In an interesting article Sally Langford, et al. [ http://adsabs.harvard.edu/abs/2009AsBio...9..305L ] describe an analysis method for earthshine images. Essentially, the Laplacian's effect on images is to detect edges and the size of the 'jump' in going from e.g. sky to lunar disc in the observed image translates into an amplitude of a characteristic signature of the derivatives of an edge. This enables the large-spatial-scale features of the scattered light halo to be removed from the short-spatial-scale features of edges and craters.
Our method for removing effects of halo is to construct a forward model of the halo, based on the BS of the Moon (caught in the same image as the DS - Sally's FOV is smaller - about 20 arc minutes; our is 60 arc minutes), and subtracting it, leaving the DS almost halo-free.
How well does the Laplacian method work on our type of images? Sally's images were well-exposed images of the DS alone - allowing large counts - 10's of thousands; ours are co-addition of 100 images taken so that the BS is not overexposed, leaving the signal on the DS typically near 10 counts. One image with 10,000 counts should thus have the SNR of 100 of ours.
We took one of our good images - from JD 2456095 - near New Moon - and plot the profile across the image near disc centre in the image that we obtain using the forward modelling method (named 'EFM' in the plot and this blog) and the same profile from the Laplacian of the image. We average 20 rows.
Inspection of the columns (approx # 100-120) where the lunar disc edge is we see a clear 'step' up from the sky level in the EFM-cleaned image but there is no sign, above the noise, in the Laplacian of the same image. Inspecting the whole Laplacian of the EFM image we do see a faint signature:
The above image is 'histogram equalized' to show the feature. Our own method - also shown when histogram-equalized is here:
We see that the halo has been only partially removed on the right.
I think we would be hard pressed to extract a signal from the Laplacian of the image. It also bothers me that the whole earthshine signal is reduced to the value of the derivatives in just edge pixels. In our own image we have hundreds of pixels to measure on - in the Laplacian we get only a signal along the edge.
We should note that the remnants of halo on the right are much less evident in the Laplacian image, suggesting that there is a kernel of a good idea in the method. For reference, the Laplacian of the raw observed image is here (histogram equalized):
It does seem as if the Laplacian helps remove a lot of the halo, but also reduces the analyzable part of the image to the DS edge.
The Langford paper mentions both smoothing images and co-adding them - the former is done at the resolution of the worst image for a sequence. The Langford paper analyses features on the disk - not the edge.
Working out how the Laplacian is best used, on realistic images of the type we have, would be a good student project!