Synthesis of Reeb Graph and Morse Operators from Level Sets of a Boundary Representation

Abstract

In the context of Industrie 4.0, it is necessary for several applications, to encode characteristics of a Boundary Representation of a manifold M in an economical manner. Two related characterizations of closed B-Reps (and the solid they represent) are (1) medial axis and (2) Reeb Graph. The medial axis of a solid region is a non-manifold mixture of 1-simplices and 2- simplices and it is expensive to extract. Because of this reason, this manuscript concentrates in the work-flow necessary to extract the Reeb Graph of the B-Rep. The extraction relies on (a) tests of geometric similarities among slices of M and (b) characterization of the topological transitions in the slice sequence of M. The process roughly includes: (1) tilt of the B-Rep to obtain an unambiguous representation of the level sets ofM,(2) identification and classification of the topological transitions that arise between consecutive level sets, (3) sample of Reeb graph vertices inside the material regions defined by the level sets, (4) creation of Reeb graph edges based on the type of topological transition and the 2D similarity among material regions of consecutive levels. Although the Reeb Graph is a topological construct, geometrical processing is central in its synthesis and compliance with the Nyquist-Shannon sampling interval is crucial for its construction. Future work is needed on the extension of our methodology to account for manifolds with internal voids or nested solids.