Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Michael K. Ng, Jianyu Pan

Research output: Contribution to journalJournal articlepeer-review

21 Citations (Scopus)
44 Downloads (Pure)


The main aim of this paper is to develop a fast algorithm for solving weighted Toeplitz regularized least squares problems arising from image restoration. Based on augmented system formulation, we develop new Hermitian and skew-Hermitian splitting (HSS) preconditioners for solving such linear systems. The advantage of the proposed preconditioner is that the blurring matrix, weighting matrix, and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use. We show that for a preconditioned system that is derived from a saddle point structure of size (m+n)×(m+n), the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 λ with |λ| > 1. We also study how to choose the HSS parameter to minimize the magnitude of λ, and therefore the Krylov subspace method applied to solving the preconditioned system converges very quickly. Experimental results for image restoration problems are reported to demonstrate that the performance of the proposed preconditioner is better than the other testing preconditioners.

Original languageEnglish
Pages (from-to)B94-B121
Number of pages28
JournalSIAM Journal on Scientific Computing
Issue number1
Publication statusPublished - 20 Feb 2014

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Colored noise
  • Image restoration
  • Least squares problems
  • Preconditioners
  • Splitting
  • Weighted Toeplitz matrices


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