Cosine transform preconditioners for high resolution image reconstruction

Michael K. Ng; Raymond H. Chan; Tony F. Chan; Andy M. Yip

doi:10.1016/S0024-3795(99)00274-8

Cosine transform preconditioners for high resolution image reconstruction

Michael K. Ng, Raymond H. Chan, Tony F. Chan, Andy M. Yip

Research output: Contribution to journal › Article › peer-review

47 Scopus citations

Abstract

This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L₂ or H₁ norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.

Original language	English (US)
Pages (from-to)	89-104
Number of pages	16
Journal	Linear Algebra and Its Applications
Volume	316
Issue number	1-3
DOIs	https://doi.org/10.1016/S0024-3795(99)00274-8
State	Published - Sep 1 2000
Externally published	Yes

Bibliographical note

Funding Information:
∗ Corresponding author. Tel.: +852-28592252; fax: +852-25592225. E-mail addresses: [email protected] (M.K. Ng), [email protected] (R.H. Chan), [email protected] (T.F. Chan), [email protected] (A.M. Yip). 1 Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and HKU CRCG Grant No. 10202720. 2 Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. 3 Research supported by ONR under grant N00014-96-1-0277 and NSF under grant DMS 96-26755.

Keywords

Discrete cosine transform
Image reconstruction
Neumann boundary condition
Toeplitz matrix

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/S0024-3795(99)00274-8

Cite this

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title = "Cosine transform preconditioners for high resolution image reconstruction",

abstract = "This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L2 or H1 norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.",

keywords = "Discrete cosine transform, Image reconstruction, Neumann boundary condition, Toeplitz matrix",

author = "Ng, {Michael K.} and Chan, {Raymond H.} and Chan, {Tony F.} and Yip, {Andy M.}",

note = "Funding Information: ∗ Corresponding author. Tel.: +852-28592252; fax: +852-25592225. E-mail addresses: [email protected] (M.K. Ng), [email protected] (R.H. Chan), [email protected] (T.F. Chan), [email protected] (A.M. Yip). 1 Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and HKU CRCG Grant No. 10202720. 2 Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. 3 Research supported by ONR under grant N00014-96-1-0277 and NSF under grant DMS 96-26755.",

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T1 - Cosine transform preconditioners for high resolution image reconstruction

AU - Ng, Michael K.

AU - Chan, Raymond H.

AU - Chan, Tony F.

AU - Yip, Andy M.

N1 - Funding Information: ∗ Corresponding author. Tel.: +852-28592252; fax: +852-25592225. E-mail addresses: [email protected] (M.K. Ng), [email protected] (R.H. Chan), [email protected] (T.F. Chan), [email protected] (A.M. Yip). 1 Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and HKU CRCG Grant No. 10202720. 2 Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. 3 Research supported by ONR under grant N00014-96-1-0277 and NSF under grant DMS 96-26755.

PY - 2000/9/1

Y1 - 2000/9/1

N2 - This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L2 or H1 norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.

AB - This paper studies the application of preconditioned conjugate gradient methods in high resolution image reconstruction problems. We consider reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. The resulting blurring matrices are spatially variant. The classical Tikhonov regularization and the Neumann boundary condition are used in the reconstruction process. The preconditioners are derived by taking the cosine transform approximation of the blurring matrices. We prove that when the L2 or H1 norm regularization functional is used, the spectra of the preconditioned normal systems are clustered around 1 for sufficiently small subpixel displacement errors. Conjugate gradient methods will hence converge very quickly when applied to solving these preconditioned normal equations. Numerical examples are given to illustrate the fast convergence.

KW - Discrete cosine transform

KW - Image reconstruction

KW - Neumann boundary condition

KW - Toeplitz matrix

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SN - 0024-3795

VL - 316

SP - 89

EP - 104

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -

Cosine transform preconditioners for high resolution image reconstruction

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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