Generalization of Strang's preconditioner with applications to iterative deconvolution

Raymond H. Chan; Michael K. Ng; Robert J. Plemmons

doi:10.1117/12.190864

Generalization of Strang's preconditioner with applications to iterative deconvolution

Raymond H. Chan^*, Michael K. Ng, Robert J. Plemmons

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

Abstract

In this paper, we proposed a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices A_n. The [n/2]th column of our circulant preconditioner S_n is equal to the [n/2]th column of the given matrix A_n. Thus if A_n is a square Toeplitz matrix, then S_n is just the Strang circulant preconditioner. When S_n is not Hermitian, our circulant preconditioner can be defined as (S^*_nS_n)^1/2. This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. Comparisons of our preconditioner S_n with other circulant-based preconditioners are carried out for some 1D Toeplitz least squares problems: min||b - Ax||₂. Preliminary numerical results show that S_n performs quite well. Test results are also reported for a 2D deconvolution problem arising in ground-based atmospheric imaging.

Original language	English
Title of host publication	SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation
Subtitle of host publication	Advanced Signal Processing: Algorithms, Architectures, and Implementations V
Editors	Franklin T. Luk
Publisher	Society of Photo-Optical Instrumentation Engineers
Pages	528-539
Number of pages	12
ISBN (Print)	0819416207
DOIs	https://doi.org/10.1117/12.190864
Publication status	Published - Jul 1994
Event	SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation - San Diego, United States Duration: 24 Jul 1994 → 29 Jul 1994 https://www.spiedigitallibrary.org/conference-proceedings-of-spie/browse/SPIE-Optics-Photonics/1994

Publication series

Name	Proceedings of SPIE - The International Society for Optical Engineering
Volume	2296
ISSN (Print)	0277-786X

Symposium

Symposium	SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation
Country/Territory	United States
City	San Diego
Period	24/07/94 → 29/07/94
Internet address	https://www.spiedigitallibrary.org/conference-proceedings-of-spie/browse/SPIE-Optics-Photonics/1994

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10.1117/12.190864

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Chan, R. H., Ng, M. K., & Plemmons, R. J. (1994). Generalization of Strang's preconditioner with applications to iterative deconvolution. In F. T. Luk (Ed.), SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation: Advanced Signal Processing: Algorithms, Architectures, and Implementations V (pp. 528-539). (Proceedings of SPIE - The International Society for Optical Engineering; Vol. 2296). Society of Photo-Optical Instrumentation Engineers. https://doi.org/10.1117/12.190864

Chan, Raymond H. ; Ng, Michael K. ; Plemmons, Robert J. / Generalization of Strang's preconditioner with applications to iterative deconvolution. SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation: Advanced Signal Processing: Algorithms, Architectures, and Implementations V. editor / Franklin T. Luk. Society of Photo-Optical Instrumentation Engineers, 1994. pp. 528-539 (Proceedings of SPIE - The International Society for Optical Engineering).

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abstract = "In this paper, we proposed a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The [n/2]th column of our circulant preconditioner Sn is equal to the [n/2]th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as (S*nSn) 1/2 . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. Comparisons of our preconditioner Sn with other circulant-based preconditioners are carried out for some 1D Toeplitz least squares problems: min||b - Ax||2. Preliminary numerical results show that S n performs quite well. Test results are also reported for a 2D deconvolution problem arising in ground-based atmospheric imaging.",

author = "Chan, {Raymond H.} and Ng, {Michael K.} and Plemmons, {Robert J.}",

year = "1994",

month = jul,

doi = "10.1117/12.190864",

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booktitle = "SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation",

note = "SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation ; Conference date: 24-07-1994 Through 29-07-1994",

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Chan, RH, Ng, MK & Plemmons, RJ 1994, Generalization of Strang's preconditioner with applications to iterative deconvolution. in FT Luk (ed.), SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation: Advanced Signal Processing: Algorithms, Architectures, and Implementations V. Proceedings of SPIE - The International Society for Optical Engineering, vol. 2296, Society of Photo-Optical Instrumentation Engineers, pp. 528-539, SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation, San Diego, California, United States, 24/07/94. https://doi.org/10.1117/12.190864

Generalization of Strang's preconditioner with applications to iterative deconvolution. / Chan, Raymond H.; Ng, Michael K.; Plemmons, Robert J.
SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation: Advanced Signal Processing: Algorithms, Architectures, and Implementations V. ed. / Franklin T. Luk. Society of Photo-Optical Instrumentation Engineers, 1994. p. 528-539 (Proceedings of SPIE - The International Society for Optical Engineering; Vol. 2296).

Research output: Chapter in book/report/conference proceeding › Conference proceeding › peer-review

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N2 - In this paper, we proposed a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The [n/2]th column of our circulant preconditioner Sn is equal to the [n/2]th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as (S*nSn) 1/2 . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. Comparisons of our preconditioner Sn with other circulant-based preconditioners are carried out for some 1D Toeplitz least squares problems: min||b - Ax||2. Preliminary numerical results show that S n performs quite well. Test results are also reported for a 2D deconvolution problem arising in ground-based atmospheric imaging.

AB - In this paper, we proposed a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The [n/2]th column of our circulant preconditioner Sn is equal to the [n/2]th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as (S*nSn) 1/2 . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. Comparisons of our preconditioner Sn with other circulant-based preconditioners are carried out for some 1D Toeplitz least squares problems: min||b - Ax||2. Preliminary numerical results show that S n performs quite well. Test results are also reported for a 2D deconvolution problem arising in ground-based atmospheric imaging.

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Chan RH, Ng MK, Plemmons RJ. Generalization of Strang's preconditioner with applications to iterative deconvolution. In Luk FT, editor, SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation: Advanced Signal Processing: Algorithms, Architectures, and Implementations V. Society of Photo-Optical Instrumentation Engineers. 1994. p. 528-539. (Proceedings of SPIE - The International Society for Optical Engineering). doi: 10.1117/12.190864

Generalization of Strang's preconditioner with applications to iterative deconvolution

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