Fast Single Image Super-Resolution Using a New Analytical Solution for $\ell _{2}$ – $\ell _{2}$ Problems

Ningning Zhao, Qi Wei, Adrian Basarab, Nicolas Dobigeon, Denis Kouame, Jean-Yves Tourneret
2016 IEEE Transactions on Image Processing   unpublished
OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : Eprints ID : 16140 To link to this article : Abstract-This paper addresses the problem of single image super-resolution (SR), which consists of recovering a high-resolution image from its blurred, decimated, and noisy version. The existing algorithms for single image SR use
more » ... fferent strategies to handle the decimation and blurring operators. In addition to the traditional first-order gradient methods, recent techniques investigate splitting-based methods dividing the SR problem into up-sampling and deconvolution steps that can be easily solved. Instead of following this splitting strategy, we propose to deal with the decimation and blurring operators simultaneously by taking advantage of their particular properties in the frequency domain, leading to a new fast SR approach. Specifically, an analytical solution is derived and implemented efficiently for the Gaussian prior or any other regularization that can be formulated into an ℓ 2-regularized quadratic model, i.e., an ℓ 2-ℓ 2 optimization problem. The flexibility of the proposed SR scheme is shown through the use of various priors/regularizations, ranging from generic image priors to learning-based approaches. In the case of non-Gaussian priors, we show how the analytical solution derived from the Gaussian case can be embedded into traditional splitting frameworks, allowing the computation cost of existing algorithms to be decreased significantly. Simulation results conducted on several images with different priors illustrate the effectiveness of our fast SR approach compared with existing techniques. Index Terms-Single image super-resolution, deconvolution, decimation, block circulant matrix, variable splitting based algorithms.