1. Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction October 2007 - IEEE Vis Filip Sadlo, Ronald Peikert @ CGL - ETH Zurich

2. Efficient Visualization of LCS by filtered AMR Ridge Extraction 2 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

3. Efficient Visualization of LCS by filtered AMR Ridge Extraction 3 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

4. Efficient Visualization of LCS by filtered AMR Ridge Extraction 4 Finite-Time Lyapunov Exponent (FTLE) FTLE: “growth of perturbation after advection time T”

5. Efficient Visualization of LCS by filtered AMR Ridge Extraction 5 FTLE Computation Advection of particle pairs: tedious Haller 2001: by pre-sampled flow map  Shadden et al. 2005 t0=t0= FTLE

6. Efficient Visualization of LCS by filtered AMR Ridge Extraction 6 FTLE Computation Advection of particle pairs: tedious Haller 2001: by pre-sampled flow map  Shadden et al. 2005 t0=t0= FTLE

7. Efficient Visualization of LCS by filtered AMR Ridge Extraction 7 FTLE Computation Advection of particle pairs: tedious Haller 2001: by pre-sampled flow map  Shadden et al. 2005 t0=t0= FTLE

8. Efficient Visualization of LCS by filtered AMR Ridge Extraction 8 FTLE Computation Advection of particle pairs: tedious Haller 2001: by pre-sampled flow map  Shadden et al. 2005 t0=t0= FTLE

9. Efficient Visualization of LCS by filtered AMR Ridge Extraction 9 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

10. Efficient Visualization of LCS by filtered AMR Ridge Extraction 10 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

11. Efficient Visualization of LCS by filtered AMR Ridge Extraction 11 Overview Related Work Height Ridges Filtered AMR Ridge Extraction Efficiency FTLE & FSLE Proposed: FTLEM FTLEM & FSLE

12. Efficient Visualization of LCS by filtered AMR Ridge Extraction 12 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

13. Efficient Visualization of LCS by filtered AMR Ridge Extraction 13 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

14. Efficient Visualization of LCS by filtered AMR Ridge Extraction 14 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” PC A min  0, min  0

15. Efficient Visualization of LCS by filtered AMR Ridge Extraction 15 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)

16. Efficient Visualization of LCS by filtered AMR Ridge Extraction 16 Filtered AMR Ridge Extraction ridge intersects cell edge Initialization: Ridge-Cell Detection

17. Efficient Visualization of LCS by filtered AMR Ridge Extraction 17 Filtered AMR Ridge Extraction ridge cell Initialization: Ridge-Cell Detection

18. Efficient Visualization of LCS by filtered AMR Ridge Extraction 18 Filtered AMR Ridge Extraction ridge cell ridge cell neighbor Iteration 1: Collect for Subdivision

19. Efficient Visualization of LCS by filtered AMR Ridge Extraction 19 Filtered AMR Ridge Extraction Iteration 1: Subdivision

20. Efficient Visualization of LCS by filtered AMR Ridge Extraction 20 Filtered AMR Ridge Extraction ridge intersects cell edge Iteration 1: Ridge-Cell Detection

21. Efficient Visualization of LCS by filtered AMR Ridge Extraction 21 Filtered AMR Ridge Extraction ridge cell Iteration 1: Ridge-Cell Detection

22. Efficient Visualization of LCS by filtered AMR Ridge Extraction 22 Filtered AMR Ridge Extraction ridge cell ridge cell 2-neighbor Iteration 1: Ridge Growing

23. Efficient Visualization of LCS by filtered AMR Ridge Extraction 23 Filtered AMR Ridge Extraction ridge cell Iteration 1: Ridge Growing

24. Efficient Visualization of LCS by filtered AMR Ridge Extraction 24 Filtered AMR Ridge Extraction ridge intersects cell edge ridge cell Iteration 1: Ridge Growing

25. Efficient Visualization of LCS by filtered AMR Ridge Extraction 25 Filtered AMR Ridge Extraction ridge cell Iteration 1: Ridge Growing

26. Efficient Visualization of LCS by filtered AMR Ridge Extraction 26 Filtered AMR Ridge Extraction ridge cell neighbor ridge cell Iteration 2: Collect for Subdivision

27. Efficient Visualization of LCS by filtered AMR Ridge Extraction 27 Filtered AMR Ridge Extraction Iteration 2: Subdivision

28. Efficient Visualization of LCS by filtered AMR Ridge Extraction 28 Filtered AMR Ridge Extraction ridge intersects cell edge Iteration 2: Ridge-Cell Detection

29. Efficient Visualization of LCS by filtered AMR Ridge Extraction 29 Filtered AMR Ridge Extraction Iteration 2: Ridge-Cell Detection ridge cell

30. Efficient Visualization of LCS by filtered AMR Ridge Extraction 30 Filtered AMR Ridge Extraction Iteration 2: Ridge Growing ridge cell ridge cell 2-neighbor

31. Efficient Visualization of LCS by filtered AMR Ridge Extraction 31 Filtered AMR Ridge Extraction Iteration 2: Ridge Growing ridge cell ridge cell 2-neighbor for  1-level difference

32. Efficient Visualization of LCS by filtered AMR Ridge Extraction 32 Filtered AMR Ridge Extraction Iteration 2: Ridge Growing ridge cell

33. Efficient Visualization of LCS by filtered AMR Ridge Extraction 33 Filtered AMR Ridge Extraction Iteration 2: Ridge Growing ridge cell ridge intersects cell edge

34. Efficient Visualization of LCS by filtered AMR Ridge Extraction 34 Filtered AMR Ridge Extraction Iteration 2: Ridge Growing ridge cell

35. Efficient Visualization of LCS by filtered AMR Ridge Extraction 35 Filtered AMR Ridge Extraction ridge cell Iteration 3: Collect for Subdivision ridge cell neighbor

36. Efficient Visualization of LCS by filtered AMR Ridge Extraction 36 Filtered AMR Ridge Extraction... Iteration 3: …

37. Efficient Visualization of LCS by filtered AMR Ridge Extraction 37 Filtered AMR Ridge Extraction Final Result

38. Efficient Visualization of LCS by filtered AMR Ridge Extraction 38 Filtered AMR Ridge Extraction from FTLE: Method video

39. Efficient Visualization of LCS by filtered AMR Ridge Extraction 39 Filtered AMR Ridge Extraction from FTLE: Francis Turbine video

40. Efficient Visualization of LCS by filtered AMR Ridge Extraction 40 Efficiency directadaptive initial grid3,613,153 nodes1,183 nodes final grid3,613,153 nodes298,964 nodes flow map [s]19,953.512,350.21 FTLE [s]10.7330.73 ridge extr. [s]278.462,337.16 total [s]20,242.744,930.72 Subdivision iterations: 4 Speed-up: > 4

41. Efficient Visualization of LCS by filtered AMR Ridge Extraction 41 Finite-Size Lyapunov Exponent (FSLE), Aurell 1997 FSLE: “time needed to separate by factor s”

42. Efficient Visualization of LCS by filtered AMR Ridge Extraction 42 FTLE & FSLE (Filtered) FTLE T = 0.1 FSLE Prescribed scale = 1.5 T max = 0.1 FSLE Prescribed scale = 4 T max = 0.1

43. Efficient Visualization of LCS by filtered AMR Ridge Extraction 43 Proposed: Finite-Time Lyapunov Exponent Maximum (FTLEM) FTLEM: “maximum FTLE over advection time T” …

44. Efficient Visualization of LCS by filtered AMR Ridge Extraction 44 FTLEM & FSLE (Filtered) FTLEM T max = 0.1 Properties of both FSLE FSLE Prescribed scale = 1.5 T max = 0.1 FSLE Prescribed scale = 4 T max = 0.1

45. Efficient Visualization of LCS by filtered AMR Ridge Extraction 45 Conclusion Efficient method for ridge extraction Applied to FTLE, FSLE and FTLEM FTLEM as a new FTLE variant Future Work –Exploit temporal coherency

46. Efficient Visualization of LCS by filtered AMR Ridge Extraction 46 Thanks for your attention

47. Efficient Visualization of LCS by filtered AMR Ridge Extraction 47 FTLE Ridge Filtering No filtering FTLE min = 3.5, 4.0 & CC min = 1000, 4000 tria