Wavelet Frame Based Image Restoration Using Sparsity, Nonlocal and Support Prior of Frame Coefficients

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.