Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction

Efficient Visualization ofLagrangian Coherent Structures by Filtered AMR Ridge Extraction

October 2007 - IEEE Vis

Filip Sadlo, Ronald Peikert @ CGL - ETH Zurich

Efficient Visualization of LCS by filtered AMR Ridge Extraction

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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


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


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


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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ridge intersects cell edge

Initialization: Ridge-Cell Detection


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ridge cell

Initialization: Ridge-Cell Detection


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ridge cell

ridge cell neighbor

Iteration 1: Collect for Subdivision


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Iteration 1: Subdivision


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Iteration 1: Ridge-Cell Detection


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ridge cell



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ridge cell

ridge cell 2-neighbor

Iteration 1: Ridge Growing


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ridge cell



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ridge intersects cell edgeridge cell



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ridge cell



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ridge cell neighbor

ridge cell



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Iteration 2: Subdivision


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ridge cell


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ridge cell

ridge cell 2-neighbor


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ridge cell

ridge cell 2-neighborfor 1-level difference


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ridge cell


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ridge cell



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ridge cell


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ridge cell


ridge cell neighbor


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. . .

Iteration 3: …


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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


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


1.5Tmax = 0.1


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

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Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction