This paper presents a new efficient approach for the solution of the ℓp-ℓq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of ℓp/ℓq-norms by weighted ℓ 2-norms. Computed image restoration examples illustrate that it suffices to carry out only a few iterations to achieve highquality restorations. The combination of a low iteration count and a modest storage requirement makes the proposed method attractive.
We introduce a Convex Non-Convex (CNC) denoising variational model for restoring images corrupted by Additive White Gaussian Noise (AWGN). We propose the use of parameterized non-convex regularizers to effectively induce sparsity of the gradient magnitudes in the solution, while maintaining strict convexity of the total cost functional. Some widely used non-convex regularization functions are evaluated and a new one is analyzed which allows for better restorations. An efficient minimization algorithm based on the Alternating Directions Methods of Multipliers (ADMM) strategy is proposed for simultaneously restoring the image and automatically selecting the regularization parameter by exploiting the discrepancy principle. Theoretical convexity conditions for both the proposed CNC variational model and the optimization sub-problems arising in the ADMM-based procedure are provided which guarantee convergence to a unique global minimizer. Numerical examples are presented which indicate how the proposed approach is particularly effective and well suited for images characterized by moderately sparse gradients.
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