  In this paper, we present a frequency domain neural network for image
super-resolution. The network employs the convolution theorem so as to cast
convolutions in the spatial domain as products in the frequency domain.
Moreover, the non-linearity in deep nets, often achieved by a rectifier unit,
is here cast as a convolution in the frequency domain. This not only yields a
network which is very computationally efficient at testing but also one whose
parameters can all be learnt accordingly. The network can be trained using back
propagation and is devoid of complex numbers due to the use of the Hartley
transform as an alternative to the Fourier transform. Moreover, the network is
potentially applicable to other problems elsewhere in computer vision and image
processing which are often cast in the frequency domain. We show results on
super-resolution and compare against alternatives elsewhere in the literature.
In our experiments, our network is one to two orders of magnitude faster than
the alternatives with an imperceptible loss of performance.
