  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 different
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 can be obtained and implemented efficiently for the
Gaussian prior or any other regularization that can be formulated into an
$\ell_2$-regularized quadratic model, i.e., an $\ell_2$-$\ell_2$ optimization
problem. Furthermore, 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
intotraditional 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 the existing techniques.
