# How do you deconvolve irregularly sampled points?

$f:\mathbb{R}^n\rightarrow\mathbb{R}$ 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.

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• Is this better for DSP.stackexchange.com ? – enthdegree Feb 9 '14 at 4:11
• Can you not resample to regularly spaced points? – AnonSubmitter85 Feb 9 '14 at 19:30
• No, this system doesn't work that way. – enthdegree Feb 9 '14 at 21:00
• 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 – enthdegree Feb 10 '14 at 0:55
• do you have the signal/image figure to illustrate what do you mean by irregular points and ugly? Thanks – lennon310 Feb 11 '14 at 0:08