Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction

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Efficient Visualization ofLagrangian Coherent
Structures by Filtered AMR Ridge Extraction

• October 2007 - IEEE Vis
• Filip Sadlo, Ronald Peikert _at_ CGL - ETH Zurich

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Lagrangian Coherent Structures (LCS)
FTLE
Shadden et al. 2005

• Vector Field Topology
• Crit. pts. streamlines
• Instantaneous view
• Fast

• Lagr. Coherent Structures
• Ridges in Lyapunov Exponent
• Transient view
• Slow (trajectory per point time)
• -gt Adaptive approach

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Lagrangian Coherent Structures (LCS)
FTLE
Shadden et al. 2005

• Vector Field Topology
• Crit. pts. streamlines
• Instantaneous view
• Fast

• Lagr. Coherent Structures
• Ridges in Lyapunov Exponent
• Transient view
• Slow (trajectory per point time)
• -gt Adaptive approach

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Finite-Time Lyapunov Exponent (FTLE)

• FTLE growth of perturbation after advection
  time T

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FTLE Computation

• Advection of particle pairs tedious
• Haller 2001 by pre-sampled flow map f

t0
FTLE
Shadden et al. 2005
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FTLE Computation

• Advection of particle pairs tedious
• Haller 2001 by pre-sampled flow map f

t0
FTLE
Shadden et al. 2005
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FTLE Computation

• Advection of particle pairs tedious
• Haller 2001 by pre-sampled flow map f

t0
FTLE
Shadden et al. 2005
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FTLE Computation

• Advection of particle pairs tedious
• Haller 2001 by pre-sampled flow map f

t0
FTLE
Shadden et al. 2005
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LCS in Nature
from www.scienceclarified.com/Ga-He/Glacier.html
from www.publicaffairs.water.ca.gov/swp/swptoday.
cfm

• Confluences
• Interfaces
• Sacramento Feather

• Glaciers
• Moraines
• Glacier Bay National Park

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Moraines and LCS

• Appearing as dark lines on the surface,
  moraines indicate how many smaller glaciers feed
  into the system
• -gt LCS, dynamical systems

from www.fs.fed.us/r10/tongass/forest_facts/resou
rces/geology/icefields.htm
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Overview

• Related Work
• Height Ridges
• Filtered AMR Ridge Extraction
• Efficiency
• FTLE FSLE
• Proposed FTLEM
• FTLEM FSLE

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Related Work

• Ridge Extraction
• Eberly 1996 Ridges in Image and Data Analysis
  (nD)
• Furst et al. 2001 Marching Ridges (2D)
• Sahner et al. 2005 Streamlines in Feature Flow
  Field (1D)
• LCS
• Hussain 1986 Based on vorticity (3D)
• Robinson 1991 Based on correlation (3D)
• Haller 2001 Ridges in FTLE, material surfaces
  (2D)
• FTLE
• Lorenz 1965 Measures predictability
• Haller 2001 Based on pre-sampled flow map
• Path Line Oriented Topology
• Theisel et al. 2004 Based on geometry of path
  lines
• Shi et al. 2006 Same for periodic fields

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Height Ridges

• Eberly 1996
• s scalar field
• ?min min. eigenvalue of Hessian (s)
• ?min eigenvector for ?min (?min?? ridge)
• 2D height ridge in 3-space
• ?min?? ?s 0 ? ? ?min ? 0

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Height Ridges

• Furst et al. 2001 Marching Ridges
• Orientate ?min at nodes of cell by PCA
• Evaluate ?min?? ?s at nodes
• Interpolate zero crossings on edges
• Use zero crossings with ?min ? 0
• Triangulate crossings
• We also filter crossings e.g. by FTLE
• We use Marching Cubes instead of triangulation

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Filtered AMR Ridge Extraction Motivation

• Avoid sampling
• in regions with no ridges (after filtering)
• Advantages
• if only few ridges are present in given data
• if data can be sampled at arbitrary locations
• if cost of sampling is high
• Accuracy
• Obtained ridges identical to those from uniform
  sampling
• Rarely small or faint ridges may get missed (see
  paper)

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Filtered AMR Ridge Extraction
Initialization Ridge-Cell Detection
ridge intersects cell edge
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Filtered AMR Ridge Extraction
Initialization Ridge-Cell Detection
ridge cell
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Filtered AMR Ridge Extraction
Iteration 1 Collect for Subdivision
ridge cell
ridge cell neighbor
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Filtered AMR Ridge Extraction
Iteration 1 Subdivision
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Filtered AMR Ridge Extraction
Iteration 1 Ridge-Cell Detection
ridge intersects cell edge
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Filtered AMR Ridge Extraction
Iteration 1 Ridge-Cell Detection
ridge cell
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Filtered AMR Ridge Extraction
Iteration 1 Ridge Growing
ridge cell
ridge cell 2-neighbor
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Filtered AMR Ridge Extraction
Iteration 1 Ridge Growing
ridge cell
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Filtered AMR Ridge Extraction
Iteration 1 Ridge Growing
ridge intersects cell edge
ridge cell
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Filtered AMR Ridge Extraction
Iteration 1 Ridge Growing
ridge cell
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Filtered AMR Ridge Extraction
Iteration 2 Collect for Subdivision
ridge cell
ridge cell neighbor
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Filtered AMR Ridge Extraction
Iteration 2 Subdivision
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Filtered AMR Ridge Extraction
Iteration 2 Ridge-Cell Detection
ridge intersects cell edge
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Filtered AMR Ridge Extraction
Iteration 2 Ridge-Cell Detection
ridge cell
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Filtered AMR Ridge Extraction
Iteration 2 Ridge Growing
ridge cell
ridge cell 2-neighbor
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Filtered AMR Ridge Extraction
Iteration 2 Ridge Growing
ridge cell
ridge cell 2-neighbor
for ? 1-level difference
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Filtered AMR Ridge Extraction
Iteration 2 Ridge Growing
ridge cell
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Filtered AMR Ridge Extraction
Iteration 2 Ridge Growing
ridge intersects cell edge
ridge cell
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Filtered AMR Ridge Extraction
Iteration 2 Ridge Growing
ridge cell
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Filtered AMR Ridge Extraction
Iteration 3 Collect for Subdivision
ridge cell
ridge cell neighbor
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Filtered AMR Ridge Extraction
Iteration 3

• . . .

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Filtered AMR Ridge Extraction
Final Result
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Filtered AMR Ridge Extraction from FTLE Method

• video

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Filtered AMR Ridge Extraction from FTLE Francis
Turbine

• video

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Efficiency
Subdivision iterations 4
Speed-up gt 4
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Finite-Size Lyapunov Exponent (FSLE), Aurell 1997

• FSLE time needed to separate by factor s

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FTLE FSLE (Filtered)

• FTLE
• T 0.1

FSLE Prescribed scale 1.5 Tmax 0.1
FSLE Prescribed scale 4 Tmax 0.1
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Proposed Finite-Time Lyapunov Exponent Maximum
(FTLEM)

• FTLEM maximum FTLE over advection time T

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FTLEM FSLE (Filtered)

• FTLEM
• Tmax 0.1
• Properties of both FSLE

FSLE Prescribed scale 1.5 Tmax 0.1
FSLE Prescribed scale 4 Tmax 0.1
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Conclusion

• Efficient method for ridge extraction
• Applied to FTLE, FSLE and FTLEM
• FTLEM as a new FTLE variant
• Future Work
• Exploit temporal coherency

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• Thanks for your attention

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FTLE Ridge Filtering

• No filtering
  FTLEmin 3.5, 4.0 CCmin 1000, 4000
  tria