Fast Computation of Neck-Like Features

Locating neck-like features, or locally narrow parts, of a surface is crucial in various applications such as segmentation, shape analysis, path planning, and robotics. Topological methods are often utilized to find the set of shortest loops around handles and tunnels. However, there are abundant neck-like features on genus-0 shapes without any handles. While 3D geometry-aware topological approaches exist to find neck loops, their construction can be cumbersome and may even lead to geometrically wide loops. Thus we propose a "topology-aware geometric approach" to compute the tightest loops around neck features on surfaces, including genus-0 surfaces. Our algorithm starts with a volumetric representation of an input surface and then calculates the distance function of mesh points to the boundary surface as a Morse function. All neck features induce critical points of this Morse function where the Hessian matrix has precisely one positive eigenvalue, i.e., type-2 saddles. As we focus on geometric neck features, we bypass a topological construction such as the Morse-Smale complex or a lower-star filtration. Instead, we directly create a cutting plane through each neck feature. Each resulting loop can then be tightened to form a closed geodesic representation of the neck feature. Moreover, we offer criteria to measure the significance of a neck feature through the evolution of critical points when smoothing the distance function. Furthermore, we speed up the detection process through mesh simplification without compromising the quality of the output loops.