Time-varying reeb graphs for continuous space-time data
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
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
Topological Landscapes: A Terrain Metaphor for Scientific Data
IEEE Transactions on Visualization and Computer Graphics
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Time-varying Reeb graphs for continuous space--time data
Computational Geometry: Theory and Applications
Efficient algorithms for computing Reeb graphs
Computational Geometry: Theory and Applications
3D Edge Detection by Selection of Level Surface Patches
Journal of Mathematical Imaging and Vision
Separatrix persistence: extraction of salient edges on surfaces using topological methods
SGP '09 Proceedings of the Symposium on Geometry Processing
Defining, contouring, and visualizing scalar functions on point-sampled surfaces
Computer-Aided Design
Flexible and topologically localized segmentation
EUROVIS'07 Proceedings of the 9th Joint Eurographics / IEEE VGTC conference on Visualization
Combining in-situ and in-transit processing to enable extreme-scale scientific analysis
SC '12 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Proceedings of the 18th ACM SIGPLAN symposium on Principles and practice of parallel programming
Exploring power behaviors and trade-offs of in-situ data analytics
SC '13 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
| Hi-index | 0.00 |
This paper introduces two efficient algorithms that compute the Contour Tree of a three-dimensional 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 pre-process 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~\ref{fig:top}. Data exploration time is reduced since the user understands the evolution of level set components with changing isovalue. The Augmented Contour Tree provides even more accurate information segmenting the range space of the scalar field into regions of invariant topology. The exploration time for a single isosurface is also improved since its genus is known in advance.Our first new algorithm augments any given Contour Tree with the Betti numbers of all possible corresponding isocontours in linear time with the size of the tree. Moreover, 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 in our previous approach from O(m log m) to O(n log n + m), where m is the number of cells and n is the number of vertices in the domain of F.Our second contribution is a new divide-and-conquer algorithm that computes the Augmented Contour Tree with improved efficiency. The approach 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 independent function that computes the tree for a single cell. We have implemented this function for the trilinear interpolant and plan to replace it with higher-order interpolants when needed. The time complexity 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 where t = O(n1–ε). We report the running times for a parallel implementation, showing good scalability with the number of processors.