Stable Hierarchical Shape Descriptions

Blum's Skeleton: Connected but unstable internal junctions
Brady, Asada, Rom, Medioni's smooth local symmetries (subset): Not connectedNon-ligature skeleton + Bends/Seeds: Connected and Stable

Although it is desirable to derive hierarchical descriptions of objects, it has long been known that these are difficult to compute reliably.  In order to understand the nature of these instabilities, we have sought the origin of the fragility of the skeleton.

We have taken two approaches.  First, motivated by natural processes underlying shape decomposition, we have considered as unstable those regions which "glue" the parts of a shape together. Second, we have undertaken a mathematical study of perturbations in the skeleton as a function of boundary motions.  These two approaches have led to the similar conclusion that a particular subset of the skeleton, what Blum called "ligature", is the source of many problems.  With this knowledge, it is hoped that ligature will also be part of the solution to the skeleton instability problem.