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 @ CGL - ETH Zurich
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Lagrangian Coherent Structures (LCS)
Vector Field Topology• Crit. pts. & streamlines• Instantaneous view• Fast
Lagr. Coherent Structures• Ridges in Lyapunov Exponent• Transient view• Slow (trajectory per point &
time)-> Adaptive approach
Shadden et al. 2005
FTLE
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Lagrangian Coherent Structures (LCS)
Vector Field Topology• Crit. pts. & streamlines• Instantaneous view• Fast
Lagr. Coherent Structures• Ridges in Lyapunov Exponent• Transient view• Slow (trajectory per point &
time)-> Adaptive approach
Shadden et al. 2005
FTLE
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Finite-Time Lyapunov Exponent (FTLE)
FTLE: “growth of perturbation after advection time T”
0
1, , ln /FTLE t T
T x
0t T
0t
x0t
0t T
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FTLE Computation
• Advection of particle pairs: tedious• Haller 2001: by pre-sampled flow map
0 0
:tt t t x x x
0tx tx
Shadden et al. 2005
t0=FTLE
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FTLE Computation
• Advection of particle pairs: tedious• Haller 2001: by pre-sampled flow map
0
0 2 t Tt x
0 0
:tt t t x x x
0tx tx
Shadden et al. 2005
t0=FTLE
max2
TA A A
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FTLE Computation
• Advection of particle pairs: tedious• Haller 2001: by pre-sampled flow map
0
0max 0 2
1, , ln t T
tFTLE t TT
x x
0 0
:tt t t x x x
max2
TA A A
0tx tx
Shadden et al. 2005
t0=FTLE
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FTLE Computation
• Advection of particle pairs: tedious• Haller 2001: by pre-sampled flow map
0
0max 0 2
1, , ln t T
tFTLE t TT
x x
0 0
:tt t t x x x
max2
TA A A
0tx tx
Shadden et al. 2005
t0=FTLE
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LCS in Nature
Confluences• Interfaces• Sacramento & Feather
Glaciers• Moraines• Glacier Bay National Park
from: www.scienceclarified.com/Ga-He/Glacier.htmlfrom: www.publicaffairs.water.ca.gov/swp/swptoday.cfm
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Moraines and LCS
“Appearing as dark lines on the surface, moraines indicate how many smaller glaciers feed into the system”
-> LCS, dynamical systems
from: www.fs.fed.us/r10/tongass/forest_facts/resources/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
min
min s = 0 , min 0
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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
Height Ridges
|, | : “min s = 0”
PCA
min 0 , min 0
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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
ridge intersects cell edge
Initialization: Ridge-Cell Detection
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Filtered AMR Ridge Extraction
ridge cell
Initialization: Ridge-Cell Detection
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Filtered AMR Ridge Extraction
ridge cell
ridge cell neighbor
Iteration 1: Collect for Subdivision
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Filtered AMR Ridge Extraction
Iteration 1: Subdivision
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Filtered AMR Ridge Extraction
ridge intersects cell edge
Iteration 1: Ridge-Cell Detection
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Filtered AMR Ridge Extraction
ridge cell
Iteration 1: Ridge-Cell Detection
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Filtered AMR Ridge Extraction
ridge cell
ridge cell 2-neighbor
Iteration 1: Ridge Growing
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Filtered AMR Ridge Extraction
ridge cell
Iteration 1: Ridge Growing
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Filtered AMR Ridge Extraction
ridge intersects cell edgeridge cell
Iteration 1: Ridge Growing
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Filtered AMR Ridge Extraction
ridge cell
Iteration 1: Ridge Growing
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Filtered AMR Ridge Extraction
ridge cell neighbor
ridge cell
Iteration 2: Collect for Subdivision
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Filtered AMR Ridge Extraction
Iteration 2: Subdivision
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Filtered AMR Ridge Extraction
ridge intersects cell edge
Iteration 2: Ridge-Cell Detection
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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-neighborfor 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 cell
ridge intersects cell edge
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Filtered AMR Ridge Extraction
Iteration 2: Ridge Growing
ridge cell
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Filtered AMR Ridge Extraction
ridge cell
Iteration 3: Collect for Subdivision
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
direct adaptive
initial grid 3,613,153 nodes
1,183 nodes
final grid 3,613,153 nodes
298,964 nodes
flow map [s] 19,953.51 2,350.21
FTLE [s] 10.73 30.73
ridge extr. [s] 278.46 2,337.16
total [s] 20,242.74 4,930.72
Subdivision iterations: 4
Speed-up: > 4
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Finite-Size Lyapunov Exponent (FSLE), Aurell
1997FSLE: “time needed to separate by factor s”
0
1, , lns
s
FSLE t T sT
x
sx
0 st T
0t
0t0 st T
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FTLE & FSLE (Filtered)
FTLET = 0.1
FSLEPrescribed scale =
1.5Tmax = 0.1
FSLEPrescribed scale =
4Tmax = 0.1
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Proposed: Finite-Time Lyapunov Exponent Maximum (FTLEM)
FTLEM: “maximum FTLE over advection time T”
01,...,
1, , , max ln /
kk n
FTLEM t T nk t
x
nx
0 t n t
0t
0t0 t n t
0 1 t t
0 1 t t1
0 2 t t
0 2 t t
2 …
0 ( 1) t n t
1n
0 ( 1) t n t
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FTLEM & FSLE (Filtered)
FTLEM
Tmax = 0.1
Properties of both FSLE
FSLEPrescribed scale =
1.5Tmax = 0.1
FSLEPrescribed scale =
4Tmax = 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