Image Super-Resolution Based on Sparsity Prior via Smoothed $l_0$ Norm

In this paper we aim to tackle the problem of reconstructing a high-resolution image from a single low-resolution input image, known as single image super-resolution. In the literature, sparse representation has been used to address this problem, where it is assumed that both low-resolution and high-resolution images share the same sparse representation over a pair of coupled jointly trained dictionaries. This assumption enables us to use the compressed sensing theory to find the jointly sparse representation via the low-resolution image and then use it to recover the high-resolution image. However, sparse representation of a signal over a known dictionary is an ill-posed, combinatorial optimization problem. Here we propose an algorithm that adopts the smoothed $l_0$-norm (SL0) approach to find the jointly sparse representation. Improved quality of the reconstructed image is obtained for most images in terms of both peak signal-to-noise-ratio (PSNR) and structural similarity (SSIM) measures.