Reeb graph computation based on a minimal contouring

Abstract

Given a manifold surface M and a continuous function f : M rarr R, the Reeb graph of (M, f) is a widely-used high-level descriptor of M and its usefulness has been demonstrated for a variety of applications, which range from shape parameterization and abstraction to deformation and comparison. In this context, we propose a novel computation of the Reeb graph that is based on the analysis of the iso-contours solely at saddle points and does not require sampling or sweeping the image of f. Furthermore, the proposed approach does not use global sorting steps of the function values and exploits only a local information on f, without handling it as a whole. By combining the minimal number of nodes in the Reeb graph with the use of a small amount of memory footprint and temporary data structures, the overall computation takes O(sn)-time, where n is the number of vertices of the triangulation of M and s is the number of saddles of f. Finally, the technique can be easily extended to compute the Reeb graphs of time-varying functions.