# GWCS Documentation¶

GWCS is a package for managing the World Coordinate System (WCS) of astronomical data.

## Introduction & Motivation for GWCS¶

The mapping from ‘pixel’ coordinates to corresponding ‘real-world’ coordinates (e.g. celestial coordinates, spectral wavelength) is crucial to relating astronomical data to the phenomena they describe. Images and other types of data often come encoded with information that describes this mapping – this is referred to as the ‘World Coordinate System’ or WCS. The term WCS is often used to refer specifically to the most widely used ‘FITS implementation of WCS’, but here unless specified WCS refers to the broader concept of relating pixel ⟷ world. (See the discussion in APE14 for more on this topic).

The FITS WCS standard, currently the most widely used method of encoding WCS in data, describes a set of required FITS header keywords and allowed values that describe how pixel ⟷ world transformations should be done. This current paradigm of encoding data with only instructions on how to relate pixel to world, separate from the transformation machinery itself, has several limitations:

• Limited flexibility. WCS keywords and their values are rigidly defined so that the instructions are unambiguous. This places limitations on, for example, describing geometric distortion in images since only a handful of distortion models are defined in the FITS standard (and therefore can be encoded in FITS headers as WCS information).

• Separation of data from transformation pipelines. The machinery that transforms pixel ⟷ world does not exist along side the data – there is merely a roadmap for how one would do the transformation. External packages and libraries (e.g wcslib, or its Python interface astropy.wcs) must be written to interpret the instructions and execute the transformation. These libraries don’t allow easy access to coordinate frames along the course of the full pixel to world transformation pipeline. Additionally, since these libraries can only interpret FITS WCS information, any custom ‘WCS’ definitions outside of FITS require the user to write their own transformation pipelines.

• Incompatibility with varying file formats. New file formats that are becoming more widely used in place of FITS to store astronomical data, like the ASDF format, also require a method of encoding WCS information. FITS WCS and the accompanying libraries are adapted for FITS only. A more flexible interface would be agnostic to file type, as long as the necessary information is present.

The GWCS package and GWCS object is a generalized WCS implementation that mitigates these limitations. The goal of the GWCS package is to provide a flexible toolkit for expressing and evaluating transformations between pixel and world coordinates, as well as intermediate frames along the course of this transformation.The GWCS object supports a data model which includes the entire transformation pipeline from input pixel coordinates to world coordinates (and vice versa). The basis of the GWCS object is astropy modeling. Models that describe the pixel ⟷ world transformations can be chained, joined or combined with arithmetic operators using the flexible framework of compound models in modeling. This approach allows for easy access to intermediate frames. In the case of a celestial output frame coordinates provides further transformations between standard celestial coordinate frames. Spectral output coordinates are instances of Quantity and can be transformed to other units with the tools in that package. Time coordinates are instances of Time. GWCS supports transforms initialized with Quantity objects ensuring automatic unit conversion.

## Pixel Conventions and Definitions¶

This API assumes that integer pixel values fall at the center of pixels (as assumed in the FITS-WCS standard, see Section 2.1.4 of Greisen et al., 2002, A&A 446, 747), while at the same time matching the Python 0-index philosophy. That is, the first pixel is considered pixel 0, but pixel coordinates (0, 0) are the center of that pixel. Hence the first pixel spans pixel values -0.5 to 0.5.

There are two main conventions for ordering pixel coordinates. In the context of 2-dimensional imaging data/arrays, one can either think of the pixel coordinates as traditional Cartesian coordinates (which we call x and y here), which are usually given with the horizontal coordinate (x) first, and the vertical coordinate (y) second, meaning that pixel coordinates would be given as (x, y). Alternatively, one can give the coordinates by first giving the row in the data, then the column, i.e. (row, column). While the former is a more common convention when e.g. plotting (think for example of the Matplotlib scatter(x, y) method), the latter is the convention used when accessing values from e.g. Numpy arrays that represent images (image[row, column]).

The GWCS object assumes Cartesian order (x, y), however the Common Interface for World Coordinate System - APE 14 accepts both conventions. The order of the pixel coordinates ((x, y) vs (row, column)) in the Common API depends on the method or property used, and this can normally be determined from the property or method name. Properties and methods containing pixel assume (x, y) ordering, while properties and methods containing array assume (row, column) ordering.

## Installation¶

gwcs requires:

To install from source:

git clone https://github.com/spacetelescope/gwcs.git
cd gwcs
python setup.py install


To install the latest release:

pip install gwcs


The latest release of GWCS is also available as part of astroconda.

## Getting Started¶

The WCS data model represents a pipeline of transformations between two coordinate frames, the final one usually a physical coordinate system. It is represented as a list of steps executed in order. Each step defines a starting coordinate frame and the transform to the next frame in the pipeline. The last step has no transform, only a frame which is the output frame of the total transform. As a minimum a WCS object has an input_frame (defaults to “detector”), an output_frame and the transform between them.

The WCS is validated using the ASDF Standard and serialized to file using the asdf package. There are two ways to save the WCS to a file:

### A step by step example of constructing an imaging GWCS object.¶

The following example shows how to construct a GWCS object equivalent to a FITS imaging WCS without distortion, defined in this FITS imaging header:

WCSAXES =                    2 / Number of coordinate axes
WCSNAME = '47 Tuc     '        / Coordinate system title
CRPIX1  =               2048.0 / Pixel coordinate of reference point
CRPIX2  =               1024.0 / Pixel coordinate of reference point
PC1_1   =   1.290551569736E-05 / Coordinate transformation matrix element
PC1_2   =  5.9525007864732E-06 / Coordinate transformation matrix element
PC2_1   =  5.0226382102765E-06 / Coordinate transformation matrix element
PC2_2   = -1.2644844123757E-05 / Coordinate transformation matrix element
CDELT1  =                  1.0 / [deg] Coordinate increment at reference point
CDELT2  =                  1.0 / [deg] Coordinate increment at reference point
CUNIT1  = 'deg'                / Units of coordinate increment and value
CUNIT2  = 'deg'                / Units of coordinate increment and value
CTYPE1  = 'RA---TAN'           / TAN (gnomonic) projection + SIP distortions
CTYPE2  = 'DEC--TAN'           / TAN (gnomonic) projection + SIP distortions
CRVAL1  =        5.63056810618 / [deg] Coordinate value at reference point
CRVAL2  =      -72.05457184279 / [deg] Coordinate value at reference point
LONPOLE =                180.0 / [deg] Native longitude of celestial pole
LATPOLE =      -72.05457184279 / [deg] Native latitude of celestial pole
RADESYS = 'ICRS'                / Equatorial coordinate system


The following imports are generally useful:

>>> import numpy as np
>>> from astropy.modeling import models
>>> from astropy import coordinates as coord
>>> from astropy import units as u
>>> from gwcs import wcs
>>> from gwcs import coordinate_frames as cf


The forward_transform is constructed as a combined model using astropy.modeling. The frames are subclasses of CoordinateFrame. Although strings are acceptable as coordinate_frames it is recommended this is used only in testing/debugging.

Using the modeling package create a combined model to transform detector coordinates to ICRS following the FITS WCS standard convention.

First, create a transform which shifts the input x and y coordinates by CRPIX. We subtract 1 from the CRPIX values because the first pixel is considered pixel 1 in FITS WCS:

>>> shift_by_crpix = models.Shift(-(2048 - 1)*u.pix) & models.Shift(-(1024 - 1)*u.pix)


Create a transform which rotates the inputs using the PC matrix.

>>> matrix = np.array([[1.290551569736E-05, 5.9525007864732E-06],
...                    [5.0226382102765E-06 , -1.2644844123757E-05]])
>>> rotation = models.AffineTransformation2D(matrix * u.deg,
...                                          translation=[0, 0] * u.deg)
>>> rotation.input_units_equivalencies = {"x": u.pixel_scale(1*u.deg/u.pix),
...                                       "y": u.pixel_scale(1*u.deg/u.pix)}
>>> rotation.inverse = models.AffineTransformation2D(np.linalg.inv(matrix) * u.pix,
...                                                  translation=[0, 0] * u.pix)
>>> rotation.inverse.input_units_equivalencies = {"x": u.pixel_scale(1*u.pix/u.deg),
...                                               "y": u.pixel_scale(1*u.pix/u.deg)}


Create a tangent projection and a rotation on the sky using CRVAL.

>>> tan = models.Pix2Sky_TAN()
>>> celestial_rotation =  models.RotateNative2Celestial(5.63056810618*u.deg, -72.05457184279*u.deg, 180*u.deg)

>>> det2sky = shift_by_crpix | rotation | tan | celestial_rotation
>>> det2sky.name = "linear_transform"


Create a detector coordinate frame and a celestial ICRS frame.

>>> detector_frame = cf.Frame2D(name="detector", axes_names=("x", "y"),
...                             unit=(u.pix, u.pix))
>>> sky_frame = cf.CelestialFrame(reference_frame=coord.ICRS(), name='icrs',
...                               unit=(u.deg, u.deg))


This WCS pipeline has only one step - from detector to sky:

>>> pipeline = [(detector_frame, det2sky),
...             (sky_frame, None)
...            ]
>>> wcsobj = wcs.WCS(pipeline)
>>> print(wcsobj)
From      Transform
-------- ----------------
detector linear_transform
icrs             None


To convert a pixel (x, y) = (1, 2) to sky coordinates, call the WCS object as a function:

>>> sky = wcsobj(1*u.pix, 2*u.pix, with_units=True)
>>> print(sky)
<SkyCoord (ICRS): (ra, dec) in deg
(5.52515954, -72.05190935)>


The invert() method evaluates the backward_transform() if available, otherwise applies an iterative method to calculate the reverse coordinates.

>>> wcsobj.invert(sky)
(<Quantity 1. pix>, <Quantity 2. pix>)


#### Save a WCS object as a pure ASDF file¶

>>> from asdf import AsdfFile
>>> tree = {"wcs": wcsobj}
>>> wcs_file = AsdfFile(tree)
>>> wcs_file.write_to("imaging_wcs.asdf")


Listing of imaging_wcs.asdf

#### Save a WCS object as an ASDF extension in a FITS file¶

>>> from astropy.io import fits
>>> from asdf import fits_embed
>>> hdul = fits.open("example_imaging.fits")
>>> hdul.info()
Filename: example_imaging.fits
No.    Name      Ver    Type      Cards   Dimensions   Format
0  PRIMARY       1 PrimaryHDU     775   ()
1  SCI           1 ImageHDU        71   (600, 550)   float32
>>> tree = {"sci": hdul[1].data,
...         "wcs": wcsobj}
>>> fa = fits.embed.AsdfInFits(hdul, tree)
>>> fa.write_to("imaging_with_wcs_in_asdf.fits")
>>> fits.info("imaging_with_wcs_in_asdf.fits")
Filename: example_with_wcs.asdf
No.    Name      Ver    Type      Cards   Dimensions   Format
0  PRIMARY       1 PrimaryHDU     775   ()
1  SCI           1 ImageHDU        71   (600, 550)   float32
2  ASDF          1 BinTableHDU     11   1R x 1C   [5086B]


#### Reading a WCS object from a file¶

ASDF is used to read a WCS object from a pure ASDF file or from an ASDF extension in a FITS file.

>>> import asdf
>>> asdf_file = asdf.open("imaging_wcs.asdf")
>>> wcsobj = asdf_file.tree['wcs']

>>> import asdf
>>> fa = asdf.open("imaging_with_wcs_in_asdf.fits")
>>> wcsobj = fa.tree["wcs"]


## gwcs.wcs Module¶

### Classes¶

 WCS([forward_transform, input_frame, …]) Basic WCS class. NoConvergence(*args[, best_solution, …]) An error class used to report non-convergence and/or divergence of numerical methods.

## gwcs.coordinate_frames Module¶

Defines coordinate frames and ties them to data axes.

### Classes¶

 Frame2D([axes_order, unit, axes_names, …]) A 2D coordinate frame. CelestialFrame([axes_order, …]) Celestial Frame Representation SpectralFrame([axes_order, reference_frame, …]) Represents Spectral Frame CompositeFrame(frames[, name]) Represents one or more frames. CoordinateFrame(naxes, axes_type, axes_order) Base class for Coordinate Frames. TemporalFrame(reference_frame[, unit, …]) A coordinate frame for time axes.

## gwcs.wcstools Module¶

### Functions¶

 wcs_from_fiducial(fiducial[, …]) Create a WCS object from a fiducial point in a coordinate frame. grid_from_bounding_box(bounding_box[, step, …]) Create a grid of input points from the WCS bounding_box. wcs_from_points(xy, world_coordinates, fiducial) Given two matching sets of coordinates on detector and sky, compute the WCS.

## gwcs.selector Module¶

The classes in this module create discontinuous transforms.

The main class is RegionsSelector. It maps inputs to transforms and evaluates the transforms on the corresponding inputs. Regions are well defined spaces in the same frame as the inputs. Regions are assigned unique labels (int or str). The region labels are used as a proxy between inputs and transforms. An example is the location of IFU slices in the detector frame.

RegionsSelector uses two structures:
• A mapping of inputs to labels - “label_mapper”

• A mapping of labels to transforms - “transform_selector”

A “label_mapper” is also a transform, a subclass of astropy.modeling.core.Model, which returns the labels corresponding to the inputs.

An instance of a LabelMapper class is passed to RegionsSelector. The labels are used by RegionsSelector to match inputs to transforms. Finally, RegionsSelector evaluates the transforms on the corresponding inputs. Label mappers and transforms take the same inputs as RegionsSelector. The inputs should be filtered appropriately using the inputs_mapping argument which is ian instance of Mapping. The transforms in “transform_selector” should have the same number of inputs and outputs.

This is illustrated below using two regions, labeled 1 and 2

+-----------+
| +-+       |
| | |  +-+  |
| |1|  |2|  |
| | |  +-+  |
| +-+       |
+-----------+

                   +--------------+
| label mapper |
+--------------+
^       |
|       V
----------|   +-------+
|             | label |
+--------+      +-------+
--->     | inputs |          |
+--------+          V
|          +--------------------+
|          | transform_selector |
|          +--------------------+
V                  |
+-----------+           |
| transform |<-----------
+------------+
|
V
+---------+
| outputs |
+---------+


The base class _LabelMapper can be subclassed to create other label mappers.

### Classes¶

 LabelMapperArray(mapper[, inputs_mapping, name]) Maps array locations to labels. LabelMapperDict(inputs, mapper[, …]) Maps a number to a transform, which when evaluated returns a label. LabelMapperRange(inputs, mapper[, …]) The structure this class uses maps a range of values to a transform. RegionsSelector(inputs, outputs, selector, …) This model defines discontinuous transforms. LabelMapper(inputs, mapper[, no_label, …]) Maps inputs to regions.

## gwcs.spectroscopy Module¶

Spectroscopy related models.

### Classes¶

 Solve the Grating Dispersion Law for the wavelength. AnglesFromGratingEquation3D(groove_density, …) Solve the 3D Grating Dispersion Law in Direction Cosine space for the refracted angle. Snell3D(**kwargs) Snell model in 3D form. SellmeierGlass(B_coef, C_coef, **kwargs) Sellmeier equation for glass. SellmeierZemax([temperature, …]) Sellmeier equation used by Zemax.

## gwcs.geometry Module¶

Models for general analytical geometry transformations.

### Classes¶

 ToDirectionCosines(**kwargs) Transform a vector to direction cosines. FromDirectionCosines(**kwargs) Transform directional cosines to vector. SphericalToCartesian([wrap_lon_at]) Convert spherical coordinates on a unit sphere to cartesian coordinates. CartesianToSpherical([wrap_lon_at]) Convert cartesian coordinates to spherical coordinates on a unit sphere.