Efficient Morse Decompositions of Vector Fields

1
Efficient Morse Decompositions of Vector Fields

• Guoning Chen and Eugene Zhang _at_ Oregon State
  University
• Konstantin Mischaikow _at_ Rutgers University
• Robert S. Laramee _at_University of Wales Swansea

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Outline

• Introduction
• Efficient Morse decomposition
• Results
• Conclusion

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Outline

• Introduction
• Efficient Morse decomposition
• Results
• Conclusion

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Introduction

• Many applications involve vector fields
• Mathematics
• Dynamical systems
• Physics
• Fluid mechanics
• Electromagnetism
• Engineering
• Weather prediction
• Tsunami and hurricane modeling
• Engineering design and testing
• Others

Airfoil data from Jim Liburdy and Dan Morse
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Introduction

• Vector field topology analysis helps to identify
  the important dynamics
• Traditional vector field topology on surfaces
  (Vector field topological skeleton, ECG)
  includes
• Fixed points nodes (source/sink) and saddles
• Periodic orbits
• Connectivity
• Graph representation (ECG)

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Introduction

• Related work
• Vector field topology analysis
• Fixed point (singularity) extraction
• Helmann and Hesselink 1989,1991
• Tricoche et al. 2001, Polthier and Preuß 2003
• Scheuermann et al. 1997, 1998
• Periodic orbit extraction
• Wischgoll and Scheuermann 2001, 2002
• Theisel et al. 2004
• Chen et al. 2007

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Introduction

• computational instability of ECG

Case 1 different mesh configurations
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Introduction

• computational instability of ECG

Case 2 noise in the data
9
Introduction

• computational instability of ECG

Case 3 different numerical integration schemes
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Introduction

• More stable interpretation is desired

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Introduction

• More stable interpretation is desired

12
Introduction

• Previous method Chen 07 produces Morse
  decompositions that are typically too coarse.

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Introduction

• Contributions
• Demonstrate the instability of vector field
  topology defined in terms of individual
  trajectories
• Propose Morse decomposition as the foundation for
  vector field topology
• Propose a means to obtain finer Morse
  decompositions efficiently

14
Outline

• Introduction
• Efficient Morse decomposition
• Results
• Conclusion

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Efficient Morse Decomposition

• Definition Morse decomposition
• A Morse decomposition of surface X for the flow
  is a finite collection of disjoint compact
  invariant sets, called Morse sets.
• The triangular regions containing the Morse sets
  are referred to as Morse neighborhoods.
• Morse set connection graph (MCG)

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Efficient Morse Decomposition

• A pipeline of Morse decomposition

Flow combinatorialization
Strongly connected component extracting
Constructing a quotient graph
Computing MCG
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Efficient Morse Decomposition

• A pipeline of Morse decomposition

Flow combinatorialization
Strongly connected component extracting
Constructing a quotient graph
Computing MCG
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Efficient Morse Decomposition

• Geometry based Flow Combinatorialization

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Efficient Morse Decomposition

• We propose map based method

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Efficient Morse Decomposition
21
Outline

• Introduction
• Efficient Morse decomposition
• Results
• Conclusion

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Results

• Analytic data 1

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Results

• Engine Simulation data sets
• Gas engine

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Results

• Diesel engine

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Results

• Performance

Dataset name Number of triangles Number of Morse sets Time for flow combinatorialization (seconds) Time for computing MCG (seconds)
Gas engine (t0.1) 26,298 50 27.8 7.9
Gas engine ( t0.3) 26,298 57 75.4 1.2
Diesel engine (t0.3) 221,574 200 1101.3 37.7
All the results are obtained in a 3.6 GHz PC with
3GB RAM.
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Outline

• Introduction
• Efficient Morse decomposition
• Results
• Conclusion

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Conclusion

• Summary
• Demonstrate the instability of vector field
  topology defined in terms of individual
  trajectories
• Propose Morse decomposition as the foundation for
  vector field topology
• Propose a means to obtain finer Morse
  decompositions efficiently

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Conclusion

• Future work
• Extend to time-varying data analysis
• Improve the performance

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Thank you!

• Questions?

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Efficient Morse Decomposition

• Geometry-based flow combinatorialization

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Efficient Morse Decomposition

• Regions of isolation

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Efficient Morse Decomposition

• Regions of isolation

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Efficient Morse Decomposition

• Regions of recurrence

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Efficient Morse Decomposition

• Regions of recurrence

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Efficient Morse Decomposition

• Regions of recurrence

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Efficient Morse Decomposition

• Regions of recurrence

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Efficient Morse Decomposition

• Regions of recurrence

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Efficient Morse decomposition

• The stability of Morse decomposition

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Implementation of Flow Combinatorialization

• Explicit Outer Approximation Computation
• Rigorous
• Computationally expensive
• Boundary approximation
• Inaccurate with large time interval
• Hybrid method
• Not suitable for surfaces

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Implementation of Flow Combinatorialization

• A forward backward mapping framework with an
  adaptive edge sampling for computing the
  sufficient outer approximations
• The basic observation
• The image of a simple object under a continuous
  map is still simple.

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Implementation of Flow Combinatorialization

• An adaptive edge sampling

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Implementation of Flow Combinatorialization
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Implementation of Flow Combinatorialization

• Trace each vertex v of a triangle T forward for
  the time t.
• If it falls in triangle Ti, add the edges
  from the triangles of the one-ring neighbors of v
  to Ti in the directed graph.
• 2) Trace each vertex v of T backward with t.
• If it falls in triangle Ti, add the edges from
  Ti to the one-ring neighbors of v.
• Compute the image of each edge following the
  original flow and inversed flow, respectively.
  The adaptive edge sampling algorithm is employed
  to guarantee a simple outer approximation. The
  directed edges are added accordingly during the
  process.

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Implementation of Flow Combinatorialization

• The exploration of spatial vs. temporal

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Results

• Analytic data 2

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Results

• Engine Simulation data sets
• Gas engine

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Results

• Engine Simulation data sets
• Gas engine

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Results

• Diesel engine

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Results

• Performance

Dataset name Number of triangles Number of Morse sets Time for flow combinatorialization (seconds) Time for computing MCG (seconds)
Gas engine (temporal t0.1) 26,298 50 27.8 7.9
Gas engine (temporal t0.3 26,298 57 75.4 1.2
Diesel engine (temporal t0.3) 221,574 200 1101.3 37.7
Diesel engine (spatial ts0.08) 221,574 201 689.4 43.2
All the results are obtained in a 3.6 Hz PC with
3GB RAM.