What did the Point Spread Function (PSF) ever do for us?

[NickK]

What did the Point Spread Function (PSF) ever do for us?

Unfortunately quite a bit. It's part of the reason my image doesn't look like hubble for a start..

Here you can see the additional PSF between bubble's image and a basic stacked image of mine. It's the reason why everything is a little bit more blurred. However even Hubble has a PSF! We can look at an ideal mathematically generated PSF and show the PSF between Hubble and my image - lets just look at the bright star in the top left of the hubble image:

Strictly speaking the ideal image is also a PSF caused by the circular aperture.. as we're all looking through circular things.. then lets just ignore that point

Lets look at each star with a plot of the image. If you imagine the difference between the plots in 3D - that's the point spread function.

First the ideal PSF:

You'll note that I've shown a 2D slice through to make it easier to see. The centre bright spot is in the middle, the brightness of the star then quickly drops creating a nice around appearance. But wait - what's the little rings around it? Well these are caused by looking through a circular aperture - known as defraction - and they appear as the little bumps at the bottom of the graph.

Next Hubble's:

The top is is flat showing that the centre of the star was too bright. however the sides quickly drop evenly away and it looks very close to the ideal plot above. You'll note the spikes caused by Hubble's secondary mirror supports - a natural additional "defraction spikes" that reflector owners put up with.

Then finally mine:

The first thing you'll note is the gentle slopes of the image that aren't circular. I'm using a refractor so the image doesn't have defraction spikes. The scope is operating at below dawes limit - this means that a point smaller than the limit means it gets blurred due to defraction .. this all adds to my PSF complexity. Guiding errors also add to the non-roundness of the stars and this is what you're seeing here. Additionally the earth's atmosphere that bubbles moves the point around causing additional blurring.

Wait? Why aren't they the same? Well the PSF is a cunning beast - it changes completely between images and even over the time you take an image. Next time you look up and see the stars twinkle then you're seeing the PSF change over time and so the star appears to change over time thus the twinkle.

The last point is that PSF doesn't just affect the stars.. it affects the *whole* image resulting in a blurred mess - including nebulosity, galaxies - that means you'll miss some lovely little things in the background hidden in the blur..

The technical term of all this badness is "Convolution".

Doooomed! We're all Doooomed!

Well not really - what if you could undo all this "Convolution" .. well "Deconvolution" is the technical term for reversing all the blurring.

Wait? You're going to list lots of maths now.. well actually you don't need to. *cheer here* but we can do away with grandiose terms such as vector spaces etc as you can use your imagination instead. Just think if it as taking sand and moving the sand to correct the shape.

The stars in the image provide a great point of reference we can use for analysis.

Undoing the blur

Imagine taking my 3D plot and pulling in the sides of the slope proportionally to match the ideal PSF - the top peak of the plot will pile up taller so the star will appear brighter and more rounded.

But we don't do this for just the stars. We do the same shifting for each pixel in the image depending on the different against the PSFs. The result of doing this is that the we uncover new detail in the target, clean up nebulosity and see new detail in the background (such as small galaxies).

Here you can see the processing taking effect:

Before:

During:

You can see the peak hitting the top.

End:

[Jessun]

Thank you for a great write up!

I enjoyed reading that. It leaves me with the question of when to deconvoulute? Before anything else?

[NickK]

Thank you for a great write up!

I enjoyed reading that. It leaves me with the question of when to deconvoulute? Before anything else?

The first thing I would do is to remove the CCD noise. I've found this to cause serious issues because the noise is not convoluted by the PSF which means that attempting to de-convolute it causes problems.