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The FITS standard defines the world coordinate system (WCS) as a mechanism to associate physical values to positions within a dataset. For example, it can be used to convert pixel coordinates in an image to celestial coordinates like the right ascension and declination. The functions in this section are mainly just wrappers over CFITSIO, WCSLIB and GSL library functions to help in common applications.

- Macro:
**GAL_WCS_DISTORTION_TPD** - Macro:
**GAL_WCS_DISTORTION_SIP** - Macro:
**GAL_WCS_DISTORTION_TPV** - Macro:
**GAL_WCS_DISTORTION_DSS** - Macro:
**GAL_WCS_DISTORTION_WAT** - Macro:
**GAL_WCS_DISTORTION_INVALID** -
Gnuastro identifiers of the various WCS distortion conventions, for more, see Calabretta et al. (2004, preprint)

^{183}. Among these, SIP is a prior distortion, the rest other are sequent distortions. TPD is a superset of all these, hence it has both prior and sequeal distortion coefficients. More information is given in the documentation of`dis.h`

, from the WCSLIB manual^{184}.

- Macro:
**GAL_WCS_FLTERROR** Limit of rounding for floating point errors.

- Function:

*struct wcsprm **

**gal_wcs_read_fitsptr***(fitsfile*`*fptr`

, size_t`hstartwcs`

, size_t`hendwcs`

, int`*nwcs`

) [

**Not thread-safe**] Return the WCSLIB`wcsprm`

structure that is read from the CFITSIO`fptr`

pointer to an opened FITS file. Also put the number of coordinate representations found into the space that`nwcs`

points to. To read the WCS structure directly from a filename, see`gal_wcs_read`

below. After processing has finished, you can free the returned structure with WCSLIB’s`wcsvfree`

keyword:status = wcsvfree(&nwcs,&wcs);

If you don’t want to search the full FITS header for WCS-related FITS keywords (for example due to conflicting keywords), but only a specific range of the header keywords you can use the

`hstartwcs`

and`hendwcs`

arguments to specify the keyword number range (counting from zero). If`hendwcs`

is larger than`hstartwcs`

, then only keywords in the given range will be checked. Hence, to ignore this feature (and search the full FITS header), give both these arguments the same value.If the WCS information couldn’t be read from the FITS file, this function will return a

`NULL`

pointer and put a zero in`nwcs`

. A WCSLIB error message will also be printed in`stderr`

if there was an error.This function is just a wrapper over WCSLIB’s

`wcspih`

function which is not thread-safe. Therefore, be sure to not call this function simultaneously (over multiple threads).

- Function:

*struct wcsprm **

**gal_wcs_read***(char*`*filename`

, char`*hdu`

, size_t`hstartwcs`

, size_t`hendwcs`

, int`*nwcs`

) [

**Not thread-safe**] Return the WCSLIB structure that is read from the HDU/extension`hdu`

of the file`filename`

. Also put the number of coordinate representations found into the space that`nwcs`

points to. Please see`gal_wcs_read_fitsptr`

for more.

- Function:

*void*

**gal_wcs_write***(struct wcsprm*`*wcs`

, char`*filename`

, char`*extname`

, gal_fits_list_key_t`*headers`

, char`*program_string`

) Write the given WCS structure into the second extension of an empty FITS header. The first/primary extension will be empty like the default format of all Gnuastro outputs. When

`extname!=NULL`

it will be used as the FITS extension name. Any set of extra headers can also be written through the`headers`

list and if`program_string!=NULL`

it will be used in a commented keyword title just above the written version information.

- Function:

*void*

**gal_wcs_write_in_fitsptr(fitsfile**`*fptr`

, struct wcsprm`*wcs`

) Convert the input

`wcs`

structure (keeping the WCS programmatically) into FITS keywords and write them into the given FITS file pointer. This is a relatively low-level function which assumes the FITS file has already been opened with CFITSIO. If you just want to write the WCS into an empty file, you can use`gal_wcs_write`

(which internally calls this function after creating the FITS file and later closes it safely).

- Function:

*struct wcsprm **

**gal_wcs_copy***(struct wcsprm*`*wcs`

) Return a fully allocated (independent) copy of

`wcs`

.

- Function:

*void*

**gal_wcs_remove_dimension***(struct wcsprm*`*wcs`

, size_t`fitsdim`

) Remove the given FITS dimension from the given

`wcs`

structure.

- Function:

*void*

**gal_wcs_on_tile***(gal_data_t*`*tile`

) Create a WCSLIB

`wcsprm`

structure for`tile`

using WCS parameters of the tile’s allocated block dataset, see Tessellation library (`tile.h`) for the definition of tiles. If`tile`

already has a WCS structure, this function won’t do anything.In many cases, tiles are created for internal/low-level processing. Hence for performance reasons, when creating the tiles they don’t have any WCS structure. When needed, this function can be used to add a WCS structure to each tile tile by copying the WCS structure of its block and correcting the reference point’s coordinates within the tile.

- Function:

*double **

**gal_wcs_warp_matrix***(struct wcsprm*`*wcs`

) Return the Warping matrix of the given WCS structure as an array of double precision floating points. This will be the final matrix, irrespective of the type of storage in the WCS structure. Recall that the FITS standard has several methods to store the matrix. The output is an allocated square matrix with each side equal to the number of dimensions.

- Function:

*void*

**gal_wcs_clean_small_errors***(struct wcsprm*`*wcs`

) Errors can make small differences between the pixel-scale elements (

`CDELT`

) and can also lead to extremely small values in the`PC`

matrix. With this function, such errors will be “cleaned” as follows: 1) if the maximum difference between the`CDELT`

elements is smaller than the reference error, it will be set to the mean value. When the FITS keyword`CRDER`

(optional) is defined it will be used as a reference, if not the default value is`GAL_WCS_FLTERROR`

. 2) If any of the PC elements differ from 0, 1 or -1 by less than`GAL_WCS_FLTERROR`

, they will be rounded to the respective value.

- Function:

*void*

**gal_wcs_decompose_pc_cdelt***(struct wcsprm*`*wcs`

) Decompose the

`PCi_j`

and`CDELTi`

elements of`wcs`

. According to the FITS standard, in the`PCi_j`

WCS formalism, the rotation matrix elements \(m_{ij}\) are encoded in the`PCi_j`

keywords and the scale factors are encoded in the`CDELTi`

keywords. There is also another formalism (the`CDi_j`

formalism) which merges the two into one matrix.However, WCSLIB’s internal operations are apparently done in the

`PCi_j`

formalism. So its outputs are also all in that format by default. When the input is a`CDi_j`

, WCSLIB will still read the matrix directly into the`PCi_j`

matrix and the`CDELTi`

values are set to`1`

(one). This function is designed to correct such issues: after it is finished, the`CDELTi`

values in`wcs`

will correspond to the pixel scale, and the`PCi_j`

will correction show the rotation.

- Function:

*int*

**gal_wcs_distortion_from_string***(char*`*distortion`

) Convert the given string (assumed to be a FITS-standard, string-based distortion identifier) to a Gnuastro’s integer-based distortion identifier (one of the

`GAL_WCS_DISTORTION_*`

macros defined above). The sting-based distortion identifiers have three characters and are all in capital letters.

- Function:

*int*

**gal_wcs_distortion_to_string***(int*`distortion`

) Convert the given Gnuastro integer-based distortion identifier (one of the

`GAL_WCS_DISTORTION_*`

macros defined above) to the string-based distortion identifier) of the FITS standard. The sting-based distortion identifiers have three characters and are all in capital letters.

- Function:

*int*

**gal_wcs_distortion_identify***(struct wcsprm*`*wcs`

) Returns the Gnuastro identifier for the distortion of the input WCS structure. The returned value is one of the

`GAL_WCS_DISTORTION_*`

macros defined above. When the input pointer to a structure is`NULL`

, or it doesn’t contain a distortion, the returned value will be`GAL_WCS_DISTORTION_INVALID`

.

- Function:

*struct wcsprm **

**gal_wcs_distortion_convert(struct***wcsprm*`*inwcs`

, int`outdisptype`

, size_t`*fitsize`

) Return a newly allocated WCS structure, where the distortion is implemented in a different standard, identified by the identifier

`outdisptype`

. The Gnuastro WCS distortion identifiers are defined in the`GAL_WCS_DISTORTION_*`

macros mentioned above.The available conversions in this function will grow. Currently it only supports converting TPV to SIP and vice versa, following the recipe of Shupe et al. (2012)

^{185}. Please get in touch with us if you need other types of conversions.For some conversions, direct analytical conversions don’t exist. It is thus necessary to model and fit the two types. In such cases, it is also necessary to specify the

`fitsize`

array that is the size of the array along each C-ordered dimension, so you can simply pass the`dsize`

element of your`gal_data_t`

dataset, see Generic data container (`gal_data_t`

). Currently this is only necessary when converting TPV to SIP. For other conversions you may simply pass a`NULL`

pointer.For example, if you want to convert the TPV coefficients of your input

`image.fits`to SIP coefficients, you can use the following functions (which are also available as a command-line operation in Fits).int nwcs; gal_data_t *data=gal_fits_img_read("image.fits", "1", -1, 1); inwcs=gal_wcs_read("image.fits", "1", 0, &nwcs); data->wcs=gal_wcs_distortion_convert(inwcs, GAL_WCS_DISTORTION_TPV, NULL); wcsfree(inwcs); gal_fits_img_write(data, "tpv.fits", NULL, NULL);

- Function:

*double*

**gal_wcs_angular_distance_deg***(double*`r1`

, double`d1`

, double`r2`

, double`d2`

) Return the angular distance (in degrees) between a point located at (

`r1`

,`d1`

) to (`r2`

,`d2`

). All input coordinates are in degrees. The distance (along a great circle) on a sphere between two points is calculated with the equation below.$$\cos(d)=\sin(d_1)\sin(d_2)+\cos(d_1)\cos(d_2)\cos(r_1-r_2)$$

However, since the pixel scales are usually very small numbers, this function won’t use that direct formula. It will be use the Haversine formula which is better considering floating point errors:

$${\sin^2(d)\over 2}=\sin^2\left( {d_1-d_2\over 2} \right)+\cos(d_1)\cos(d_2)\sin^2\left( {r_1-r_2\over 2} \right)$$

- Function:

*double **

**gal_wcs_pixel_scale***(struct wcsprm*`*wcs`

) Return the pixel scale for each dimension of

`wcs`

in degrees. The output is an allocated array of double precision floating point type with one element for each dimension. If its not successful, this function will return`NULL`

.

- Function:

*double*

**gal_wcs_pixel_area_arcsec2***(struct wcsprm*`*wcs`

) Return the pixel area of

`wcs`

in arcsecond squared. If the input WCS structure is not two dimensional and the units (`CUNIT`

keywords) are not`deg`

(for degrees), then this function will return a NaN.

- Function:

*gal_data_t **

**gal_wcs_world_to_img***(gal_data_t*`*coords`

, struct wcsprm`*wcs`

, int`inplace`

) Convert the linked list of world coordinates in

`coords`

to a linked list of image coordinates given the input WCS structure.`coords`

must be a linked list of data structures of float64 (‘double’) type, seeLinked lists (`list.h`) and List of`gal_data_t`

. The top (first popped/read) node of the linked list must be the first WCS coordinate (RA in an image usually) etc. Similarly, the top node of the output will be the first image coordinate (in the FITS standard).If

`inplace`

is zero, then the output will be a newly allocated list and the input list will be untouched. However, if`inplace`

is non-zero, the output values will be written into the input’s already allocated array and the returned pointer will be the same pointer to`coords`

(in other words, you can ignore the returned value). Note that in the latter case, only the values will be changed, things like units or name (if present) will be untouched.

- Function:

*gal_data_t **

**gal_wcs_img_to_world***(gal_data_t*`*coords`

, struct wcsprm`*wcs`

, int`inplace`

) Convert the linked list of image coordinates in

`coords`

to a linked list of world coordinates given the input WCS structure. See the description of`gal_wcs_world_to_img`

for more details.

https://www.atnf.csiro.au/people/mcalabre/WCS/dcs_20040422.pdf

https://www.atnf.csiro.au/people/mcalabre/WCS/wcslib/dis_8h.html

Proc. of SPIE Vol. 8451 84511M-1. https://doi.org/10.1117/12.925460, also available at http://web.ipac.caltech.edu/staff/shupe/reprints/SIP_to_PV_SPIE2012.pdf.

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GNU Astronomy Utilities 0.13 manual, September 2020.