Cosine transform preconditioners for high resolution image reconstruction

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

Research output: Contribution to journalArticlepeer-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 languageEnglish (US)
Pages (from-to)89-104
Number of pages16
JournalLinear Algebra and Its Applications
Volume316
Issue number1-3
DOIs
StatePublished - Sep 1 2000
Externally publishedYes

Bibliographical note

Funding Information:
∗ Corresponding author. Tel.: +852-28592252; fax: +852-25592225. E-mail addresses: mng@maths.hku.hk (M.K. Ng), rchan@math.cuhk.edu.hk (R.H. Chan), chan@math.ucla.edu (T.F. Chan), mhyipa@hkusua.hku.hk (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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