A nonlinear second-order parabolic partial di erential equation (PDE) - based model for image interpolation is proposed in this article. The anisotropic di usion scheme introduced here is also investigated mathematically, a rigorous mathematical treatment being provided for it. The interpolation (inpainting, completion) techniques, which reconstruct the missing regions by using the known image information around them, are divided into structure-based and texturebased methods. The structure-based interpolation approaches use PDE and variational models to perform the image completion tasks. Some in uential variational inpainting schemes are MumfordShah Inpainting and Total Variation (TV) Inpainting models. The second-order PDE-based algorithms also follow the variational principles, while the higher-order PDE interpolation models, such as CDD Inpainting or Cahn-Hillard Inpainting, do not derive from variational schemes, being directly given as evolutionary equations. The novel second-order PDE inpainting approach proposed here is derived from our past nonlinear di usion-based restoration models, by introducing an image mask corresponding to the missing or highly deteriorated image zones. The introduced structural completion model is composed of a nonlinear parabolic equation and some boundary conditions. It is based on a positive and monotonically decreasing edge-stopping function that is properly chosen for an e ective restoration. A robust mathematical treatment of the well-posedness of this di erential model is then performed, the existence and uniqueness of a weak solution of the PDE being investigated. A consistent and fast-converging explicit nite-di erence based numerical approximation scheme is then constructed for it. The successfully performed inpainting experiments and method comparison are also described in this paper. Our anisotropic di usion approach provides an e ective structure-based reconstruction, outperforming many existing interpolation methods, but cannot inpaint properly the missing textures. It also works successfully in noisy conditions, reducing the amount of Gaussian noise.