Image Super-Resolution via Adaptive $\ell _{p} (0 Regularization and Sparse Representation

Previous studies have shown that image patches can be well represented as a sparse linear combination of elements from an appropriately selected over-complete dictionary. Recently, single-image super-resolution (SISR) via sparse representation using blurred and downsampled low-resolution images has attracted increasing interest, where the aim is to obtain the coefficients for sparse representation by solving an $\ell _{0}$ or $\ell _{1}$ norm optimization problem. The $\ell _{0}$ optimization is a nonconvex and NP-hard problem, while the $\ell _{1}$ optimization usually requires many more measurements and presents new challenges even when the image is the usual size, so we propose a new approach for SISR recovery based on $\ell _{p} (0 regularization nonconvex optimization. The proposed approach is potentially a powerful method for recovering SISR via sparse representations, and it can yield a sparser solution than the $\ell _{1}$ regularization method. We also consider the best choice for $\ell _{p}$ regularization with all $p$ in (0, 1), where we propose a scheme that adaptively selects the norm value for each image patch. In addition, we provide a method for estimating the best value of the regularization parameter $\lambda $ adaptively, and we discuss an alternate iteration method for selecting $p$ and $\lambda $ . We perform experiments, which demonstrates that the proposed $\ell _{p} (0 regularization nonconvex optimization method can outperform the convex optimization method and generate higher quality images.