Image Super-Resolution via Adaptive $\ell _{p} (0<p<1)$ 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 ℓ
₀
or ℓ
₁
norm optimization problem. The ℓ
₀
optimization is a nonconvex and NP-hard problem, while the ℓ
₀
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 ℓ
_p
0 <; p <; 1) 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 ℓ
₁
regularization method. We also consider the best choice for ℓ
_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 λ adaptively, and we discuss an alternate iteration method for selecting p and λ. We perform experiments, which demonstrates that the proposed ℓ
_p
(0 <; p <; 1) regularization nonconvex optimization method can outperform the convex optimization method and generate higher quality images.