A data structure for dynamic trees
Journal of Computer and System Sciences
Maintenance of a minimum spanning forest in a dynamic plane graph
Journal of Algorithms
VIS '97 Proceedings of the 8th conference on Visualization '97
Contour trees and small seed sets for isosurface traversal
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Near-optimal fully-dynamic graph connectivity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Introduction to Algorithms
Surface Coding Based on Morse Theory
IEEE Computer Graphics and Applications
Constructing a Reeb graph automatically from cross sections
IEEE Computer Graphics and Applications
Computing contour trees in all dimensions
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Dynamic generators of topologically embedded graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Morse-smale complexes for piecewise linear 3-manifolds
Proceedings of the nineteenth annual symposium on Computational geometry
GRIN'01 No description on Graphics interface 2001
Topological quadrangulations of closed triangulated surfaces using the Reeb graph
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
Removing excess topology from isosurfaces
ACM Transactions on Graphics (TOG)
Parallel Computation of the Topology of Level Sets
Algorithmica
Augmented Reeb Graphs for Content-Based Retrieval of 3D Mesh Models
SMI '04 Proceedings of the Shape Modeling International 2004
Loops in Reeb Graphs of 2-Manifolds
Discrete & Computational Geometry
Simplifying Flexible Isosurfaces Using Local Geometric Measures
VIS '04 Proceedings of the conference on Visualization '04
Feature-based surface parameterization and texture mapping
ACM Transactions on Graphics (TOG)
Topology-Controlled Volume Rendering
IEEE Transactions on Visualization and Computer Graphics
Robust on-line computation of Reeb graphs: simplicity and speed
ACM SIGGRAPH 2007 papers
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
Automated analysis of remyelination therapy for spinal cord injury
Proceedings of the Seventh Indian Conference on Computer Vision, Graphics and Image Processing
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on Computational geometry
Graph-based representations of point clouds
Graphical Models
A deterministic o(m log m) time algorithm for the reeb graph
Proceedings of the twenty-eighth annual symposium on Computational geometry
Simplified reeb graph as effective shape descriptor for the striatum
MeshMed'12 Proceedings of the 2012 international conference on Mesh Processing in Medical Image Analysis
An efficient computation of handle and tunnel loops via Reeb graphs
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
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The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)^3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)^3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n^2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.