On the Evolution of the Skeleton

It is commonly held that skeletons are sensitive to noise. It is also believed that smoothing, typically invoked to combat noise, obeys the causality principle that no new structures are created via smoothing. We demonstrate that both views are incorrect. We characterize how smooth points of the skeleton evolve under a general boundary evolution, with the corollary that, when the boundary is smoothed by a geometric heat equation, the skeleton evolves according to a related geometric heat equation. The surprise is that, while certain aspects of the skeleton simplify, as one would expect, others can behave wildly, including the creation of new skeleton branches. Fortunately such sections can be flagged as ligature, or those portions of the skeleton related to shape concavities. Our analysis also includes junctions and an explicit model for boundary noise. Provided a smoothness condition is met, the skeleton can often reduce noise. However, when the smoothness condition is violated, the skeleton can change violently, which, we speculate, corresponds to situations in which “parts” are created, e.g., when the handle appears on a rotating cup.