  The wavelet frame systems have been widely investigated and applied for image
restoration and many other image processing problems over the past decades,
attributing to their good capability of sparsely approximating piece-wise
smooth functions such as images. Most wavelet frame based models exploit the
$l_1$ norm of frame coefficients for a sparsity constraint in the past. The
authors in \cite{ZhangY2013, Dong2013} proposed an $l_0$ minimization model,
where the $l_0$ norm of wavelet frame coefficients is penalized instead, and
have demonstrated that significant improvements can be achieved compared to the
commonly used $l_1$ minimization model. Very recently, the authors in
\cite{Chen2015} proposed $l_0$-$l_2$ minimization model, where the nonlocal
prior of frame coefficients is incorporated. This model proved to outperform
the single $l_0$ minimization based model in terms of better recovered image
quality. In this paper, we propose a truncated $l_0$-$l_2$ minimization model
which combines sparsity, nonlocal and support prior of the frame coefficients.
The extensive experiments have shown that the recovery results from the
proposed regularization method performs better than existing state-of-the-art
wavelet frame based methods, in terms of edge enhancement and texture
preserving performance.
