Image inpainting consists in recovering lost or deteriorated parts of an image. For this task, successful numerical methods include diffusion-based schemes. More specifically, this type of algorithm can apply an anisotropic partial differential equation (PDE) to fill in the gaps of the incomplete image. The quality of the restored image depends on the choice of the PDE. The most common approaches employ the gradient or other first-order differential operators. In this project, our goal is to investigate whether the use of higher-order differential operators can restore more details in the solution. In particular, we aim at generalizing current first-order-based PDEs to also include second-order operators based on the Hessian matrix.