There are a few different ways to come at the problem theoretically.

One is to work on post-decode data in spatial domain. These approaches basically work by explicitly trying to detect block edges and just filter them. This is the approach, for example, of the H264 "in loop deblocking filter" which there is a lot of literature on. See for example "Adaptive Deblocking Filter" by List, Joch, et.al. For an example of the filter-based approach on the 8x8 DCT case see "DCT-Based Image Compression using Wavelet-Based Algorithm with Efficient Deblocking Filter" by Yan and Chen. (BTW the JPEG standard contains a "block smoother" which basically predicts AC1 as a linear function from neighboring block coefficients. This is okay for the specific case of smooth images and very high quantization, but is generally not awesome and is an ancient technique. Ignore.)

A more hardcore version of the filtering approach is "Combined Frequency and Spatial Domain Algorithm for the Removal of Blocking Artifacts" which does adaptively-offset and adaptively-directed gaussian filters ; this is sort of like the image denoising stuff that creates pixel gradient flow vectors - the filters are local gradient adaptive so they don't go across real edges. This appears to perform quite well but is very expensive.

The other general approach is a more abstract maximum-likelihood idea. You received a lossy compressed image I. You know the original image was one of the many which when compressed produces I. You want to output the one that was most likely the true original image. This is a maximum likelihood problem, and requires some a-priori model of what you think "natural" images look like. In particular, for the case of quantized DCT coefficients, you have a quantized DCT coefficient C ; instead of just reproducing Q*C you can reproduce anything in the range { Q*C - Q/2 , Q*C + Q/2 } , and you should choose the thing in that range that makes the "best" image.

"Optimal JPEG Decoding" (1998) by Jung, Antonini, Barlaud takes this approach directly. Their results are not awesome though; presumably because their prior is not good. A more modern version of the same idea is "Block Artifact Reduction Using a Transform-Domain Markov Random Field Model" by Li and Delp which uses a better model for image likelihood, but is in the same vein of doing a brute force search in the allowed coefficient space to find the maximum-likelihood reproduction.

A related method that was popular for a while is "Projection onto Convex Sets". This is basically just a method of satisfying simple convex constraints in an optimization. Here our constraint is that the quantized coefficient stay the same, that is, repro in { Q*C - Q/2 , Q*C + Q/2 } . You then apply some target function, such as you want smoothness or something, and take iterative steps towards that goal and project onto the constraints one by one. There are a lot more details to this, I haven't paid too much attention to it because these are all crazy expensive and I want something realtime.

"Blocking Artifact Detection and Reduction in Compressed Data" by Triantafyllidis etal (2002) is in the same vein but simpler and more analytical. It again worse directly in DCT space on coefficients within their quantization range, but it directly solves for the ideal reconstruction value as a function of neighbors based on minimization of specific simple deblocking metric. You wind up with just some equations for how to modify each coefficient in terms of neighbor coefficients. While the paper is good, I think one of their base assumptions - that the frequencies can be dealt with independently - is not sound, and most other people do not make that assumption.

"Derivation of Prediction Equations for Blocking Effect Reduction" by Gopal Lakhani and Norman Zhong (1999) is an older, simpler still version of the Triantafyllidis paper. They only correct the first few coefficients and solve for optimal reconstruction to minimize MSDS (mean squared difference of slopes). You can actually look at the equations here and they're very intuitively obviously right. For example, the first AC coefficient should be corrected using the difference of the neighboring DC coefficients. In case you don't see that that's obviously right, if you have DC's like [8],[16],[24] after dequantization at Q=8, and your AC's all got quantized to zero, obviously the original image most likely had a smooth slope, so the first AC in the middle block should be predicted to be the linear interpolation.

An interesting one I found that's related to the stuff I tried with smooth reconstruction of the DC band is : "Improvement of DCT-based Compression Algorithms Using Poisson�s Equation" by Yamatani and Saito (2006) .

BTW a related issue that often comes up is the incorrectness of center dequantization of AC coefficients. I've written about this before and lots of these papers mention it; the best full note on it is : "Biased Reconstruction for JPEG Decoding" by Price.

The very modern stuff has gotten quite arcane. People now are doing things like directional overcomplete wavelets on the reproduced image; with this they can detect both block artifacts and also ringing and other quantized transform artifacts. They then use maximum-likelihood markov models to guess what the source image was that produced this output. This stuff is extremely complex and I haven't really followed it because it's nowhere near realtime, but probably the best solution for offline very high quality JPEG decoders.

An interesting outlier is John Costella's Unblock . It's based on a clever simple idea that I've never seen anywhere else. Unblock is based on the assumption that pixels near the block boundaries come from the same model as pixels in the centers of blocks. That sounds obvious but it's quite profound. It means that pixels near the edges of blocks should have the same statistics as pixels in the centers (in the maximum likelihood lingo, this is a prior we can use to choose an optimal output). In particular, it's useful because in the DCT the interior pixels are much more accurate than the edge pixels. What Unblock does is looks at the statistics of the decompressed interior pixels and assumes those are our goal, and then it forces the pixels near the edge to match the statistics of the interior. The corrections are applied as wide smooth filters.