Discretization schemes and numerical approximations of PDE impainting models and a comparative evaluation on novel real world MRI reconstruction applications

Abstract

While various PDE models are in discussion since the last ten years and are widely applied nowadays in image processing and computer vision tasks, including restoration, filtering, segmentation and object tracking, the perspective adopted in the majority of the relevant reports is the view of applied mathematician, attempting to prove the existence theorems and devise exact numerical methods for solving them. Unfortunately, such solutions are exact for the continuous PDEs but due to the discrete approximations involved in image processing, the results yielded might be quite unsatisfactory. The major contribution of This work is, therefore, to present, from an engineering perspective, the application of PDE models in image processing analysis, from the algorithmic point of view, the discretization and numerical approximation schemes used for solving them. It is of course impossible to tackle all PDE models applied in image processing in this report from the computational point of view. It is, therefore, focused on image impainting PDE models, that is on PDEs, including anisotropic diffusion PDEs, higher order non-linear PDEs, variational PDEs and other constrained/regularized and unconstrained models, applied to image interpolation/ reconstruction. Apart from this novel computational critical overview and presentation of the PDE image impainting models numerical analysis, the second major contribution of This work is to evaluate, especially the anisotropic diffusion PDEs, in novel real world image impainting applications related to MRI.