Sparse representation based image deblurring model under random-valued impulse noise

In this article, we introduce a new patch-based model for restoring images simultaneously corrupted by blur and random-valued impulse noise. The model involves a $$l_0$$l0-norm data-fidelity term, a sparse representation prior over learned dictionaries, and the total variation (TV) regularization. Unlike previous works Cai et al. (Inverse Probl Imaging 2(2):187–204, 2008), Ma et al. (SIAM J Imaging Sci 6(4):2258–2284, 2013), one-phase approach is utilized for random-valued impulse noise. As in Yuan and Ghanem (IEEE conference on computer vision and pattern recognition (CVPR), pp 5369–5377, 2015), the $$l_0$$l0 data-fitting term plays an influential role for removing random-valued impulse noise. Moreover, the sparse representation prior enables to preserve textures and details efficiently, whereas TV regularization locally smoothes images while keeping sharp edges. To handle nonconvex and nondifferentiable terms, we adopt a variable splitting scheme, and then the penalty method and alternating minimization algorithm are employed. This results in an efficient iterative algorithm for solving our model. Numerical results are reported to show the effectiveness of the proposed model compared with the state-of-the-art methods.