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Article: A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions

TitleA Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
Authors
KeywordsDeblurring
Boundary conditions
Toeplitz matrix
Circulant matrix
Hankel matrix
Issue Date1999
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISC
Citation
SIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 851-866 How to Cite?
AbstractBlur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.
Persistent Identifierhttp://hdl.handle.net/10722/42994
ISSN
2015 Impact Factor: 1.792
2015 SCImago Journal Rankings: 2.166
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorNg, MKPen_HK
dc.contributor.authorChan, RHen_HK
dc.contributor.authorTang, WCen_HK
dc.date.accessioned2007-03-23T04:36:27Z-
dc.date.available2007-03-23T04:36:27Z-
dc.date.issued1999en_HK
dc.identifier.citationSIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 851-866en_HK
dc.identifier.issn1064-8275en_HK
dc.identifier.urihttp://hdl.handle.net/10722/42994-
dc.description.abstractBlur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.en_HK
dc.format.extent384679 bytes-
dc.format.extent26112 bytes-
dc.format.mimetypeapplication/pdf-
dc.format.mimetypeapplication/msword-
dc.languageengen_HK
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISCen_HK
dc.relation.ispartofSIAM Journal on Scientific Computing-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectDeblurringen_HK
dc.subjectBoundary conditionsen_HK
dc.subjectToeplitz matrixen_HK
dc.subjectCirculant matrixen_HK
dc.subjectHankel matrixen_HK
dc.titleA Fast Algorithm for Deblurring Models with Neumann Boundary Conditionsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1064-8275&volume=21&issue=3&spage=851&epage=866&date=1999&atitle=A+Fast+Algorithm+for+Deblurring+Models+with+Neumann+Boundary+Conditionsen_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.doi10.1137/S1064827598341384en_HK
dc.identifier.hkuros52942-
dc.identifier.isiWOS:000084272500004-

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