Shearlet-Wavelet Regularized Semismooth Newton Iteration for Image Restoration

Abstract

Image normally has both dots-like and curve structures. But the traditional wavelet or multidirectional wave (ridgelet, contourlet, curvelet, etc.) could only restore one of these structures efficiently so that the restoration results for complex images are unsatisfactory. For the image restoration, this paper adopted a strategy of combined shearlet and wavelet frame and proposed a new restoration method. Theoretically, image sparse representation of dots-like and curve structures could be achieved by shearlet and wavelet, respectively. Under theL1regularization, the two frame-sparse structures could show their respective advantages and efficiently restore the two structures. In order to achieve superlinear convergence, this paper applied semismooth Newton method based on subgradient to solve objective functional without differentiability. Finally, through numerical results, the effectiveness of this strategy was validated, which presented outstanding advantages for any individual frame alone. Some detailed information that could not be restored in individual frame could be clearly demonstrated with this strategy.