  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 $\ell_0$ minimization model, where the
$\ell_0$ "norm" of the frame coefficients is penalized. We adapt the penalty
decomposition (PD) method to solve the proposed optimization problem. 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. Some convergence analysis of
the PD method will also be provided.
