Earlier quoted context omitted.
> I don't think this method has anything to do with Markov Chains. Oh, it absolutely does. I think it's fair to say that Efros launched the field of nearest neighbor texture synthesis, and his abstract states: "The texture synthesis process grows a new image outward from an initial seed, one pixel at a time. A Markov random field model is assumed, and the conditional distribution of a pixel given all its neighbors sy…
Unless I horribly misread the paper, this is not based on the Efros' quilting method, which indeed uses Markov fields. The method linked here seems to interpolate every pixel independently from its surroundings (neighbor means a close-by pixel in the training set in the feature space, not a spatially close pixel). And I didn't mean that Markov processes are abstract in any "distant" sense, but that they are an abstra…
"To address these limitations, we appeal to a classic learning architecture that can naturally allow for multiple outputs and user-control: non-parametric models, or nearest-neighbors (NN). Though quite a classic approach [11, 15, 20, 24], it has largely been abandoned in recent history with the advent of deep architectures. Intuitively, NN works by requiring a large training set of pairs of (incomplete inputs, high-quality outputs), and works by simply matching the an incomplete query to the training set and returning the corresponding output. This trivially generalizes to multiple outputs through K-NN and allows for intuitive user control through on-the-fly modification of the training set..."
Note the first reference #11 is Efros' non-parametric sampling, and that the authors state this is the "classic approach" that they apply here.
What you call "interpolate every pixel independently from its surroundings" could be another way to describe a Markov chain, because 1: it is sampled according to the conditional probability distribution (which is what you get by using the K nearest matches.) and 2: the process is repeated - one pixel (or patch) is added using the best match, then it becomes part of the neighborhood in the search for the pixel/patch next door. The name for that is "Markov process", or in the discrete case, "Markov chain", if you take an unbiased random sample from the conditional distribution. If you always choose the best sample, then it's the same as a Markov chain, but biased.
> (neighbor means a close-by pixel in the training set in the feature space, not a spatially close pixel)
That's right, and that's why it's misleading to talk about nearest neighbor interpolation, because that phrase is a graphics phrase that means interpolate from spatially close pixels. Hardly anyone else calls it interpolation, they call it sampling, point sampling, and other terms.
*EDIT:
I'm going to relax a little bit on this. "Non-parametric sampling" is a tiny bit different from a Markov process in that a Markov process attempts to simulate a distribution in an unbiased way. By using the best match instead of a random sample from the conditional distribution, the output may produce a biased version of the original distribution. This is why it's called non-parametric sampling instead of calling it a Markov chain, but the distinction is pretty small and subtle -- texture synthesis using non parametric sampling is extremely similar to a Markov chain, but not necessarily exactly the same.
Side note, it's really unfortunate they used the abbreviation "NN" to talk about "nearest neighbor" in a paper that also builds on "neural networks".