Deconvolution

1 Vanilla deconvolution

Wiener filtering! Deconvolving a signal convolved with a known kernel. Say, reconstructing the pure sound of an instrument, or, the sound of the echo in a church, from a recording made in a reverberant church. It’s not purely acoustic, though; applies to images, abstract wacky function spaces etc. The procedure is to presume your signal has been blurred by (or generally convolved with) some filter, and then to find a new filter that undoes the effects of the previous filter, or as close as possible to that, since not all filters are invertible. In the basic case, then, this is approximately the same thing as filter inversion, although there are some fiddly cases when the kernel is noninvertible, or the inverse transform is unstable.

Linear versions are (apparently) straightforward Wiener filters (more-or-less generalized Kalman filters; although I think they are historically prior. 🏗 make this precise.) Clearly you get deconvolution-like behaviour in state filters sometimes too. I should inspect the edges of these definitions to work out the precise intersection. Markovian case?

Non-linear “deconvolutions” are AFAIK not strictly speaking deconvolution since convolution is a linear operation. Anyway, that usage seems to be in the literature. c.f. the “iterative Richardson-Lucy algorithm”.

See also compressive sensing, convolution kernels.

• Tim Cornwell & Alan Bridle, 1996, Deconvolution Tutorial

• linear example in python skimage with perfunctory explanation

  …inverse filter based on the PSF (Point Spread Function), the prior regularisation (penalisation of high frequency) and the tradeoff between the data and prior adequacy.

• the usual invocation of the DSPguide

• practical pedagogic example in octave

2 Deconvolution method in statistics

Curses! I thought that this weird obvious idea was actually a new idea of mine. Turns out it’s old. “Density deconvolution” is a keyword here, and it’s reasonably common in hierarchical models.

3 References