Why Does Non-binary Mask Optimisation Work for Diffusion-Based Image Compression?

Finding optimal data for inpainting is a key problem for image-compression with partial differential equations. Not only the location of important pixels but also their values should be optimal to maximise the quality gain. The position of important data is usually encoded in a binary mask. Recent studies have shown that allowing non-binary masks may lead to tremendous speedups but comes at the expense of higher storage costs and yields prohibitive memory requirements for the design of competitive image compression codecs. We show that a recently suggested heuristic to eliminate the additional storage costs of the non-binary mask has a strong theoretical foundation in finite dimension. Binary and non-binary masks are equivalent in the sense that they can both give the same reconstruction error if the binary mask is supplemented with optimal data which does not increase the memory footprint. Further, we suggest two fast numerical schemes to obtain this optimised data. This provides a significant building block in the conception of efficient data compression schemes with partial differential equations.