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In electrical engineering and [[applied mathematics]], '''blind deconvolution''' is [[deconvolution]] without explicit knowledge of the [[impulse response function]] used in the [[convolution]]. This is usually achieved by making appropriate assumptions of the input to estimate the impulse response by analyzing the output. Blind deconvolution is not solvable without making assumptions on input and impulse response. Most of the algorithms to solve this problem are based on assumption that both input and impulse response live in respective known subspaces. However, blind deconvolution remains a very challenging non-convex optimization problem even with this assumption.
In [[electrical engineering]] and [[applied mathematics]], '''blind deconvolution''' is [[deconvolution]] without explicit knowledge of the [[impulse response function]] used in the [[convolution]]. This is usually achieved by making appropriate assumptions of the input to estimate the impulse response by analyzing the output. Blind deconvolution is not solvable without making assumptions on input and impulse response. Most of the algorithms to solve this problem are based on assumption that both input and impulse response live in respective known subspaces. However, blind deconvolution remains a very challenging non-[[convex optimization]] problem even with this assumption.


[[File:Blind deconvolution illustration.png|thumb|right|alt=Blind deconvolution illustration|Top left image: NGC224 by [[Hubble Space Telescope]]. Top right contour: best fit of the [[point spread function]] (PSF) (a priori).<ref name="barmby2007">
[[File:Blind deconvolution illustration.png|thumb|right|alt=Blind deconvolution illustration|Top left image: NGC224 by [[Hubble Space Telescope]]. Top right contour: best fit of the [[point spread function]] (PSF) (a priori).<ref name="barmby2007">
{{cite journal
{{cite journal
|last1=Barmby |first1=Pauline
| bibcode=2007AJ....133.2764B
|last2=McLaughlin |first2=Dean E.
| author=P. Barmby
|last3=Harris |first3=William E.
|author2=D. E. McLaughlin
|last4=Harris |first4=Gretchen L. H.
| title=Structural Parameters for Globular Clusters in M31 and Generalizations for the Fundamental Plane
|last5=Forbes |first5=Duncan A.
| journal=[[The Astronomical Journal]]
|year=2007
| volume=133
|title=Structural Parameters for Globular Clusters in M31 and Generalizations for the Fundamental Plane
| issue=6
|url=http://www.inf.fu-berlin.de/lehre/WS05/Mustererkennung/infomax/infomax.pdf
| pages=2764–2786
|journal=[[The Astronomical Journal]]
| year=2007
|volume=133|issue=6 |pages=2764–2786
| doi=10.1086/516777
|arxiv=0704.2057
|arxiv = 0704.2057 |url=http://www.inf.fu-berlin.de/lehre/WS05/Mustererkennung/infomax/infomax.pdf |display-authors=etal}}
|bibcode=2007AJ....133.2764B
</ref> Middle left image: Deconvolution by [[maximum a posteriori estimation]] (MAP), the 2nd iteration. Middle right contour: Estimate of the PSF by MAP, the 2nd iteration. Bottom left image: Deconvolution by MAP, the final result. Bottom right contour: Estimate of the PSF by MAP, the final result.]]
|doi=10.1086/516777
|s2cid=58913061
}}</ref> Middle left image: Deconvolution by [[maximum a posteriori estimation]] (MAP), the 2nd iteration. Middle right contour: Estimate of the PSF by MAP, the 2nd iteration. Bottom left image: Deconvolution by MAP, the final result. Bottom right contour: Estimate of the PSF by MAP, the final result.]]


== In image processing ==
==In image processing==
In [[image processing]], blind deconvolution is a deconvolution technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown [[point spread function]] (PSF).
In [[image processing]], blind deconvolution is a deconvolution technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown [[point spread function]] (PSF).<ref name="blinddeconv">
<ref name="blinddeconv">
{{cite journal
{{cite journal
|last1=Lam |first1=Edmund Y.
| url=http://www.opticsinfobase.org/abstract.cfm?uri=josaa-17-7-1177
|last2=Goodman |first2=Joseph W.
| author=E. Lam
|year=2000
|author2=J.W. Goodman
| title=Iterative statistical approach to blind image deconvolution
|title=Iterative statistical approach to blind image deconvolution
| journal=[[Journal of the Optical Society of America A]]
|journal=[[Journal of the Optical Society of America A]]
| volume=17
|volume=17 |issue=7 |pages=1177–1184
|bibcode=2000JOSAA..17.1177L
| issue=7
|doi=10.1364/JOSAA.17.001177
| pages=1177–1184
|pmid=10883969
| year=2000
}}</ref> Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.
| doi=10.1364/JOSAA.17.001177
|bibcode = 2000JOSAA..17.1177L }}
</ref>
Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.


Most of the work on blind deconvolution started in early 1970s. Blind deconvolution is used in astronomical imaging and medical imaging.
Most of the work on blind deconvolution started in early 1970s. Blind deconvolution is used in astronomical imaging and medical imaging.


Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF.
Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF. Iterative methods include [[maximum a posteriori estimation]] and [[expectation-maximization algorithm]]s. A good estimate of the PSF is helpful for quicker convergence but not necessary.
Iterative methods include [[maximum a posteriori estimation]] and [[expectation-maximization algorithm]]s. A good estimate of the PSF is helpful for quicker convergence but not necessary.


Examples of non-iterative techniques include SeDDaRA,<ref>
Examples of non-iterative techniques include SeDDaRA,<ref>{{cite journal|url=https://www.osapublishing.org/ao/abstract.cfm?uri=ao-41-32-6884|title=Noniterative blind data restoration by use of an extracted filter function|last=Caron|first=James N.|last2=Namazi|first2=Nader M.|date=2002-11-11|website=www.osapublishing.org|language=EN|doi=10.1364/AO.41.006884|access-date=2016-03-21|last3=Rollins|first3=Chris J.|bibcode=2002ApOpt..41.6884C}}</ref> the [[cepstrum]] transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the information about the scene is provided in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.
{{cite journal
|last1=Caron |first1=James N.
|last2=Namazi |first2=Nader M.
|last3=Rollins |first3=Chris J.
|date=2002
|title=Noniterative blind data restoration by use of an extracted filter function
|journal=Applied Optics
|volume=41 |issue=32 |pages=6884–9
|bibcode=2002ApOpt..41.6884C
|doi=10.1364/AO.41.006884
|pmid=12440543
}}</ref> the [[cepstrum]] transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the information about the scene is provided in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.


'''<big>Examples</big>'''
Limitation of Blind deconvolution is that both input image and blur kernel must live in fixed subspace. That means input image, represented by '''w,''' has to be written as '''w =Bh,'''where '''B''' is random matrix of size L by K (K<L) and '''h''' is of size K by 1, whereas blur kernel, if represented by '''x,''' has to be written as '''x=Cm,''' where C is random matrix of size L by N '''(N<L)''' and '''m''' is of size N by 1.Observed image, if represented by '''y''', given by '''y=w*x,''' can only be reconstructed if L>=K +N.


Any blurred image can be given as input to blind deconvolution algorithm, it can deblur the image, but essential condition for working of this algorithm must not be violated as discussed above. In the first example (picture of shapes), recovered image was very fine, exactly similar to original image because L > K + N. In the second example (picture of a girl), L < K + N, so essential condition is violated, hence recovered image is far different from original image.
'''<big>Examples</big>'''

Any blurred image can be given as input to blind deconvolution algorithm, it can deblur the image, but essential condition for working of this algorithm must not be violated as discussed above. In the first example (picture of shapes),recovered image was very fine, exactly similar to original image because L< K+N. In the second example,(picture of a girl), L> K+N , so essential condition is violated, hence recovered image is far different from original image.


[[File:Blurred_img.png|thumb|Blurred Image, obtained by convolution of original image with blur kernel. Input image lies in fixed subspace of wavelet transform and blur kernel lies in random subspace.]]
[[File:Blurred_img.png|thumb|Blurred Image, obtained by convolution of original image with blur kernel. Input image lies in fixed subspace of wavelet transform and blur kernel lies in random subspace.]]
== In signal processing ==


=== Seismic data ===
==In signal processing==
In the case of [[Deconvolution#Seismology|deconvolution of seismic data]], the original unknown signal is made of spikes hence is possible to characterize with [[sparsity]] constraints<ref>{{cite journal | title=Sparse seismic deconvolution by method of orthogonal matching pursuit |
author=Broadhead, Michael|
year=2010}}</ref> or [[Regularization (mathematics)|regularizations]] such as '''''l''<sub>1</sub> norm'''/'''''l''<sub>2</sub> norm''' norm ratios,<ref>{{cite journal | title=Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization|author1=A. Repetti |author2=M. Q. Pham |author3=L. Duval |author4=E. Chouzenoux |author5=J.-C. Pesquet |
journal=IEEE Signal Processing Letters|
pages=539–543|
year=2015|
arxiv=1407.5465|bibcode=2015ISPL...22..539R|doi=10.1109/LSP.2014.2362861}}</ref> suggested by W. C. Gray in 1978.<ref>{{cite journal| title=Variable norm deconvolution| author=W. C. Gray| year=1978| url=http://sepwww.stanford.edu/oldreports/sep14/14_19.pdf| deadurl=yes| archiveurl=https://web.archive.org/web/20150409220356/http://sepwww.stanford.edu/oldreports/sep14/14_19.pdf| archivedate=2015-04-09| df=}}</ref>


=== Audio deconvolution ===
===Seismic data===
In the case of [[Deconvolution#Seismology|deconvolution of seismic data]], the original unknown signal is made of spikes hence is possible to characterize with [[sparsity]] constraints<ref>{{cite conference
Audio deconvolution (often referred to as ''dereverberation'') is a [[reverberation]] reduction in audio mixtures. It is part of audio processing of recordings in ill-posed cases such as the [[cocktail party effect]]. One possibility is to use [[Independent component analysis|ICA]].<ref>{{cite journal |
| last1 = L. Tonellot | first1 = T.
title=Time-domain blind audio source separation using advanced ICA methods |author1=Koldovsky, Zbynek |author2=Tichavsky, Petr |lastauthoramp=yes |
| last2 = K. Broadhead | first2 = M.
booktitle=the Proceedings of the 8th Annual Conference of the International Speech Communication Association (Interspeech 2007)|
| contribution = Sparse Seismic Deconvolution by Method of Orthogonal Matching Pursuit
pages=846–849,|
| doi = 10.3997/2214-4609.201401250
year=2007
| publisher = European Association of Geoscientists & Engineers
| title = 72nd EAGE Conference and Exhibition incorporating SPE EUROPEC 2010
| year = 2010}}</ref> or [[Regularization (mathematics)|regularizations]] such as '''''l''<sub>1</sub> norm'''/'''''l''<sub>2</sub> norm''' norm ratios,<ref>
{{cite journal
|last1=Barmby |first1=P.
|last2=McLaughlin |first2=D. E.
|last3=Harris |first3=W. E.
|last4=Harris |first4=G. L. H.
|last5=Forbes |first5=D. A.
|year=2015
|title=Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization
|journal=IEEE Signal Processing Letters
|volume=22 |issue=5 |pages=539–543
|arxiv=1407.5465
|bibcode=2015ISPL...22..539R
|doi=10.1109/LSP.2014.2362861
|s2cid=9605797
}}</ref> suggested by W. C. Gray in 1978.<ref>
{{cite thesis
|last=Gray |first=W. C.
|year=1978
|title=Variable norm deconvolution
| type = Ph.D. thesis
| publisher = Stanford University
|url=http://sepwww.stanford.edu/oldreports/sep14/14_19.pdf
|archive-url=https://web.archive.org/web/20150409220356/http://sepwww.stanford.edu/oldreports/sep14/14_19.pdf
|archive-date=2015-04-09
|url-status=dead
}}</ref>
}}</ref>


=== In general ===
===Audio deconvolution===
Audio deconvolution (often referred to as ''dereverberation'') is a [[reverberation]] reduction in audio mixtures. It is part of audio processing of recordings in ill-posed cases such as the [[cocktail party effect]]. One possibility is to use [[Independent component analysis|ICA]].<ref>
{{cite conference
|last1=Koldovsky |first1=Zbynek
|last2=Tichavsky |first2=Petr
|year=2007
|title=Time-domain blind audio source separation using advanced ICA methods
|book-title=The Proceedings of the 8th Annual Conference of the International Speech Communication Association (Interspeech 2007)
|pages=846–849
}}</ref>

===In general===
Suppose we have a signal transmitted through a channel. The channel can usually be modeled as a [[LTI system theory|linear shift-invariant system]], so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an [[Equalization (communications)|equalizer]].
Suppose we have a signal transmitted through a channel. The channel can usually be modeled as a [[LTI system theory|linear shift-invariant system]], so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an [[Equalization (communications)|equalizer]].
[[File:Recovered_img.png|thumb|Recovered image after applying algorithm of blind deconvolution. This algorithm basically solves optimization problem using nuclear norm minimization. L=65536, K= 65 and N=44838,]]
[[File:Recovered_img.png|thumb|Recovered image after applying algorithm of blind deconvolution. This algorithm basically solves optimization problem using nuclear norm minimization. L=65536, K=65 and N=44838,]]
If we are given the original signal, we can use a supervising technique, such as finding a [[Wiener filter]], but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired [[spectral power density]]. This is what happens, for example, when the original signal is known to have no [[auto correlation]], and we "[[Whitening transform|whiten]]" the received signal.
If we are given the original signal, we can use a supervising technique, such as finding a [[Wiener filter]], but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired [[spectral power density]]. This is what happens, for example, when the original signal is known to have no [[auto correlation]], and we "[[Whitening transform|whiten]]" the received signal.


Whitening usually leaves some [[phase (waves)|phase]] distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equalizer to obtain a signal with a PSF approximating what we know about the original PSF.
Whitening usually leaves some [[phase (waves)|phase]] distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equalizer to obtain a signal with a PSF approximating what we know about the original PSF.
[[File:Original_img.png|thumb|Original Image]]
[[File:Original_img.png|thumb|Original image]]
[[File:Blur_img.png|thumb|Blurred Image: obtained after the convolution of original image with blur kernel. Original image lies in fixed subspace of wavelet transform and blur lies in random subspace. L=65536, K=200, N =65400]]
[[File:Blur_img.png|thumb|Blurred image: obtained after the convolution of original image with blur kernel. Original image lies in fixed subspace of wavelet transform and blur lies in random subspace. L=65536, K=200, N=65400]]
[[File:Rcvrd_img.png|thumb|Recovered Image. recovered image is very different from original image, because essential condition for the algorithm of blind deconvolution using nuclear norm minimization is violated. L= L=65536, K=200, N =65400]]
[[File:Rcvrd_img.png|thumb|Recovered image: very different from original image, because essential condition for the algorithm of blind deconvolution using nuclear norm minimization is violated. L=65536, K=200, N=65400]]


=== High-order statistics ===
===High-order statistics===
Blind deconvolution algorithms often make use of [[high-order statistics]], with moments higher than two. This can be implicit or explicit.<ref>{{cite journal | title=Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors|
Blind deconvolution algorithms often make use of [[high-order statistics]], with moments higher than two. This can be implicit or explicit.<ref>
{{cite book
|last1=Cardoso |first1=J.-F.
|year=1991
|chapter=Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors
|title=&#91;Proceedings&#93; ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing
author=Cardoso, J-F|
|volume=5 |pages=3109–3112
booktitle=Acoustics, Speech, and Signal Processing, 1991. ICASSP-91., 1991 International Conference on|
|doi=10.1109/ICASSP.1991.150113
pages=3109–3112|
|isbn=978-0-7803-0003-3
year=1991|
|citeseerx=10.1.1.8.9380
work=IEEE|url=http://ieeexplore.ieee.org/document/150113/
|s2cid=7972548
}}</ref>
}}</ref>


== See also ==
==See also==
* [[Channel model]]
* [[Channel model]]
* [[Inverse problem]]
* [[Inverse problem]]
Line 97: Line 141:
*[http://bigwww.epfl.ch/algorithms/deconvolutionlab/ ImageJ plugin for deconvolution]
*[http://bigwww.epfl.ch/algorithms/deconvolutionlab/ ImageJ plugin for deconvolution]


== References ==
==References==


<!-- ---------------------------------------------------------------
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Latest revision as of 05:19, 4 September 2024

In electrical engineering and applied mathematics, blind deconvolution is deconvolution without explicit knowledge of the impulse response function used in the convolution. This is usually achieved by making appropriate assumptions of the input to estimate the impulse response by analyzing the output. Blind deconvolution is not solvable without making assumptions on input and impulse response. Most of the algorithms to solve this problem are based on assumption that both input and impulse response live in respective known subspaces. However, blind deconvolution remains a very challenging non-convex optimization problem even with this assumption.

Blind deconvolution illustration
Top left image: NGC224 by Hubble Space Telescope. Top right contour: best fit of the point spread function (PSF) (a priori).[1] Middle left image: Deconvolution by maximum a posteriori estimation (MAP), the 2nd iteration. Middle right contour: Estimate of the PSF by MAP, the 2nd iteration. Bottom left image: Deconvolution by MAP, the final result. Bottom right contour: Estimate of the PSF by MAP, the final result.

In image processing

[edit]

In image processing, blind deconvolution is a deconvolution technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown point spread function (PSF).[2] Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.

Most of the work on blind deconvolution started in early 1970s. Blind deconvolution is used in astronomical imaging and medical imaging.

Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF. Iterative methods include maximum a posteriori estimation and expectation-maximization algorithms. A good estimate of the PSF is helpful for quicker convergence but not necessary.

Examples of non-iterative techniques include SeDDaRA,[3] the cepstrum transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the information about the scene is provided in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.

Examples

Any blurred image can be given as input to blind deconvolution algorithm, it can deblur the image, but essential condition for working of this algorithm must not be violated as discussed above. In the first example (picture of shapes), recovered image was very fine, exactly similar to original image because L > K + N. In the second example (picture of a girl), L < K + N, so essential condition is violated, hence recovered image is far different from original image.

Blurred Image, obtained by convolution of original image with blur kernel. Input image lies in fixed subspace of wavelet transform and blur kernel lies in random subspace.

In signal processing

[edit]

Seismic data

[edit]

In the case of deconvolution of seismic data, the original unknown signal is made of spikes hence is possible to characterize with sparsity constraints[4] or regularizations such as l1 norm/l2 norm norm ratios,[5] suggested by W. C. Gray in 1978.[6]

Audio deconvolution

[edit]

Audio deconvolution (often referred to as dereverberation) is a reverberation reduction in audio mixtures. It is part of audio processing of recordings in ill-posed cases such as the cocktail party effect. One possibility is to use ICA.[7]

In general

[edit]

Suppose we have a signal transmitted through a channel. The channel can usually be modeled as a linear shift-invariant system, so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an equalizer.

Recovered image after applying algorithm of blind deconvolution. This algorithm basically solves optimization problem using nuclear norm minimization. L=65536, K=65 and N=44838,

If we are given the original signal, we can use a supervising technique, such as finding a Wiener filter, but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired spectral power density. This is what happens, for example, when the original signal is known to have no auto correlation, and we "whiten" the received signal.

Whitening usually leaves some phase distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equalizer to obtain a signal with a PSF approximating what we know about the original PSF.

Original image
Blurred image: obtained after the convolution of original image with blur kernel. Original image lies in fixed subspace of wavelet transform and blur lies in random subspace. L=65536, K=200, N=65400
Recovered image: very different from original image, because essential condition for the algorithm of blind deconvolution using nuclear norm minimization is violated. L=65536, K=200, N=65400

High-order statistics

[edit]

Blind deconvolution algorithms often make use of high-order statistics, with moments higher than two. This can be implicit or explicit.[8]

See also

[edit]
[edit]

References

[edit]
  1. ^ Barmby, Pauline; McLaughlin, Dean E.; Harris, William E.; Harris, Gretchen L. H.; Forbes, Duncan A. (2007). "Structural Parameters for Globular Clusters in M31 and Generalizations for the Fundamental Plane" (PDF). The Astronomical Journal. 133 (6): 2764–2786. arXiv:0704.2057. Bibcode:2007AJ....133.2764B. doi:10.1086/516777. S2CID 58913061.
  2. ^ Lam, Edmund Y.; Goodman, Joseph W. (2000). "Iterative statistical approach to blind image deconvolution". Journal of the Optical Society of America A. 17 (7): 1177–1184. Bibcode:2000JOSAA..17.1177L. doi:10.1364/JOSAA.17.001177. PMID 10883969.
  3. ^ Caron, James N.; Namazi, Nader M.; Rollins, Chris J. (2002). "Noniterative blind data restoration by use of an extracted filter function". Applied Optics. 41 (32): 6884–9. Bibcode:2002ApOpt..41.6884C. doi:10.1364/AO.41.006884. PMID 12440543.
  4. ^ L. Tonellot, T.; K. Broadhead, M. (2010). "Sparse Seismic Deconvolution by Method of Orthogonal Matching Pursuit". 72nd EAGE Conference and Exhibition incorporating SPE EUROPEC 2010. European Association of Geoscientists & Engineers. doi:10.3997/2214-4609.201401250.
  5. ^ Barmby, P.; McLaughlin, D. E.; Harris, W. E.; Harris, G. L. H.; Forbes, D. A. (2015). "Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization". IEEE Signal Processing Letters. 22 (5): 539–543. arXiv:1407.5465. Bibcode:2015ISPL...22..539R. doi:10.1109/LSP.2014.2362861. S2CID 9605797.
  6. ^ Gray, W. C. (1978). Variable norm deconvolution (PDF) (Ph.D. thesis). Stanford University. Archived from the original (PDF) on 2015-04-09.
  7. ^ Koldovsky, Zbynek; Tichavsky, Petr (2007). "Time-domain blind audio source separation using advanced ICA methods". The Proceedings of the 8th Annual Conference of the International Speech Communication Association (Interspeech 2007). pp. 846–849.
  8. ^ Cardoso, J.-F. (1991). "Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors". [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 3109–3112. CiteSeerX 10.1.1.8.9380. doi:10.1109/ICASSP.1991.150113. ISBN 978-0-7803-0003-3. S2CID 7972548.
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