Removal of blocking artifacts in images due to compression
If you talk about JPEG-like blocking artifacts, an interesting appproach consists in the re-applications of the same JPEG algorithm on shifted versions of the image, see Enhancement of JPEG-Compressed Images by
Re-application of JPEG, by Aria Nosratinia, 1999. It has a software implementation. It implement a sort of shift-invariant processing. He did the same with the JPEG-2000 coder, in Post-Processing of JPEG-2000 Images to Remove Compression Artifacts. You can find an overview of these artifacts here.

Improved methods include sparsifying priors like total variation, see for instance Adapted Total Variation for Artifact Free Decompression of JPEG Images. Recent references include Reducing Artifacts in JPEG Decompression Via a Learned Dictionary and Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion.