# Deconvolution

April 20, 2015 — April 11, 2016

convolution
density
functional analysis
linear algebra
nonparametric
probability
signal processing
sparser than thou
statistics

I wish, for a project of my own, to know about how to deconvolve with

1. High dimensional data
2. irregularly sampled data
3. inhomogenous (although known) convolution kernels

This is in a signal processing setting; for the (closely-related) kernel-density estimation in a statistical setting, see kernel approximation. If you don’t know your noise spectrum, see blind 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”.

## 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

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