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.

## 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.

## Reading

- AhRR12
- Ahmed, A., Recht, B., & Romberg, J. (2012) Blind Deconvolution using Convex Programming.
*arXiv:1211.5608 [cs, Math]*. - AlAl08
- Almeida, M. S. C., & Almeida, L. B.(2008) Blind deblurring of natural images. In IEEE International Conference on Acoustics, Speech and Signal Processing, 2008. ICASSP 2008 (pp. 1261–1264). DOI.
- AlAl10
- Almeida, M. S. C., & Almeida, L. B.(2010) Blind and Semi-Blind Deblurring of Natural Images.
*IEEE Transactions on Image Processing*, 19(1), 36–52. DOI. - AlFi13
- Almeida, M. S., & Figueiredo, M. A.(2013) Blind image deblurring with unknown boundaries using the alternating direction method of multipliers. In ICIP (pp. 586–590). Citeseer
- BMDK12
- Babacan, S. D., Molina, R., Do, M. N., & Katsaggelos, A. K.(2012) Bayesian Blind Deconvolution with General Sparse Image Priors. In A. Fitzgibbon, S. Lazebnik, P. Perona, Y. Sato, & C. Schmid (Eds.), Computer Vision – ECCV 2012 (pp. 341–355). Springer Berlin Heidelberg
- BeVG13
- Benichoux, A., Vincent, E., & Gribonval, R. (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.
- BPGD13
- Bilen, C., Puy, G., Gribonval, R., & Daudet, L. (2013) Blind sensor calibration in sparse recovery using convex optimization. In SAMPTA-10th International Conference on Sampling Theory and Applications-2013.
- BrHu00
- Brooks, T. F., & Humphreys Jr, W. M.(n.d.) Extension of DAMAS Phased Array Processing for Spatial Coherence Determination (DAMAS-C). In 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference). American Institute of Aeronautics and Astronautics
- BrHu06
- Brooks, T. F., & Humphreys, W. M.(2006) A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays.
*Journal of Sound and Vibration*, 294(4–5), 856–879. DOI. - BuCo09
- Butucea, C., & Comte, F. (2009) Adaptive estimation of linear functionals in the convolution model and applications.
*Bernoulli*, 15(1), 69–98. DOI. - CaFl11
- Carrasco, M., & Florens, J.-P. (2011) Spectral Method for Deconvolving a Density.
*Econometric Theory*, 27(Special Issue 03), 546–581. DOI. - CaHa88
- Carroll, R. J., & Hall, P. (1988) Optimal Rates of Convergence for Deconvolving a Density.
*Journal of the American Statistical Association*, 83(404), 1184–1186. DOI. - Cuca08
- Cucala, L. (2008) Intensity Estimation for Spatial Point Processes Observed with Noise.
*Scandinavian Journal of Statistics*, 35(2), 322–334. DOI. - DeHa15
- Delaigle, A., & Hall, P. (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. - DeHM08
- Delaigle, A., Hall, P., & Meister, A. (2008) On Deconvolution with Repeated Measurements.
*The Annals of Statistics*, 36(2), 665–685. DOI. - DeMe08
- Delaigle, A., & Meister, A. (2008) Density estimation with heteroscedastic error.
*Bernoulli*, 14(2), 562–579. - DPWL13
- Dokmanić, I., Parhizkar, R., Walther, A., Lu, Y. M., & Vetterli, M. (2013) Acoustic echoes reveal room shape.
*Proceedings of the National Academy of Sciences*, 110(30), 12186–12191. DOI. - Efro14
- Efron, B. (2014) The Bayes deconvolution problem.
- Fan91
- Fan, J. (1991) On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems.
*The Annals of Statistics*, 19(3), 1257–1272. DOI. - GrGS10
- Griffa, A., Garin, N., & Sage, D. (2010) Comparison of deconvolution software in 3D microscopy: A user point of view—Part 1.
*GIT Imaging & Microscopy*, 12(EPFL-ARTICLE-163617), 43–45. - HaMe07
- Hall, P., & Meister, A. (2007) A ridge-parameter approach to deconvolution.
*The Annals of Statistics*, 35(4), 1535–1558. DOI. - Hard13
- Hardy, S. J.(2013) Direct deconvolution of radio synthesis images using L 1 minimisation.
*Astronomy & Astrophysics*, 557, A134. DOI. - HaKD13
- Hawe, S., Kleinsteuber, M., & Diepold, K. (2013) Analysis operator learning and its application to image reconstruction.
*IEEE Transactions on Image Processing*, 22(6), 2138–2150. DOI. - HMLH07
- Hom, E. F. Y., Marchis, F., Lee, T. K., Haase, S., Agard, D. A., & Sedat, J. W.(2007) AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data.
*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 24(6), 1580–1600. - HuSa90
- Hua, Y., & Sarkar, T. K.(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. - Lair78
- Laird, N. (1978) Nonparametric Maximum Likelihood Estimation of a Mixing Distribution.
*Journal of the American Statistical Association*, 73(364), 805–811. DOI. - LiCM12
- Liu, G., Chang, S., & Ma, Y. (2012) Blind Image Deblurring by Spectral Properties of Convolution Operators.
*arXiv:1209.2082 [cs]*. - Mall86
- Mallet, A. (1986) A maximum likelihood estimation method for random coefficient regression models.
*Biometrika*, 73(3), 645–656. DOI. - Meis08
- Meister, A. (2008) Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions.
*Inverse Problems*, 24(1), 015003. DOI. - OrGR10
- Orieux, F., Giovannelli, J.-F., & Rodet, T. (2010) Bayesian estimation of regularization and PSF parameters for Wiener-Hunt deconvolution.
*Journal of the Optical Society of America A*, 27(7), 1593. DOI. - Penc00
- Pencil, M. (n.d.) Method for Estimating Parameters of bxponentially DampedKJndamped Sinusoids in Noise.
- RoKa89
- Roy, R., & Kailath, T. (1989) ESPRIT-estimation of signal parameters via rotational invariance techniques.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*, 37(7), 984–995. DOI. - SHHS14
- Schuler, C. J., Hirsch, M., Harmeling, S., & Schölkopf, B. (2014) Learning to Deblur.
*arXiv:1406.7444 [cs]*. - StCa90
- Stefanski, L. A., & Carroll, R. J.(1990) Deconvolving kernel density estimators.
*Statistics*, 21(2), 169–184. DOI. - StCI75
- Stockham, J., T. G., Cannon, T. M., & Ingebretsen, R. B.(1975) Blind deconvolution through digital signal processing.
*Proceedings of the IEEE*, 63(4), 678–692. DOI. - WöHK13
- Wörmann, J., Hawe, S., & Kleinsteuber, M. (2013) Analysis Based Blind Compressive Sensing.
*IEEE Signal Processing Letters*, 20(5), 491–494. DOI.