A class of nonlinear parabolic PDEs with variable growth structure applied to multi-frame MRI super-resolution

This research paper proposes a novel parabolic model driven by a nonlinear operator with a variable exponent applied to multi-frame image super-resolution. Our idea is based essentially on enhancing the classical image super-resolution models by considering novel regularized terms involving $p\left(x\right)$-growth structure. This regularization leads to deriving a new nonlinear parabolic PDE with nonstandard growth conditions. We start initially by examining the theoretical solvability of our model. We employ the so-called variable exponents Lebesgue-Sobolev spaces to establish an appropriate functional framework for the theoretical investigation of our proposed model. We then apply the Faedo–Galerkin method to establish both the existence and uniqueness of a weak solution for the proposed model. To validate the effectiveness of our model in the multi-frame super resolution (SR) context, we conduct numerical experiments on Magnetic Resonance Images (MRI) featuring diverse characteristics, including corners and edges, while applying different warping, decimation and blurring matrices with noises on the low-resolution (LR) images. We initiate the evaluation by introducing an adaptive discrete scheme of the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise while conducting many behavior tests on some parameters with major contributions. Additionally, we perform simulations on real data (videos) to show the superiority of the proposed model. The obtained high resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming the competitive models visually and quantitatively.