  In this paper we focus on learning optimized partial differential equation
(PDE) models for image filtering. In this approach, the grey-scaled images are
represented by a vector field of two real-valued functions and the image
restoration problem is modelled by an evolutionary process such that the
restored image at any time satisfies an initial-boundary-value problem of
cross-diffusion with reaction type. The coupled evolution of the two components
of the image is determined by a nondiagonal matrix that depends on those
components. A critical question when designing a good-performing filter lies in
the selection of the optimal coefficients and influence functions which define
the cross-diffusion matrix. We propose the use of deep learning techniques in
order to optimize the parameters of the model. In particular, we use a back
propagation technique in order to minimize a cost function related to the
quality of the denoising processe, while we ensure stability during the
learning procedure. Consequently, we obtain improved image restoration models
with solid mathematical foundations. The learning framework and resulting
models are presented along with related numerical results and image
comparisons.

Numerical Analysis;Numerical Analysis