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.

翻译：近年来，基于修补的压缩方法已被证明是传统编解码器（如JPEG和JPEG2000）的可行替代方案。与在变换域中存储系数的基于变换的编解码器不同，基于修补的方法仅存储原始图像像素的一个小子集，并通过使用合适的修补算子从中重建图像。均匀扩散修补是此类修补算子的一个良好候选，因为它简单、理论依据充分，且能在优化数据下实现良好的重建质量。然而，一个主要挑战在于设计能够扩展至4K图像分辨率（$3840 \times 2160$像素）并具备实时能力的均匀扩散修补快速求解器。我们通过精心调整并融合数值分析中两个最高效的概念——多重网格和区域分解——来克服这一挑战。我们的区域分解算法通过在小重叠块上求解修补问题，高效利用了GPU并行性。与JPEG中采用的简单块分解策略不同，我们的方法可生成无块状伪影的重建结果。此外，将区域分解嵌入全多重网格方案可提供全局交互，并通过以相同速率同时降低低频和高频误差来实现最优收敛。即使面对非常稀疏的数据，我们也能以超过每秒60帧的速度实现4K彩色图像重建——这在以往是无法实现的。