Constructing a Reeb graph automatically from cross sections
IEEE Computer Graphics and Applications
Loops in Reeb Graphs of 2-Manifolds
Discrete & Computational Geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics)
Robust on-line computation of Reeb graphs: simplicity and speed
ACM SIGGRAPH 2007 papers
Reeb graphs for shape analysis and applications
Theoretical Computer Science
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Time-varying Reeb graphs for continuous space--time data
Computational Geometry: Theory and Applications
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
Efficient Output-Sensitive Construction of Reeb Graphs
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Efficient algorithms for computing Reeb graphs
Computational Geometry: Theory and Applications
Cut locus and topology from surface point data
Proceedings of the twenty-fifth annual symposium on Computational geometry
Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees
IEEE Transactions on Visualization and Computer Graphics
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
A deterministic o(m log m) time algorithm for the reeb graph
Proceedings of the twenty-eighth annual symposium on Computational geometry
The hitchhiker's guide to the galaxy of mathematical tools for shape analysis
ACM SIGGRAPH 2012 Courses
Feature-Preserving Reconstruction of Singular Surfaces
Computer Graphics Forum
Graph induced complex on point data
Proceedings of the twenty-ninth annual symposium on Computational geometry
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Given a continuous function f:X - S on a topological space X, its level set f-1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H1-homology of the Reeb graph from P. It takes O(n log n) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm to compute the rank of H1(MM) from point data. The best known previous algorithm for this problem takes O(n3) time for point data. The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H1-homology for the Reeb graphs in O(n ne3) time, where n is the size of the 2-skeleton and ne is the number of edges in K.