Our observed images are a convolution of a point-spread function (PSF) and the Object (the Moon). We know from analysis of images of various sources – the Moon itself, Jupiter, and stars, that the PSF has the character of a power law along the lines of a core with wings that follow 1/r^alfa with alfa near 2.9.
It is in principle also possible to determine the PSF by de-convolution. Clasically we would expect that
Image = Object * PSF
Where the ‘*’ implies spatial convolution. Taking FT of both sides we get
F(I) = F(O) x F(P)
where F is the forward Fourier transform, and the ‘x’ implies multiplication. Rearranging and taking inverse Fourier transforms (f) we get:
P = f(F(P)) = f(F(I)/F(O))
We have I (the image) and if we had O we could calculate the PSF P. Usually P is very noisy because of amplification of small noisy signals at high frequencies due to the division above. We can average over several estimates of P, however.
For us, this is easy because our data come in the form of 100-image stacks. Unfortunately we do not know O, the object. But we can make synthetic images of O that are highly realistic!
We have applied the above procedure on an image-stack of the Moon, for which we also have a synthetic image. We average the 100 PSFs that are generated. The radial profile from the peak looks like this (black curve):
The blue line is a power law with alfa=2.1 and the red line is a power law with slope 0.3. For this particular night we know that a single power law (plus a narrow table-lookup core) of alfa=2.6 has been estimated for this image using forward modelling techniques where we fit parts of the scattered halo at about 100-150 pixels distance from the photo-centre of the bright side of the Moon.
The two estimates of the PSF are thus quite different, and I wonder why.
Possibly it is a sign that the edge-fitting can be performed well with a large family of PSFs. We know that the fit is excellent at the edge of the disc.
I suggest that a next step could be to test both PSFs or types of PSFs and see how they perform in various situations. How close to the observed image is the ideal image once it is convolved with the above profile?
Did some thinking about this — I am not sure we are comparing the same indices in the two power laws…
for the forward fitting, we start with a PSF with powerlaw fallloff in the outer parts with a power law index = -1.6, then raise this profile to the power -alpha — typically, 1.3 to 1.7, very much depending on the night.
The resulting power law is then the sum of these indices, so we are getting a falloff typically of 2.9 to 3.3 in the final fitted PSF. This should be the one that inverse convolution recovers, so I’d expect a falloff of order -3 in the outer parts???
REALLY interesting! I’ll try this myself – I got part way into it at one point and then stopped for some distraction… the difference in the power law of the PSF in the outer parts is quite surprising. Great that the edge light can be fit much better!