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Hierarchical Line Integration TVCG Papers

Hierarchical Line Integration TVCG Papers. Marcel Hlawatsch , Filip Sadlo, Daniel Weiskopf University of Stuttgart, Germany. Motivation. Dense sets of trajectories required for, e.g.,. delocalized  2 < -5000. [Fuchs et al. 2008]. Line integral convolution (LIC).

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Hierarchical Line Integration TVCG Papers

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  1. Hierarchical Line IntegrationTVCG Papers Marcel Hlawatsch, Filip Sadlo, Daniel Weiskopf University of Stuttgart, Germany

  2. Motivation Dense sets of trajectories required for, e.g., delocalized 2 < -5000 [Fuchs et al. 2008] Line integral convolution (LIC) Finite-Time Lyapunov Exponent (FTLE) Other Lagrangian concepts(delocalized vortex criteria)

  3. Motivation • Integration of many trajectories is expensive! • Different trajectories pass same region • Reuse parts of trajectories • Fast LIC[Stalling and Hege 1995, 1998] • Shift convolution kernel on long trajectories • Collect LIC contributions at pixels • Grid Advection for FTLE computation[Sadlo et al. 2010] • Reuse part of path lines for FTLE time series  advectsampling grid • Fast Computation of FTLE Fields[Brunton and Rowley, Chaos 2010] • Concurrent to this work, similar idea • No quantities along trajectories (no LIC etc.) • Higher memory consumption, no proven error order

  4. Concept • Coordinate maps : : start point of traj. : end point of traj. : hierarchylevel • obtained, e.g., by integration • constructedby „concatenation“ • All levels have same resolution (no pyramid) • Overwrite (store only highest level) : : : nodes : • general case: end points not at nodes  interpolation • repeated interpolation  source of error (see later)

  5. Procedure traditional approach (n integration steps) our approach(h levels) integration of initial trajectories  one catenation (s = 2) for next level O(n)  O(h) = O(log n)

  6. Computational Cost Better than “optimum”? • Theory: speedup >2 if steps >16 • Concatenation steps: no integration 2D Time-independent FTLE(in milliseconds)

  7. Computation of Quantities: LIC • Perform operations inside hierarchical scheme • Combine quantities • , min, max, etc. • LIC: convolutionofGaussianwithGaussianisGaussian straightforward hierarchical

  8. Scheme with Time-Dependent Data • level 0: from data set (by integration, blue) • green: required for result at time t1 (at level 3) • bold outlines: blocks kept in memory (overwrite) • hatched: next time blocks • integration range  number of blocks in memory • scheme pays off for time series, i.e., dense trajectory seeding in time • no temporal interpolation needed

  9. Results: FTLE in Time-Dependent 2D Flow FTLE ridge error • Lagrangian coherent structures (LCS) • avg. error = 0.014 cells FTLE error • 95th percentile norm. error (max. = 1.16%) • max. error = 47.33%(atisolatedpoints) hierarchical straightforward • speedup  61 • 512 x 512 resol. • 100 time frames

  10. Results: FTLE in Time-Dependent 3D Flow FTLE ridge error hierarchical straightforward FTLE error • speedup  22 • 1283resol. • 64 time frames

  11. Error Analysis Error order of scheme: : numberof hier. levels : global erroratnode : cellsize : maximumsecond derivative over all coordinatemaps : Lipschitzconstantfromcontinuityofcoordinatemaps • Scheme is second order in cell size (see 2-page proofinsidepaper ;-) )

  12. Conclusion • Acceleration scheme for spacetime-dense sets of solutions (end points of traj.) • Accelerated computation of quantities along trajectories • Logarithmic computational complexity • Well suited for modern multicore or many-core architectures • Proven error order • Future work • Higher-order interpolation schemes  better error order? • Costly integrators for higher-order data  higher acceleration

  13. Hierarchical Line Integration Thank you for your attention! Acknowledgements:

  14. Results: Comparison to IBFV IBFV straightforward hierarchical

  15. Results: LIC coordinate map error • 95th percentile error (max. = 0.1%) • max. error = 1.23% • longer advection time than LIC straightforward hierarchical

  16. Illustration • We produce end points, not complete trajectories • colored points: our approach • white curves: trajectories • background: coordinate map error • larger error in regions of high FTLE(predictability …)

  17. Boundaries • Closed Boundaries / Periodic Domains • No problems (no accesses outside coordinate map) • Open Boundaries (outflow) • Design choice: stop trajectories or continue? • Stopping often preferred • Achieved by adding a zero-velocity border • Repeated interpolation against zero border affects maps • Conservative approach: propagate flag, check flag

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