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A topological hierarchy for functions on triangulated surfaces
Peer-Timo Bremer1, Herbert Edelsbrunner, Bernd Hamann
1Center for Image Processing and Integrated Computing, Department of Science, University of California, Davis, CA 95616, USA. tbremer@ucdavis.edu
IEEE Transactions on Visualization and Computer Graphics
|June 27, 2008
Summary
This study introduces a novel multiresolution function representation using topology and geometry. It enables creating accurate approximations with guaranteed error bounds for 2D domains.
Area of Science:
- Computational geometry
- Topology
- Data structures
Background:
- Representing functions over 2D domains is crucial for various computational tasks.
- Existing methods may lack topological guarantees or efficient multiresolution capabilities.
Purpose of the Study:
- To develop a novel multiresolution representation for functions on two-dimensional domains.
- To integrate topological and geometric approaches for robust function approximation.
Main Methods:
- Constructing the Morse-Smale complex of the function.
- Progressively simplifying the complex's topology by cancelling critical point pairs.
- Building a hierarchical data structure based on cancellation dependencies.
Main Results:
- A hierarchical data structure supporting traversal and reconstruction.
- Topologically valid function approximations.
- Approximations that satisfy user-defined error bounds at runtime.
Conclusions:
- The proposed method effectively combines topological and geometric techniques.
- The resulting data structure provides a powerful tool for multiresolution function analysis.
- This approach offers a robust way to extract accurate function approximations.
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