Mean Curvature Skeletons

Abstract

Inspired by recent developments in contraction-based curve skeleton extraction, we formulate the skeletonization problem via mean curvature flow (MCF). While the classical application of MCF is surface fairing, we take advantage of its area-minimizing characteristic to drive the curvature flow towards the extreme so as to collapse the input mesh geometry and obtain a skeletal structure. By analyzing the differential characteristics of the flow, we reveal that MCF locally increases shape anisotropy. This justifies the use of curvature motion for skeleton computation, and leads to the generation of what we call "mean curvature skeletons". To obtain a stable and efficient discretization, we regularize the surface mesh by performing local remeshing via edge splits and collapses. Simplifying mesh connectivity throughout the motion leads to more efficient computation and avoids numerical instability arising from degeneracies in the triangulation. In addition, the detection of collapsed geometry is facilitated by working with simplified mesh connectivity and monitoring potential non-manifold edge collapses. With topology simplified throughout the flow, minimal post-processing is required to convert the collapsed geometry to a curve. Formulating skeletonization via MCF allows us to incorporate external energy terms easily, resulting in a constrained flow. We define one such energy term using the Voronoi medial skeleton and obtain a medially centred curve skeleton. We call the intermediate results of our skeletonization motion meso-skeletons; these consist of a mixture of curves and surface sheets as appropriate to the local 3D geometry they capture.