gc_2mass_k.fits')
    >>> hdulist = fits.open(filename)
    >>> hdu = hdulist[0]
    >>> wcs = WCS(hdu.header)
    >>> wcs
    WCS Keywords
    Number of WCS axes: 2
    CTYPE : 'RA---TAN'  'DEC--TAN'
    CRVAL : 266.4  -28.93333
    CRPIX : 361.0  360.5
    NAXIS : 721  720

We can check how many pixel and world axes are in the transformation as well as the shape of the data the WCS applies to:

    >>> wcs.pixel_n_dim
    2
    >>> wcs.world_n_dim
    2
    >>> wcs.array_shape
    (720, 721)

Note that the array shape should match that of the data:

    >>> hdu.data.shape
    (720, 721)

As mentioned in Pixel Conventions and Definitions, what would normally be considered the ‘y-axis’ of the image (when looking at it visually) is the first dimension, while the ‘x-axis’ of the image is the second dimension. Thus `array_shape` returns the shape in the opposite order to the NAXIS keywords in the FITS header (in the case of FITS-WCS). If you are interested in the data shape in the reverse order (which would match the NAXIS order in the case of FITS-WCS), then you can use `pixel_shape`:

    >>> wcs.pixel_shape
    (721, 720)

Let’s now check what the physical type of each axis is:

    >>> wcs.world_axis_physical_types
    ['pos.eq.ra', 'pos.eq.dec']

This is indeed an image with two celestial axes.

The main part of the new interface defines standard methods for transforming coordinates. The most convenient way is to use the high-level methods `pixel_to_world()` and `world_to_pixel()`, which can transform directly to astropy objects:

    >>> coord = wcs.pixel_to_world([1, 2], [4, 3])
    >>> coord
    <SkyCoord (FK5: equinox=2000.0): (ra, dec) in deg
        [(266.97242993, -29.42584415), (266.97084321, -29.42723968)]>

Similarly, we can transform astropy objects back - we can test this by creating Galactic coordinates and these will automatically be converted:

    >>> from astropy.coordinates import SkyCoord
    >>> coord = SkyCoord('00h00m00s +00d00m00s', frame='galactic')
    >>> pixels = wcs.world_to_pixel(coord)
    >>> pixels
    (array(356.85179997), array(357.45340331))

If you are looking to index the original data using these pixel coordinates, be sure to instead use `world_to_array_index()` which returns the coordinates in the correct order to index Numpy arrays, and also rounds to the nearest integer values:

    >>> index = wcs.world_to_array_index(coord)
    >>> index
    (357, 357)
    >>> hdu.data[index]
    563.7532
    >>> hdulist.close()

## Advanced usage

Let’s now take a look at a WCS for a spectral cube (two celestial axes and one spectral axis):

    >>> filename = get_pkg_data_filename('l1448/l1448_13co.fits')
    >>> hdulist = fits.open(filename)
    >>> hdu = hdulist[0]
    >>> wcs = WCS(hdu.header)
    >>> wcs
    WCS Keywords
    Number of WCS axes: 3
    CTYPE : 'RA---SFL'  'DEC--SFL'  'VOPT'
    CRVAL : 57.6599999999  0.0  -9959.44378305
    CRPIX : -799.0  -4741.913  -187.0
    PC1_1 PC1_2 PC1_3  : 1.0  0.0  0.0
    PC2_1 PC2_2 PC2_3  : 0.0  1.0  0.0
    PC3_1 PC3_2 PC3_3  : 0.0  0.0  1.0
    CDELT : -0.006388889  0.006388889  66.42361
    NAXIS : 105  105  53

As before we can check how many pixel and world axes are in the transformation as well as the shape of the data the WCS applies to, as well as the physical types of each axis:

    >>> wcs.pixel_n_dim
    3
    >>> wcs.world_n_dim
    3
    >>> wcs.array_shape
    (53, 105, 105)
    >>> wcs.world_axis_physical_types
    ['pos.eq.ra', 'pos.eq.dec', 'spect.dopplerVeloc.opt']

This is indeed a spectral cube, with RA/Dec and a velocity axis.

As before, we can convert between pixels and high-level Astropy objects:

    >>> celestial, spectral = wcs.pixel_to_world([1, 2], [4, 3], [2, 3])
    >>> celestial
    <SkyCoord (ICRS): (ra, dec) in deg
        [(51.73115731, 30.32750025), (51.72414268, 30.32111136)]>
    >>> spectral
    <SpectralCoord
       (target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, kpc)
                    (57.66, 0., 1000.)
                 (pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
                    (0., 0., 0.)>)
      [2661.04211695, 2727.46572695] m / s>

and back:

    >>> from astropy import units as u
    >>> coord = SkyCoord('03h26m36.4901s +30d45m22.2012s')
    >>> pixels = wcs.world_to_pixel(coord, 3000 * u.m / u.s)
    >>> pixels
    (array(8.11341207), array(71.0956641), array(7.10297292))

And as before we can index array values using:

    >>> index = wcs.world_to_array_index(coord, 3000 * u.m / u.s)
    >>> index
    (7, 71, 8)
    >>> hdu.data[index]
    0.22262384
    >>> hdulist.close()

If you are interested in converting to/from world values as simple Python scalars or Numpy arrays without using high-level astropy objects, there are methods such as `pixel_to_world_values()` to do this - see Reference/API section for more details.

## Extending the physical types in FITS-WCS

As shown above, the `world_axis_physical_types` property returns the list of physical types for each axis. For FITS-WCS, this is determined from the CTYPE values in the header. In cases where the physical type is not known, `None` is returned. However, it is possible to override the physical types returned by using the `custom_ctype_to_ucd_mapping` context manager. Consider a WCS with the following CTYPE:

    >>> from astropy.wcs import WCS
    >>> wcs = WCS(naxis=1)
    >>> wcs.wcs.ctype = ['SPAM']
    >>> wcs.world_axis_physical_types
    [None]

We can specify that for this CTYPE, the physical type should be `'food.spam'`:

    >>> from astropy.wcs.wcsapi.fitswcs import custom_ctype_to_ucd_mapping
    >>> with custom_ctype_to_ucd_mapping({'SPAM': 'food.spam'}):
    ...     wcs.world_axis_physical_types
    ['food.spam']

## Slicing of WCS objects

A common operation when dealing with data with WCS information attached is to slice the WCS - this can be either to extract the WCS for a sub-region of the data, preserving the overall number of dimensions (e.g. a cutout from an image) or it can be reducing the dimensionality of the data and associated WCS (e.g. extracting a slice from a spectral cube).

The `SlicedLowLevelWCS` class can be used to slice any WCS object that conforms to the `BaseLowLevelWCS` API. To demonstrate this, let’s start off by reading in a spectral cube file:

    >>> filename = get_pkg_data_filename('l1448/l1448_13co.fits')
    >>> wcs = WCS(fits.getheader(filename, ext=0))

The `wcs` object is an instance of `WCS` which conforms to the `BaseLowLevelWCS` API. We can then use the `SlicedLowLevelWCS` class to slice the cube:

    >>> from astropy.wcs.wcsapi import SlicedLowLevelWCS
    >>> slices = [10, slice(30, 100), slice(30, 100)]
    >>> subwcs = SlicedLowLevelWCS(wcs, slices=slices)

The `slices` argument takes any combination of slices, integer values, and ellipsis which would normally slice a Numpy array. In the above case, we are extracting a spectral slice, and in that slice we are extracting a sub-region on the sky.

If you are implementing your own WCS class, you could choose to implement `__getitem__` and have it internally use `SlicedLowLevelWCS`. In fact, the `WCS` class does this - the example above can be written more succinctly as:

    >>> wcs[10, 30:100, 30:100]
    <...>
    SlicedFITSWCS Transformation

    This transformation has 2 pixel and 2 world dimensions

    Array shape (Numpy order): (70, 70)

    Pixel Dim  Axis Name  Data size  Bounds
            0  None              70  None
            1  None              70  None

    World Dim  Axis Name  Physical Type  Units
            0  None       pos.eq.ra      deg
            1  None       pos.eq.dec     deg

    Correlation between pixel and world axes:

               Pixel Dim
    World Dim    0    1
            0  yes  yes
            1  yes  yes

This slicing infrastructure is able to deal with slicing of WCS objects which have correlated axes - in this case, you may end up with a WCS that has a different number of pixel and world coordinates. For example, if we slice a spectral cube to extract a 1D dataset corresponding to a row in the image plane of a spectral slice, the final WCS will have one pixel dimension and two world dimensions (since both RA/Dec vary over the extracted 1D slice):

    >>> wcs[10, 40, :]
    <...>
    SlicedFITSWCS Transformation

    This transformation has 1 pixel and 2 world dimensions

    Array shape (Numpy order): (105,)

    Pixel Dim  Axis Name  Data size  Bounds
            0  None             105  None

    World Dim  Axis Name  Physical Type  Units
            0  None       pos.eq.ra      deg
            1  None       pos.eq.dec     deg

    Correlation between pixel and world axes:

               Pixel Dim
    World Dim    0
            0  yes
            1  yes