Efficient Parallel Algorithms for Inpainting-Based Representations of 4K Images -- Part I: Homogeneous Diffusion Inpainting

In recent years inpainting-based compression methods have been shown to be a viable alternative to classical codecs such as JPEG and JPEG2000. Unlike transform-based codecs, which store coefficients in the transform domain, inpainting-based approaches store a small subset of the original image pixels and reconstruct the image from those by using a suitable inpainting operator. A good candidate for such an inpainting operator is homogeneous diffusion inpainting, as it is simple, theoretically well-motivated, and can achieve good reconstruction quality for optimized data. However, a major challenge has been to design fast solvers for homogeneous diffusion inpainting that scale to 4K image resolution ($3840 \times 2160$ pixels) and are real-time capable. We overcome this with a careful adaptation and fusion of two of the most efficient concept from numerical analysis: multigrid and domain decomposition. Our domain decomposition algorithm efficiently utilizes GPU parallelism by solving inpainting problems on small overlapping blocks. Unlike simple block decomposition strategies such as the ones in JPEG, our approach yields block artifact-free reconstructions. Furthermore, embedding domain decomposition in a full multigrid scheme provides global interactions and allows us to achieve optimal convergence by reducing both low- and high-frequency errors at the same rate. We are able to achieve 4K color image reconstruction at more than $60$ frames per second even from very sparse data - something which has been previously unfeasible.