Introduction to 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
Construction of contour trees in 3D in O(n log n) steps
Proceedings of the fourteenth annual symposium on Computational geometry
Geometric Separators for Finite-Element Meshes
SIAM Journal on Scientific Computing
Computing contour trees in all dimensions
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Simplifying Flexible Isosurfaces Using Local Geometric Measures
VIS '04 Proceedings of the conference on Visualization '04
Simple and optimal output-sensitive construction of contour trees using monotone paths
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
The Feature Tree: Visualizing Feature Tracking in Distributed AMR Datasets
PVG '03 Proceedings of the 2003 IEEE Symposium on Parallel and Large-Data Visualization and Graphics
Extraction of Topologically Simple Isosurfaces from Volume Datasets
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
Volume Tracking Using Higher Dimensional Isosurfacing
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
IEEE Transactions on Visualization and Computer Graphics
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Families of cut-graphs for bordered meshes with arbitrary genus
Graphical Models
Reeb graphs for shape analysis and applications
Theoretical Computer Science
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree
Computational Geometry: Theory and Applications
Simple and optimal output-sensitive construction of contour trees using monotone paths
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Defining, contouring, and visualizing scalar functions on point-sampled surfaces
Computer-Aided Design
Adjustment for discrepancies between ALS data strips using a contour tree algorithm
ACIVS'06 Proceedings of the 8th international conference on Advanced Concepts For Intelligent Vision Systems
Description of digital images by region-based contour trees
ICIAR'05 Proceedings of the Second international conference on Image Analysis and Recognition
Tessellation of quadratic elements
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Efficient isosurface tracking using precomputed correspondence table
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
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This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to preprocess the domain mesh to allow optimal computation of isosurfaces with minimal overhead storage. The Contour Tree can also be used to build user interfaces reporting the complete topological characterization of a scalar field, as shown in Figure 1.The first part of the paper presents a new scheme that augments the Contour Tree with the Betti numbers of each isocontour in linear time. We show how to extend the scheme introduced in [3] with the Betti number computation without increasing its complexity. Thus, we improve on the time complexity from our previous approach [10] from O(m log m) to O(n log n + m), where m is the number of tetrahedra and n is the number of vertices in the domain of F.The second part of the paper introduces a new divide-and-conquer algorithm that computes the Augmented Contour Tree with improved efficiency. The central part of the scheme computes the output Contour Tree by merging two intermediate Contour Trees and is independent of the interpolant. In this way we confine any knowledge regarding a specific interpolant to an oracle that computes the tree for a single cell. We have implemented this oracle for the trilinear interpolant and plan to replace it with higher order interpolants when needed. The complexity of the scheme is O(n + t log n), where t is the number of critical points of F. For the first time we can compute the Contour Tree in linear time in many practical cases when t = O(n1 - ε).Lastly, we report the running times for a parallel implementation of our algorithm, showing good scalability with the number of processors.