World Coordinates

World coordinates are the ``real'' equatorial coordinates of objects in catalogs: right ascension, declination and their epoch.^5.1 Given an image frame, we reffer to the transformation between $x$ and $y$ pixel coordinates and their world coordinate counterparts as the World Coordinte System (WCS for short) of the frame.

The transformation between the spherical equatorial and the ``flat'' image coordinates cannot be done without choosing a projection system. GCX uses the plane-tangent projection system, which is appropiate for relatively narrow fields.^5.2

World Coordinate System Parameters

In the plane-tangent system, the WCS is specified by the following values:

1. The frame coordinates of a reference pixel in the image (usually the center of the frame) in the CRPIX1 and CRPIX2 fits header fields;
2. The world coordinates (r.a.  and dec) of the reference pixel in the CRVAL1 and CRVAL2 fields;
3. The epoch of the coordinates in the EQUINOX header field;
4. The horisontal and vertical scale of the image in degrees per pixel in the CDELT1 and CDELT2 fields;
5. The rotation of the frame in the CROTA1 field.

A slightly different form of these parameters is presented in the WCS editing dialog: the scale parameters are expressed in the more friendly arc seconds per pixel units, and the coordinates are expressed in the HMS and DMS formats.

World Coordinate System States

A given frame's WCS can be in one of the following states:

Unset

When the WCS is unset, the program has no idea about the WCS. It will refuse to do any operation that requires the WCS.

Initial

An initial WCS is an approximate set of values for the WCS parameters. It enables the program to load catalog stars and display them on the image (more or less around their true positions). It also provides a starting point for WCS fitting. GCX will not use an initial WCS for any operation that requires precise coordinates (like aperture photometry).

Fitted

The WCS has been successfully fitted, but the quality of the fit was not enough to allow it to be validated. A fitted WCS is treated very much like an initial WCS.

Valid

If a fit was good enough (enough stars were fitted, and the error was low enough), the WCS is deemed valid. All operations that use the WCS are enabled in this situation.

Obtaining an Initial WCS

When a frame is loaded, the WCS is initially unset. The header of the frame is searched for information about the initial WCS. The following fields are searched, in order:^5.3

1. CRVAL1/2, CDELT1/2, CROTA1, CRPIX1/2, EQUINOX. The bare minimum set consists ofCRVAL1, CRVAL2 and one of the CDELTs.
2. RA or OBJCTRA, DEC or OBJCTDEC, PIXSCALE or SECPIX. If neither of the scale fields is found, a default scale values is taken from Wcs Fitting Options/Default image scale;
3. OBJECT If this field is present, the object's name is searched in the catalog, and its coordinates used. The image scale is set from Wcs Fitting Options/Default image scale;

When neither of the above fields are found, the WCS is left in the unset state. An initial WCS can be set in this case by either entering the parameters in the WCS edit dialog (Wcs/Edit Wcs), loading a catalog object using Stars/Add From Catalog or loading a recipy file that has the target object or field center specified. In the last two cases, the default scale is used.

Fitting the WCS to an Image

By WCS fitting we understand the process of comparing the positions of stars extracted from the image frame versus the projected positions of catalog stars, and the subsequent adjustment of the WCS for the best match.

The fitting process consists of the following steps:

1. Detecting frame stars. This step is described in section 4.1;
2. Obtaining catalog stars for the match. These can come from either a recipy file or one of the field stars catalogs. The program will load stars from the Tycho2 and GSC catalogs. All the stars from a loaded recipy file that have the ``astrimetric'' flag set will also be used for WCS fitting;
3. Finding star pairs. This step tries to find similar asterism in the detected and catalog sets and match the corresponding stars.

   The algorithm tolerates frame rotation and changes in scale. If some bounds can be placed on initial errors (for instance if we know that only a limited rotation range is expected) it is possible to pass that information to the algorithm in order to narrow the search.

4. Fitting the solution. This is an iterative step consisting of calculating the required offset, scale and rotation in the frame coordinates, then adjusting the WCS accordingly. After that, the image coordinates of the catalog stars are recalculated and the step repeated until there is no significant change in the WCS. The iterative approach is necessary because the projection operation is non-linear. At the end of the fitting step, a rms position error is calculated, and compared to the value of the Max error for WCS validation. If the error is lower and enough pairs have been used in the fit (more than Min pars for WCS validation), the WCS is marked ``valid''.

The Scale tolerance option sets the maximum initial error of the image scale for the pairing alogorithm. A value of 0.1 specifies that the scale of the initial WCS has an error of at most $\pm 10\%$. The Rotation tolerance specifies how much field rotation is expected by the pairs matching algorithm. A value of 180 will let the algorithm match frames of any rotation. A third important parameter is Minimum number of pairs. This specifies the number of pairs at which the algorithm decides it has found a match. The default values for these parameters almost never generate a bad match, even for quite dense fields. If one increases the scale tolerance, there is an increased risk of having a bad match, and the minimum pairs should be increased as well.

The pairing algorithm requires the initial WCS to have the correct mirroring. When the initial WCS's scale comes from the CDELT1/2 fields, their signs will determine the mirroring: when both have the same sign, the frame is ``normal'', i.e. W is to the right when N is up. If the signs are different, the field is flipped.

When the initial WCS's scale comes from a single scale parameter, the mirroring will be set by the program according to the value of the General Observation Setup Data/Flipped field option.

WCS Fitting Commands

The WCS fitting steps can be performed one at a time, or all together. The Wcs/Auto Wcs operation will do the following steps: Stars/Detect sources, File/Load Field Stars/From Tycho2 Catalog, Wcs/Auto pairs, Wcs/Fit Wcs from pairs. The Wcs/Quiet Auto Wcs variant will also delete the detected and field stars at the end of the fit.

Selecting Wcs/Reload from frame will revert the WCS to the parameters before the fit. The pairs will remain marked.

In the unlikely event that the pairing algorithm fails,^5.4 it is possible to create pairs ``by hand''. Select a detected star, then right-click on the catalog star you want to pair it with and select Create Pair from the pop-up menu. When at least 2 pairs have been marked, we can fit the wcs with Wcs/Fit Wcs from Pairs. Note that the fit will not be marked as ``valid'' unless at least Minimum number of pairs have been marked.