$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a signal and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ is a known point-spread function (say, a Gaussian).

A system samples $f\star g$ at a known sequence of irregular points, $(\mathbf{x}_n)_n$. Are there any methods, analytic or numerical, to deconvolve the samples from $g$ to reveal samples of $f$?

If $f$ were band-limited and the sequence of sample points $(\mathbf{x}_n)_n$ were periodic and finer than the sampling theorem's frequency, then I believe by the convolution theorem sampled-$g$'s Fourier transform could be divided out pointwise from that of sampled-$f\ast g$, yielding sampled-$f$'s Fourier transform.

  • $\begingroup$ Is this better for DSP.stackexchange.com ? $\endgroup$ – Christian Chapman Feb 9 '14 at 4:11
  • $\begingroup$ Can you not resample to regularly spaced points? $\endgroup$ – AnonSubmitter85 Feb 9 '14 at 19:30
  • $\begingroup$ No, this system doesn't work that way. $\endgroup$ – Christian Chapman Feb 9 '14 at 21:00
  • $\begingroup$ A possible solution is to take the FFT of the samples and PSF, divide then take the inverse FFT. There is a library called NFFT that will do this. The math it uses is documented. But this can get ugly and slow, I imagine $\endgroup$ – Christian Chapman Feb 10 '14 at 0:55
  • $\begingroup$ do you have the signal/image figure to illustrate what do you mean by irregular points and ugly? Thanks $\endgroup$ – lennon310 Feb 11 '14 at 0:08

You can use a graph-based Signal Processing framework to apply regular deconvolution algorithms. The whole point is that traditional operators (such as FFT) can be defined on graph structures, and graph structures do handle "by design" without any overhead irregular sampling patterns.

Some useful reference scan be found on my former office mate's webpage.


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