It is possible to compress/inpaint images from very little data. In order to obtain reconstructions that are comparable to the original image it is necessary to optimize the underlying interpolation data. Several such strategies have been discussed in my PhD Thesis. In a series of subsequent publications (here, here and here) we showed the relationship to other partial differential equations and also stated conditions under which the existence of a unique solution can be asserted. Extensions to higher dimensional setups, such as videos, have been discussed here and here.
An implementation of my data optimization algorithms for PDE-based image compression is available on gitlab. The implementations are in modern Fortran (2018 Standard). The code is still under active maintenance as I try to fix bugs and make the code easier to maintain. The code is released under GPLv3.
A simple and straightforward implementation of the PDE-based inpainting with the Laplacian is available here. The code is MIT Licensed and implemented in julia. The application is kept short and simple on purpose as it is meant to show the principle behind the inpainting process. The zip folder also contains a sample image and a mask with 5% density such that simply executing the code should give you a result.
The source code (in Matlab) for the optical flow extensions are available here. The code is licensed GPLv3.