Robust on-line computation of Reeb graphs: simplicity and speed

Reeb graphs are a fundamental data structure for understanding and representing the topology of shapes. They are used in computer graphics, solid modeling, and visualization for applications ranging from the computation of similarities and finding defects in complex models to the automatic selection of visualization parameters. We introduce an on-line algorithm that reads a stream of elements (vertices, triangles, tetrahedra, etc.) and continuously maintains the Reeb graph of all elements already reed. The algorithm is robust in handling non-manifold meshes and general in its applicability to input models of any dimension. Optionally, we construct a skeleton-like embedding of the Reeb graph, and/or remove topological noise to reduce the output size. For interactive multi-resolution navigation we also build a hierarchical data structure which allows real-time extraction of approximated Reeb graphs containing all topological features above a given error threshold. Our extensive experiments show both high performance and practical linear scalability for meshes ranging from thousands to hundreds of millions of triangles. We apply our algorithm to the largest, most general, triangulated surfaces available to us, including 3D, 4D and 5D simplicial meshes. To demonstrate one important application we use Reeb graphs to find and highlight topological defects in meshes, including some widely believed to be "clean."