## 4. Registration of HST and ground based CCD imagesFour 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 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 This method assumes implicitly that these two images, the degraded
HST image and the ground based one, are In the one-dimensional case, the cross correlation between two identical images is given by where the integral is extended to the whole image. Its maximum is located at where For small values of is equal to zero because the cross correlation has a maximum for ; on the other hand Then If the images are not identical, the cross correlation function is given by where . For small values of where and The maximum of the correlation function is then located at This approximation is valid as long as . 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 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 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.
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 (Cousins R) filter, might explain this result. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |