We are using stacks of 100 images. These are acquired by the CCD camera in rapid succession – 100 images can be obtained in less than a minute. During that time there may be small motions of the telescope, and the turbulence in the air, as well as the slight change in refraction due to airmass changes may cause image drifts in the stack.

In one approach we ignored the possible drift and just averaged the 100 images. In the other approach we use image alignment techniques to iteratively improve the alignment of the stack images: First we calculate the regular average, then we align all stack images against that average image, then we calculate a new average image on the basis of the alignments and re-align all images against this average and so on. This procedure turns out to converge, and after 3 iterations we can stop and save the last average image.

We would like to know the effect of doing this on result quality. We therefore generated two sets of averaged images – the first using the simple first kind above, and the second the iterative method.

We estimate ‘errors’ in the bootstrapping way. That is, we extract MHM averages (mean-half-median, as explained elsewhere) of the DS intensity on raw and cleaned-up image patches and also estimate the statistical error on these values by bootstrapping the pixels inside the patch, with replacement. This bootstrap procedure gives us a histogram og MHM values and the width of this distribution is a measure of the ‘error on the mean’. We express this error as a percentage of the mean itself, for RAW images and images cleaned with the BBSO-lin, BBSO-log and EFM methods.

we now compare the results to see the ‘effect’ of performing alignment of stack images:

PSF Alignment RAW EFM BBSOlin BBSOlog
1: one without 0.73 1.25 0.74 0.79
2: one with 0.57 1.03 0.62 0.68
3: two with 0.58 0.99 0.63 0.69

Table showing errors (in percent of the mean). Lines labelled 1: and 2: show results for ‘one-alfa PSFs’ with and without alignment. The third line shows the effect of using a ‘two-alfa PSF’ and alignment.

We see that there has been a large reduction in errors by using alignment. Raw images improved by 20%, EFM images by 17%, BBSO-lin by 16% and BBSO-log by 14%.

The effect of using a two-alfa PSF on aligned images is small – indeed, all images except EFM experience a small increase in the error (probably not significant).

The effect of alignment on single-alfa PSFs is not investigated.

We conclude that alignment is a beneficial operation. We note that EFM has the largest errors but other arguments imply EFM is the better method to use – this is related to the stronger phase-dependence seen in non-EFM images, and is discussed elsewhere.