  Homogeneous diffusion inpainting can reconstruct missing image areas with
high quality from a sparse subset of known pixels, provided that their location
as well as their gray or color values are well optimized. This property is
exploited in inpainting-based image compression, which is a promising
alternative to classical transform-based codecs such as JPEG and JPEG2000.
However, optimizing the inpainting data is a challenging task. Current
approaches are either quite slow or do not produce high quality results. As a
remedy we propose fast spatial and tonal optimization algorithms for
homogeneous diffusion inpainting that efficiently utilize GPU parallelism, with
a careful adaptation of some of the most successful numerical concepts. We
propose a densification strategy using ideas from error-map dithering combined
with a Delaunay triangulation for the spatial optimization. For the tonal
optimization we design a domain decomposition solver that solves the
corresponding normal equations in a matrix-free fashion and supplement it with
a Voronoi-based initialization strategy. With our proposed methods we are able
to generate high quality inpainting masks for homogeneous diffusion and
optimized tonal values in a runtime that outperforms prior state-of-the-art by
a wide margin.
