Super-Resolution Reconstruction of Remote Sensing Images Based on an Adaptive Fourth-Order PDE

The super-resolution reconstruction of remote sensing images is a typical ill-posed problem, and partial differential equations (PDEs) can effectively address such issues. Since second-order PDEs excel primarily in capturing piecewise linear structural features, which are insufficient for the complex scenes in remote sensing images, fourth-order PDEs are introduced for the super-resolution reconstruction of these images. This article proposes an adaptive fourth-order PDE image reconstruction model, integrating PDE theory. The model effectively exploits the fourth-order term to significantly reduce noise, while the second and third-order terms are utilized to preserve edge integrity effectively. An adaptive adjustment factor dynamically regulates the weights of the second-, third-, and fourth-order terms, achieving an optimal balance between noise reduction and edge retention. In areas of image uniformity, the fourth-order term predominantly contributes to noise reduction, while near image edges, the slower diffusion of the second-order term and the reverse diffusion of the third-order term play a pivotal role in preserving edge details. The Rothe method and finite difference method are employed for the temporal and spatial discretization of the model, respectively, and an analysis of the stability and convergence of the algorithm is conducted. Numerical experiments demonstrate significant improvements over traditional super-resolution methods. Compared to the Projection Onto Convex Sets (POCS) method, our model achieves approximately a 7% improvement in peak signal-to-noise ratio (PSNR) and a 14% improvement in Structural Similarity Index Measure (SSIM).