adjusting an image

Let A be a matrix of zeros and ones. We imagine A to be extended in all
directions by zeros, so we don't need to worry about edge effects. For
each entry x of A, we look at a 3x3 window centred at x, see whether
zeros or ones predominate, and replace x by whichever entry
predominates.

This seems to be a standard procedure used in image processing, to
"clean up" an image with ragged boundary. (But I might have it slightly
wrong---please correct me if I am.)

Since the situation is finite, the procedure must terminate in a finite
cycle. My question is: does it always terminate in a cycle of length
one (that is, no change)? What about other shapes than a 3x3 square?
One could do this with any shape with an odd number of "pixels", and a
marked pixel that would play the role of a centre.

One can also generalize this to the continuous case, as an operation on
measurable subsets (up to sets of measure zero) provided the window
shape is nice enough. Then the question would be whether the iterated
operation leads to a convergent sequence of subsets, in some measure
theoretic measure of convergence.

Meta questions: is this sort of question more appropriate to some
specialized mathematical newsgroup?
I wanted to look this topic up on Google, but I didn't know any
keywords to try. Are there any?

Thanks for any pointers.
David Epstein

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