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Robust Morse decompositions of piecewise constant vector fields.

Andrzej Szymczak¹, Eugene Zhang

¹Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401-1887, USA. aszymcza@mines.edu

IEEE Transactions on Visualization and Computer Graphics

|July 13, 2011

Summary

This study presents a novel, efficient method for computing Morse decompositions of vector fields on surfaces. The approach, utilizing piecewise constant vector fields, achieves subtriangle precision and significantly faster computation times.

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Area of Science:

Computational geometry
Differential geometry
Computer graphics

Background:

Morse decomposition is crucial for analyzing vector fields on surfaces.
Existing methods for computing Morse decomposition can be computationally intensive and lack precision.
Handling discontinuities in piecewise constant vector fields presents a significant challenge.

Purpose of the Study:

To introduce a novel and efficient algorithm for computing Morse decomposition of vector fields on triangulated manifold surfaces.
To achieve subtriangle precision for Morse sets.
To develop a framework for classifying and visualizing Morse sets.

Main Methods:

Conversion of the input vector field to a piecewise constant (PC) vector field.
Application of differential inclusion theory to handle discontinuities in PC vector fields.
Development of a robust algorithm for computing Morse decompositions of PC vector fields.

Main Results:

The proposed method produces Morse decompositions similar or finer than existing techniques.
The algorithm is over an order of magnitude faster than current methods.
Subtriangle precision for Morse sets is achieved.
A classification framework for enhanced visualization of Morse sets is introduced.

Conclusions:

The new approach offers a significant speedup and improved precision for Morse decomposition computation.
The method is robust and applicable to various simulation datasets.
The visualization framework enhances the understanding of vector field topology.