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The theory of (tight) wavelet frames has been extensively studied in the pasttwenty years and they are currently widely used for image restoration and otherimage processing and analysis problems. The success of wavelet frame basedmodels, including balanced approach and analysis based approach, is due totheir capability of sparsely approximating piecewise smooth functions likeimages. Motivated by the balanced approach and analysis based approach, weshall propose a wavelet frame based $\ell_0$ minimization model, where the$\ell_0$ "norm" of the frame coefficients is penalized. We adapt the penaltydecomposition (PD) method to solve the proposed optimization problem. Numericalresults showed that the proposed model solved by the PD method can generateimages with better quality than those obtained by either analysis basedapproach or balanced approach in terms of restoring sharp features as well asmaintaining smoothness of the recovered images. Some convergence analysis ofthe PD method will also be provided.

$\ell_0$ Minimization for Wavelet Frame Based Image Restoration