K-SVD with a Real ℓ0 Optimization: Application to Image Denoising

This paper deals with sparse coding for dictionary learning in sparse representations. Because sparse coding involves an ℓ0-norm, most, if not all, existing solutions only provide an approximate solution. Instead, in this paper, a real ℓ0 optimization is considered for the sparse coding problem providing a global optimal solution. The proposed method reformulates the optimization problem as a Mixed-Integer Quadratic Program (MIQP), allowing then to obtain the global optimal solution by using an off-the-shelf solver. Because computing time is the main disadvantage of this approach, two techniques are proposed to improve its computational speed. One is to add suitable constraints and the other to use an appropriate initialization. The results obtained on an image denoising task demonstrate the feasibility of the MIQP approach for processing well-known benchmark images while achieving good performance compared with the most advanced methods.