Transcript Visualization of Lagrangian Coherent Structures

Efficient Visualization of
Lagrangian Coherent
Structures by Filtered AMR
Ridge Extraction
October 2007 - IEEE Vis
Filip Sadlo, Ronald Peikert @ CGL - ETH
Zurich
Lagrangian Coherent
Structures (LCS)
FTLE
Shadden et al. 2005
Vector Field Topology
Lagr. Coherent Structures
•
•
•
•
•
•
Crit. pts. & streamlines
Instantaneous view
Fast
Efficient Visualization of LCS by filtered AMR Ridge Extraction
Ridges in Lyapunov Exponent
Transient view
Slow (trajectory per point &
time)
-> Adaptive approach
2
Lagrangian Coherent
Structures (LCS)
FTLE
Shadden et al. 2005
Vector Field Topology
Lagr. Coherent Structures
•
•
•
•
•
•
Crit. pts. & streamlines
Instantaneous view
Fast
Efficient Visualization of LCS by filtered AMR Ridge Extraction
Ridges in Lyapunov Exponent
Transient view
Slow (trajectory per point &
time)
-> Adaptive approach
3
Finite-Time Lyapunov
Exponent (FTLE)
FTLE: “growth of perturbation after advection time T”
t0  T
t0
x 

t0
t0  T
FTLE  x, t0 , T  
1
ln   /  
T
Efficient Visualization of LCS by filtered AMR Ridge Extraction
4
FTLE Computation
• Advection of particle pairs: tedious
• Haller 2001: by pre-sampled flow map f
tt  x : xt
0
0
xt
FTLE
t0=
xt
xt0
Shadden et al. 2005
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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FTLE Computation
• Advection of particle pairs: tedious
• Haller 2001: by pre-sampled flow map f
tt  x : xt
0
0
xt
FTLE
t0=
xt
xt0
Shadden et al. 2005
tt00 T  x 
Efficient Visualization of LCS by filtered AMR Ridge Extraction
6
2
A
2
 max  AT A 

FTLE Computation
• Advection of particle pairs: tedious
• Haller 2001: by pre-sampled flow map f
tt  x : xt
0
0
xt
FTLE
t0=
xt
xt0
Shadden et al. 2005
FTLEmax  x, t0 , T  

1
ln tt00 T  x 
T
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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2

A
2
 max  AT A 

FTLE Computation
• Advection of particle pairs: tedious
• Haller 2001: by pre-sampled flow map f
tt  x : xt
0
0
xt
FTLE
t0=
xt
xt0
Shadden et al. 2005
FTLEmax  x, t0 , T  

1
ln tt00 T  x 
T
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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2

A
2
 max  AT A 

LCS in Nature
from: www.publicaffairs.water.ca.gov/swp/swptoday.cfm
from: www.scienceclarified.com/Ga-He/Glacier.html
Confluences
Glaciers
•
•
•
•
Interfaces
Sacramento & Feather
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Moraines
Glacier Bay National Park
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
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Overview
Related Work
Height Ridges
Filtered AMR Ridge Extraction
Efficiency
FTLE & FSLE
Proposed: FTLEM
FTLEM & FSLE
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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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
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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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

Efficient Visualization of LCS by filtered AMR Ridge Extraction
min  0
13
min s = 0 ,
min  0
min
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
PC
A
|, | : “min s =
0”
min  0 , min  0
– We also filter crossings e.g. by FTLE
– We use Marching Cubes instead of triangulation
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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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)
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Initialization: Ridge-Cell Detection
ridge intersects cell edge
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Initialization: Ridge-Cell Detection
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Collect for Subdivision
ridge cell
ridge cell neighbor
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Subdivision
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge-Cell Detection
ridge intersects cell edge
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge-Cell Detection
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge Growing
ridge cell
ridge cell 2-neighbor
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge Growing
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge Growing
ridge intersects cell edge
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 1: Ridge Growing
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Collect for Subdivision
ridge cell
ridge cell neighbor
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Subdivision
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge-Cell Detection
ridge intersects cell edge
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge-Cell Detection
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge Growing
ridge cell
ridge cell 2-neighbor
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge Growing
ridge cell
ridge cell 2-neighbor
for  1-level difference
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge Growing
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge Growing
ridge intersects cell edge
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 2: Ridge Growing
ridge cell
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 3: Collect for Subdivision
ridge cell
ridge cell neighbor
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Iteration 3: …
...
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge
Extraction
Final Result
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge Extraction
from FTLE: Method
video
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Filtered AMR Ridge Extraction
from FTLE: Francis Turbine
video
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Efficiency
Subdivision iterations: 4
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
Speed-up: > 4
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Finite-Size Lyapunov Exponent
(FSLE), Aurell 1997
FSLE: “time needed to separate by factor s”
t0  Ts
t0
x 
s
t0
t0  Ts
FSLE  x, t0 , Ts  
1
ln  s 
Ts
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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FTLE & FSLE (Filtered)
FTLE
FSLE
FSLE
T = 0.1
Prescribed scale =
1.5
Tmax = 0.1
Prescribed scale = 4
Tmax = 0.1
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Proposed: Finite-Time
Lyapunov Exponent Maximum
(FTLEM)
FTLEM: “maximum FTLE over advection time T”
t0
x 
t0
t0 1t
t0  2t
1
2
t0 1t
t0  2t
t0  (n  1)t
…
n
n1
t0  (n  1)t
FTLEM  x, t0 , T , n   max
k 1,..., n
Efficient Visualization of LCS by filtered AMR Ridge Extraction
t0  nt
t0  nt
1
ln   k /  
k t
43
FTLEM & FSLE (Filtered)
FTLEM
Tmax = 0.1
Properties of both
FSLE
FSLE
FSLE
Prescribed scale =
1.5
Tmax = 0.1
Prescribed scale = 4
Tmax = 0.1
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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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
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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Thanks for your attention
Efficient Visualization of LCS by filtered AMR Ridge Extraction
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FTLE Ridge Filtering
No filtering
4000 tria
Efficient Visualization of LCS by filtered AMR Ridge Extraction
FTLEmin = 3.5, 4.0
47
& CCmin = 1000,