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Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction

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 Crit. pts. & streamlines

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Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction

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

  2. 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) -> Adaptive approach Efficient Visualization of LCS by filtered AMR Ridge Extraction

  3. 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) -> Adaptive approach Efficient Visualization of LCS by filtered AMR Ridge Extraction

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

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

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

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

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

  9. LCS in Nature from: www.publicaffairs.water.ca.gov/swp/swptoday.cfm from: www.scienceclarified.com/Ga-He/Glacier.html Confluences • Interfaces • Sacramento & Feather Glaciers • Moraines • Glacier Bay National Park 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 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 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 Efficient Visualization of LCS by filtered AMR Ridge Extraction

  13. min s = 0 , min  0 min 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 Efficient Visualization of LCS by filtered AMR Ridge Extraction

  14. PCA |, | : “min s = 0” min  0 ,min  0 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 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) Efficient Visualization of LCS by filtered AMR Ridge Extraction

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  40. Efficiency Subdivision iterations: 4 Speed-up: > 4 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” Efficient Visualization of LCS by filtered AMR Ridge Extraction

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

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

  44. 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 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 Efficient Visualization of LCS by filtered AMR Ridge Extraction

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

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

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