An Approach to the Decomposition of Solids with Voids via Morse Theory

Abstract

The decomposition of solids is a problem of interest in areas of engineering such as feature recognition or manufacturing planning. The problem can be stated as finding a set of smaller and simpler pieces that glued together amount to the initial solid. This decomposition can be guided by geometrical or topological criteria and be applied to either surfaces or solids (embedded manifolds). Most topological decompositions rely on Morse theory to identify changes in the topology of a manifold. A Morse function f is defined on the manifold and the manifold's topology is studied by studying the behaviour of the critical points of f . A popular structure used to encode this behaviour is the Reeb graph. Reeb graph-based decompositions have proven to work well for surfaces and for solids without inner voids, but fail to consider solids with inner voids. In this work we present a methodology based on the handle-decomposition of a manifold that can encode changes in the topology of solids both with and without inner voids. Our methodology uses the Boundary Representation of the solid and a shape similarity criteria to identify changes in the topology of both the outer and inner boundary(ies) of the solid. Our methodology is defined for Morse functions that produce parallel planar level sets and we do not consider the case of annidated solids (i.e. solids within other solids). We present an algorithm to implement our methodology and execute experiments on several datasets. Future work includes the testing of the methodology with functions different to the height function and the speed up of the algorithm's data structure.