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Astron. Astrophys. 318, 997-1002 (1997)

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4. Registration of HST and ground based CCD images

Four parameters are needed to determine the orientation of the X,Y coordinate systems of HST and ground based images, namely the rotation angle, the scale factor and the shifts in the X and Y direction. They have been estimated using a cross correlation method. First, the deconvolved HST image has been convolved with the observed ground based point spread function. Then, the cross correlation function between this degraded HST image and the ground based image has been computed. The best estimates of the transformation parameters are obtained when the peak of the cross correlation function reaches the maximum amplitude.

For a successful application of this method, and particularly to determine the scale factor and the rotation angle, structures much greater than the resolution of the ground based image must be present in the HST image like, for example, a second object in the field; however, for nearby galaxies, the structure of the galaxy itself may be used. In this case the HST field of view also included NGC 5930.

This method assumes implicitly that these two images, the degraded HST image and the ground based one, are identical which, due to statistics and electronic noise, is never strictly the case. Also, the filter transmission, and therefore the point spread function, is slightly different. This leads to uncertainties in the registration which can be estimated as follow.

In the one-dimensional case, the cross correlation between two identical images is given by

[EQUATION]

where the integral is extended to the whole image. Its maximum is located at [FORMULA] where

[EQUATION]

For small values of x the cross correlation function can be approximated as

[EQUATION]

[FORMULA] is equal to zero because the cross correlation has a maximum for [FORMULA] ; on the other hand

[EQUATION]

Then

[EQUATION]

If the images are not identical, the cross correlation function is given by

[EQUATION]

where [FORMULA].

For small values of x, the derivative of [FORMULA] is given by

[EQUATION]

where

[EQUATION]

and

[EQUATION]

The maximum of the correlation function is then located at

[EQUATION]

This approximation is valid as long as [FORMULA].

All these quantities can be derived directly from the images and from these we obtain an estimate of the displacement of the peak of the correlation function [FORMULA] with respect to the optimal alignment, i.e. the error in the registration. In our case the error is 0.10 pixels in the ground based image, corresponding to 47 mas. The reliability of this method has been checked empirically by shifting the images from their optimal alignment by [FORMULA] 1 pixels at 0.2 pixels steps in the X direction and then estimating the errors, which are given in Table 3. The close correspondence of the offsets with the errors testifies to the reliability of this method.


[TABLE]

Table 3. Stability of the cross-correlation between HST and ground based images. Offset represents the shift from the calculated optimal alignment between the two frames


The error estimate has also been performed on the other two ground based CCD images taken with shorter exposure time. The alignment errors are 48 and 40 mas, similar to those obtained with the longer exposure. It therefore appears that the statistical errors are not the main source of uncertainty. Systematic differences between HST and ground based images, as for example the transmission curve of the F718M filter slightly narrower than the standard [FORMULA] (Cousins R) filter, might explain this result.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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