A novel multi-frame image super-resolution model based on regularized nonlinear diffusion with Caputo time fractional derivative

In this work, we introduce an innovative fractional nonlinear parabolic model using a time-fractional order derivative, specifically employing the Caputo sense for fractional differentiation. This model aims to enhance traditional super-resolution models, particularly in the context of multi-frame image super-resolution. Additionally, we incorporate a regularized Perona–Malik diffusion mechanism to control the speed and direction of diffusion at each image location. We begin our study by exploring the theoretical solvability of our proposed model. Firstly, we employ the Faedo–Galerkin approach to establish the existence and uniqueness of a weak solution for an auxiliary fractional super-resolution model. Subsequently, we use the Schauder fixed point method to demonstrate the existence and uniqueness of a weak solution for our model. To validate the effectiveness of our model in the multi-frame super-resolution (SR) context, we conduct numerical experiments on images featuring diverse characteristics, including corners and edges, while applying various warping, decimation, and blurring matrices to the low-resolution (LR) images. We start the evaluation by introducing an adaptive discrete scheme tailored to the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise. Additionally, we perform simulations on real data (videos). The obtained high-resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming competitive models both visually and quantitatively.