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I have a measurement technique that results in a frequency distribution of the results. Let's say x is the parameter of interest, the method gives you a distribution F(x). I have the idea that the various physical processes in the measurement technique means that F(x) is really a convolution of the true distribution - let's call that f(x) - and the Gaussian function. Very similarly to a Gaussian blur - but the width of the Gaussian function actually varies with x (in a known way).

What I would like to do is be able to take the measurement F(x) and transform it back to f(x) - the true result. I've looked into this and my understanding is that effectively the measurement is making a generalised Weierstrass Transform of f(x).

My problem is that I can understand how to numerically do f(x) -> F(x), it's a simple averaging of f(x) with the Gaussian function, but I've no idea how to do the reverse numerically (the measurement doesn't give a result that can be written as some simple function). I have two ideas:

  1. Do the transform directly using some appropriate method.
  2. Solve the issue the other way - i.e. iterate over f(x)s until I find one that gives close to the measured F(x).

In both cases I'm at a bit of a loss as to the best approach - let alone which of the 2 options is most sensible.

For option 1, I've read the Wikipedia article and it gives three alternatives for the inverse transform, but all involve higher order differentiations - which I would guess lead to lots of numerical stability issues.

For option 2, I guess there are lots of ways to do this - I'm thinking I could try to modify some Monte-Carlo methods or something. But I'd really appreciate it if anyone has some pointers / knows of existing algorithms that can do this.

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This is known as a deblurring problem. It is said to be supervised as you know the blurring transformation. One approach is by working in the Fourier domain to perform a deconvolution, i.e. turning the filtering in an ordinary product, which is easy to invert. As this is an unstable process, there are variants, such as Wiener filtering.

Though in your case, the filter width varies, making it a harder problem.

Have a look here and there.

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    $\begingroup$ Wow, thanks, that's really helpful. As soon as I read "point spread function" 15 year old lectures about image/signal processing came flooding back! Should have realised that myself given that I had already realised the mathematical process was so close to Gaussian blurring (albeit the physical processes are completely different). Thanks a lot. $\endgroup$ – Mooks May 3 at 12:46

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