What does "convergence" mean for Thinning?

I've been using Thinning recently on line-like structures in an attempt to generate skeleton-like chains of pixels connected through their Moore neighborhoods.

The description of Thinning says that the algorithm "finds the skeletons of foreground regions in image by applying morphological thinning until convergence". What does "convergence" mean? If I start with a cluster of pixels connected through their Moore neighborhoods, what am I guaranteed after "convergence"? Will the product always be a chain of pixels also connected through their Moore neighborhoods?

• Off-topic on this site, I'm afraid. I do not see how it pertains to Mathematica.
– Sasha
Sep 9 '13 at 1:22
• @Sasha You have a valid point with respect to my comment about "implementing Thinning in other languages", and I deleted this comment. I didn't mean it in the sense of "I want to rewrite the procedure in C, etc." but more of a "what's going on here?" remark. However, I would argue that this question is on-topic here since it's directly asking about the behavior of a particular Mathematica function for image processing, and what the description of that function means?
– LCook
Sep 9 '13 at 1:35
• @Sasha I've also changed the title to better focus the question for the purpose I claim.
– LCook
Sep 9 '13 at 1:38