Partial differential equations (PDEs) are able to reconstruct images accurately from a small fraction of their image points. The inpainting capabilities of sophisticated anisotropic PDEs allow compression codecs with suboptimal data selection approaches to compete with transform-based methods like JPEG2000. For simple linear PDEs, optimal known data can be found with powerful optimisation strategies. However, the potential of these linear methods for compression has so far not yet been determined. As a remedy, we present a compression framework with homogeneous, biharmonic, and edge-enhancing diffusion (EED) that supports different strategies for data selection and storage: on the one hand, we find exact masks with optimal control or stochastic sparsification and store them with a combination of PAQ and block coding. On the other hand, we propose a new probabilistic strategy for the selection of suboptimal known data that can be efficiently stored with binary trees and entropy coding. This new framework allows us a detailed analysis of the strengths and weaknesses of the three PDEs. Our investigation leads to surprising results: at low compression rates, the simple harmonic diffusion can surpass its more sophisticated PDE-based competitors and even JPEG2000. For high compression rates, we find that EED yields the best result due to its robust inpainting performance under suboptimal conditions.