An Efficient Algorithm for ℓ 0 Minimization in Wavelet Frame Based Image Restoration

Wavelet frame based models for image restoration have been extensively studied for the past decade [1, 2, 3, 4, 5, 6]. The success of wavelet frames in image restoration is mainly due to their capability of sparsely approximating piecewise smooth functions like images. Most of the wavelet frame based models designed in the past are based on the penalization of the l1 norm of wavelet frame coefficients, which, under certain conditions, is the right choice, as supported by theories of compressed sensing [7, 8, 9]. However, the assumptions of compressed sensing may not be satisfied in practice (e.g. for image deblurring and CT image reconstruction). Recently in [10], the authors propose to penalize the l0 “norm” of the wavelet frame coefficients instead, and they have demonstrated significant improvements of their method over some commonly used l1 minimization models in terms of quality of the recovered images. In this paper, we propose a new algorithm, called the mean doubly augmented Lagrangian (MDAL) method, for l0 minimizations based on the classical doubly augmented Lagrangian (DAL) method [11]. Our numerical experiments show that the proposed MDAL method is not only more efficient than the method proposed by [10], but can also generate recovered images with even higher quality. This study reassures the feasibility of using the l0 “norm” for image restoration problems.