Inpainting-based compression has been suggested as a qualitative competitor to the JPEG family of transform-based codecs, specifically for high compression ratios. However, it also requires sophisticated interpolation, data optimisation and encoding tasks that are both slow and hard to implement. We propose a fast and simple alternative that combines Shepard interpolation with a novel joint inpainting and prediction approach. It represents the image by a fraction of its pixel values on a sparse regular subgrid that are selected by an efficient optimisation strategy. Experiments show that our codec is up to five orders of magnitude faster than traditional inpainting-based approaches. Qualitatively, it can surpass transform-based codecs, in particular for high compression ratios and cartoon-like images.