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### Determining transformation parameters

The routine which determines the transformations between labelled position lists is called:

Labelled position lists are those in which the same objects (image features) have the same identification number. So for instance if a star say no. 100 is present on several datasets it should be labelled no. 100 (or any other unique value) in all the datasets in which it appears regardless of its coordinates. In many ways getting labelled position lists may be considered as most of the process of object-matching-based registration, the rest being essentially straight-forward.

The main mapping used by REGISTER is the linear mapping:

XX = A + B * X + C * Y
YY = D + E * X + F * Y
where the A-F are the coefficients which are to be determined, X and Y the current coordinates and XX and YY the new coordinates. REGISTER supports various types of linear transformations namely:
• a shift of origin
• a shift of origin and rotation
• a shift of origin and magnification
• a shift of origin, rotation and magnification (solid body)
• or a full six parameter fit.

When using a linear fit you can register a whole list of datasets in one go. So for instance if you have a set of position lists in which corresponding objects have been identified, you may pick any list as the reference set (the first is chosen by default) and all the mappings between this and the other datasets will be derived. If the position lists are `associated' with images then a new coordinate system, by default labelled `CCD_REG', will be added to each image: the coordinates are the same as the pixel coordinates of the reference image.

A general transformation between two datasets can also be determined. These are entered using suitably parameterised algebraic expressions. A general least squares fitting algorithm is used to find a solution which gives satisfactory values for the parameters.

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CCDPACK