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I can't see how any algorith could determine what is the picture of interest, and what is the artifact noise itself, to be able extract it form there; both seems as having similar intensity and 'color'. In my oppinion this shouldn't be managed in post-processing, but rather at the image capture stage in the optical scope, perhaps by adding front lenses with a non-reflexive finishing.
There exists the possibility that coeficients taken form a picture could not be applied to another picture in having few changes on illumination or target object. Anyway, considering that OP asked for a post-processing solution to the specific above picture, I admit that my reply do not cover the issue as requested.
Thank you very much for the reply. Here is another picture of the diffraction.
In the image I photograph a single dot ( bottom right) but three other dots appear around it. By taking a 2D FFT and working out FIR coefficients will that help remove the higher diffraction orders and preserve the main region of Interest?
Graphics editors have effects such as Sharpen. It does an imperfect job yet it's remarkable such an algorithm can be undertaken at all. It needs to evaluate how pixels near a sharp dot change in brightness if that sharp dot expands outward. Anyway perhaps you can adapt some tool from the Effects menu? For example, Swap Colors... by using the eyedropper tool you can replace the three unwanted grey dots with a neighboring background color. The colors are similar but not identical, therefore you need to remove each dot one at a time. This same tactic could work on your first image (post #1). Of course it gets tedious if you must do it to a lot of images.
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