Surely convex 'peeling' can be viewed as a crudely discretised PDE, the heat equation and Riccian flow of submanifold (levels) are linked. Perhaps this 2009 paper is relevant? http://www.ipol.im/news/20100402_curvature_microscope_seminar/JMM_ImageCurvatureMicroscope.pdf

This reminds me of this process:

I was surprised to see that we get nice smooth shapes even if we apply a square-shaped filter — but it is important here that the input is random and not adversarial.

Is this similar to coarse graining?

+James Prichard but the heat equation gives the usual curve-shortening flow, not the affine one.

It's different but similar according to "Contrast invariant image analysis and PDE's" which references [14] S. B. Angenent, G. Sapiro, and A. Tannenbaum, On the affine heat flow for nonconvex curves, J. Amer. Math. Soc., 11 (1998), pp. 601–634.

+Jukka Suomela That's very interesting! Do you have any write-up with more info about it?

I knew that there is a CA rule called "anneal" that behaves like the curve-shortening flow. See Wikipedia here:

+Gabriel Nivasch Sorry, no, I don't have anything written up on this, I did this just for fun.

+Jukka Suomela Did you try (2n+1)*(2n+1) for different values of n?

+Gabriel Nivasch Yes, the magic value 11 is not that important here. Something like 9x9 or 13x13 looks pretty similar to 11x11.

However, for very small values of the window size it gets more boring: you will get fairly fine-grained structures and the process converges quickly. So something like 3x3 or 5x5 isn't that interesting yet.