0 minimization for wavelet frame based image restoration

Yong Zhang, Bin Dong, Zhaosong Lu

Research output: Contribution to journalArticlepeer-review

84 Scopus citations


The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach and analysis based approach, is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based ℓ0 minimization model, where the ℓ0 "norm" of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method of Lu and Zhang to solve the proposed optimization problem. Some convergence analysis of the adapted PD method will also be provided. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images.

Original languageEnglish (US)
Pages (from-to)995-1015
Number of pages21
JournalMathematics of Computation
Issue number282
StatePublished - 2013
Externally publishedYes


  • Hard thresholding
  • Image restoration.
  • Wavelet frame
  • ℓ minimization


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