Robust on-line computation of Reeb graphs

Abstract

Peer-Timo BremerCASC -LLNL ptbremer@acm.org Ajith Mascarenhas CASC -LLNL mascarenhas1@llnl.gov Figure 1: (Top row) Simplified Reeb graphs of the Dancer, Malaysian Goddess, Happy Buddha; and David together with two close-ups showing a tiny tunnel at the base of David's leg. The pseudo-colored surfaces show the function used for computing the Reeb graph. The transparent models show the structure of the Reeb graph and its embedding. (Bottom row) The Heptoroid model and two levels of resolution for the Reeb graph of the Asian Dragon model. AbstractReeb 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 read. 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 more information about the project see: http://pascucci.org/research/topology/reeb-graph/ 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."

Reeb Graphs: Approximation and Persistence

“…Observe that the complexity of each Reeb graph R i , for i ∈ [1, n], is bounded by O(n e ). The set of Reeb graphs R i s can be computed in O(nn v ) time using the incremental algorithm from [21]. We use this algorithm as it can also maintain the image of each edge from…”

“…Indeed, r = O(n e ) and the complexity of each cycle α i is bounded by O(n v ) (by representing them as a sequence of vertices). Computing the images of all α i s takes O(rn 2 v ) = O(n e n 2 v ) time using the incremental algorithm from [21], and the independence test for these r cycles take O(rn 2 e ) = O(n 3 e ) time. Finally, there are n 2 pairs of i and j that we need to test, giving rise to O(n 2 n 3 e ) total time complexity.…”

“…, γ g k } are sorted by their birth-times. At the beginning of the k-th step, we first use the incremental algorithm from [21] to compute the Reeb graph R k+1 from R k . We next need to update G (k) to G (k+1) for R k+1 so that G (k+1) is consistent with each G (i) for i ∈ [1, k].…”

“…First, we compute the image of each e i in R k , which is necessarily a monotone path (i.e, monotonic in function values) denoted by π i = Φ k (e i ). These images can be computed in O(n v ) time using the incremental algorithm and the data structure of [21]. By our assumption of f (p) ≤ f (q) ≤ f (r), π 1 and π 2 are disjoint in their interiors, while π 3 may share subcurves with π 1 and π 2 .…”

Feature Tracking Using Reeb Graphs

“…Its arcs are formed by the remaining contours, i.e., by the family of contours that do not change topology. For more details on Reeb graphs and algorithms to construct them efficiently, we refer the reader to the description by Pascucci et al [4].…”