Choking Loops on Surfaces

摘要

We present a method for computing "choking” loops—a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an offset of the original surface would get pinched. Our generalized loops naturally include the usual $(2g)$ handles/tunnels computed based on the topology of the genus-$(g)$ surface, but also include loops that identify chokepoints or bottlenecks, i.e., boundaries of small membranes separating the inside or outside volume of the surface into disconnected regions. Our definition is based on persistent homology theory, which gives a measure to topological structures, thus providing resilience to noise and a well-defined way to determine topological feature size. More precisely, the persistence computed here is based on the lower star filtration of the interior or exterior 3D domain with the distance field to the surface being the associated 3D Morse function.