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

- High dimensional data
- irregularly sampled data
- 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.

## 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 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. TODO: 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

## Deconvolution method in statistics

Dammit, 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 because of course.

## Refs

- BeZa76: A. J. Berkhout, P. R. Zaanen (1976) A Comparison Between Wiener Filtering, Kalman Filtering, and Deterministic Least Squares Estimation*.
*Geophysical Prospecting*, 24(1), 141–197. DOI - BeVG13: Alexis Benichoux, Emmanuel Vincent, Rémi Gribonval (2013) A fundamental pitfall in blind deconvolution with sparse and shift-invariant priors. In ICASSP-38th International Conference on Acoustics, Speech, and Signal Processing-2013.
- Mall86: A. Mallet (1986) A maximum likelihood estimation method for random coefficient regression models.
*Biometrika*, 73(3), 645–656. DOI - HaMe07: Peter Hall, Alexander Meister (2007) A ridge-parameter approach to deconvolution.
*The Annals of Statistics*, 35(4), 1535–1558. DOI - DPWL13: Ivan Dokmanić, Reza Parhizkar, Andreas Walther, Yue M. Lu, Martin Vetterli (2013) Acoustic echoes reveal room shape.
*Proceedings of the National Academy of Sciences*, 110(30), 12186–12191. DOI - WöHK13: Julian Wörmann, Simon Hawe, Martin Kleinsteuber (2013) Analysis Based Blind Compressive Sensing.
*IEEE Signal Processing Letters*, 20(5), 491–494. DOI - HaKD13: S. Hawe, M. Kleinsteuber, K. Diepold (2013) Analysis operator learning and its application to image reconstruction.
*IEEE Transactions on Image Processing*, 22(6), 2138–2150. DOI - BMDK12: S. Derin Babacan, Rafael Molina, Minh N. Do, Aggelos K. Katsaggelos (2012) Bayesian Blind Deconvolution with General Sparse Image Priors. In Computer Vision – ECCV 2012 (pp. 341–355). Springer Berlin Heidelberg
- StCI75: Jr. Stockham T.G., T.M. Cannon, R.B. Ingebretsen (1975) Blind deconvolution through digital signal processing.
*Proceedings of the IEEE*, 63(4), 678–692. DOI - AhRR12: Ali Ahmed, Benjamin Recht, Justin Romberg (2012) Blind Deconvolution using Convex Programming.
*ArXiv:1211.5608 [Cs, Math]*. - LiCM12: Guangcan Liu, Shiyu Chang, Yi Ma (2012) Blind Image Deblurring by Spectral Properties of Convolution Operators.
*ArXiv:1209.2082 [Cs]*. - AlFi13: M. S. C. Almeida, M. A. T. Figueiredo (2013) Blind image deblurring with unknown boundaries using the alternating direction method of multipliers. In 2013 IEEE International Conference on Image Processing (pp. 586–590). DOI
- Meis08: Alexander Meister (2008) Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions.
*Inverse Problems*, 24(1), 015003. DOI - Fan92: Jianqing Fan (1992) Deconvolution with supersmooth distributions.
*Canadian Journal of Statistics*, 20(2), 155–169. DOI - StCa90: Leonard A. Stefanski, Raymond J. Carroll (1990) Deconvolving kernel density estimators.
*Statistics*, 21(2), 169–184. DOI - DeMe08: Aurore Delaigle, Alexander Meister (2008) Density estimation with heteroscedastic error.
*Bernoulli*, 14(2), 562–579. - RoKa89: R. Roy, T. Kailath (1989) ESPRIT-estimation of signal parameters via rotational invariance techniques.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*, 37(7), 984–995. DOI - Cuca08: Lionel Cucala (2008) Intensity Estimation for Spatial Point Processes Observed with Noise.
*Scandinavian Journal of Statistics*, 35(2), 322–334. DOI - SHHS14: Christian J. Schuler, Michael Hirsch, Stefan Harmeling, Bernhard Schölkopf (2014) Learning to Deblur.
*ArXiv:1406.7444 [Cs]*. - HuSa90: Yingbo Hua, Tapan K. Sarkar (1990) Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise.
*IEEE Transactions on Acoustics, Speech and Signal Processing*, 38(5), 814–824. DOI - DeHa15: Aurore Delaigle, Peter Hall (2015) Methodology for non-parametric deconvolution when the error distribution is unknown.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, n/a-n/a. DOI - Lair78: Nan Laird (1978) Nonparametric Maximum Likelihood Estimation of a Mixing Distribution.
*Journal of the American Statistical Association*, 73(364), 805–811. DOI - DeHM08: Aurore Delaigle, Peter Hall, Alexander Meister (2008) On Deconvolution with Repeated Measurements.
*The Annals of Statistics*, 36(2), 665–685. DOI - Fan91: Jianqing Fan (1991) On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems.
*The Annals of Statistics*, 19(3), 1257–1272. DOI - CaHa88: Raymond J. Carroll, Peter Hall (1988) Optimal Rates of Convergence for Deconvolving a Density.
*Journal of the American Statistical Association*, 83(404), 1184–1186. DOI - CaFl11: Marine Carrasco, Jean-Pierre Florens (2011) Spectral Method for Deconvolving a Density.
*Econometric Theory*, 27(Special Issue 03), 546–581. DOI - Efro14: Bradley Efron (2014)
*The Bayes deconvolution problem*