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Computing Morse-Smale Complexes with Accurate Geometry.

A Gyulassy1, P Bremer, V Pascucci

  • 1SCI Institute, University of Utah, USA. jediati@sci.utah.edu

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|September 11, 2015
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Summary
This summary is machine-generated.

New algorithms improve topological data analysis by accurately computing Morse-Smale complexes. These methods ensure correct connectivity and geometry, overcoming limitations of existing techniques for scientific visualization and analysis.

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Area of Science:

  • Computational Topology
  • Scientific Visualization
  • Data Analysis

Background:

  • Topological techniques are vital for scientific data analysis and visualization.
  • Existing methods for computing structures like Morse-Smale (MS) complexes often yield incorrect connectivity and poor geometry.
  • These inaccuracies can invalidate subsequent analyses, even with increased mesh resolution.

Purpose of the Study:

  • To introduce novel algorithms for robust and accurate computation of MS complexes.
  • To address the limitations of current topological analysis methods regarding connectivity and geometry.
  • To provide algorithms that improve with increasing mesh resolution.

Main Methods:

  • A randomized algorithm to compute the discrete gradient of a scalar field, designed to converge under refinement.
  • A deterministic variant utilizing two ordered function traversals to integrate probabilities and extract accurate geometry and connectivity.
  • Extensive empirical studies on synthetic and real-world data to validate algorithm performance.

Main Results:

  • The randomized algorithm shows convergence, with average correctness increasing and standard deviation decreasing as mesh resolution rises.
  • The deterministic algorithm successfully computes accurate geometry and correct MS complex connectivity.
  • Demonstrated advantages over several popular existing approaches in empirical evaluations.

Conclusions:

  • The developed algorithms offer significant improvements for computing Morse-Smale complexes.
  • These methods provide topologically consistent results with accurate geometry and connectivity.
  • The new approaches enhance the reliability of topological data analysis in scientific applications.