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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). - PowerPoint PPT Presentation
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Hierarchical Line IntegrationTVCG Papers
Marcel Hlawatsch, Filip Sadlo, Daniel WeiskopfUniversity of Stuttgart, Germany
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Motivation
Dense sets of trajectories required for, e.g.,
delocalized 2 < -5000
Line integral convolution (LIC) Finite-Time Lyapunov Exponent (FTLE) Other Lagrangian concepts(delocalized vortex criteria)
[Fuchs et al. 2008]
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 advect sampling 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
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Concept
• Coordinate maps: : start point of traj. : end point of traj.
: hierarchy level
• obtained, e.g., by integration• constructed by „concatenation“• All levels have same resolution (no pyramid)• Overwrite (store only highest level)
:
:
:
• general case: end points not at nodes interpolation• repeated interpolation source of error (see later)
: nodes
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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)
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Computational Cost
Better than “optimum”?• Theory: speedup >2 if steps >16• Concatenation steps: no integration
2D Time-independent FTLE (in milliseconds)
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• Perform operations inside hierarchical scheme• Combine quantities• , min, max, etc.• LIC: convolution of Gaussian with Gaussian is Gaussian
Computation of Quantities: LIC
straightforward hierarchical
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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
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Results: FTLE in Time-Dependent 2D Flow
straightforward hierarchical FTLE error• 95th percentile norm.
error (max. = 1.16%)• max. error = 47.33%
(at isolated points)
FTLE ridge error• Lagrangian coherent
structures (LCS)• avg. error = 0.014 cells
speedup 61• 512 x 512 resol.• 100 time frames
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Results: FTLE in Time-Dependent 3D Flow
straightforward hierarchical FTLE error FTLE ridge error
speedup 22• 1283 resol.• 64 time frames
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Error Analysis
Error order of scheme:
: number of hier. levels: global error at node
: cell size: maximum second derivative over all coordinate
maps: Lipschitz constant from continuity of coordinate
maps
® Scheme is second order in cell size (see 2-page proof inside paper ;-) )
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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
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Hierarchical Line Integration
Thank you for your attention!
Acknowledgements:
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Results: Comparison to IBFV
IBFVstraightforward hierarchical
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Results: LIC
coordinate map error• 95th percentile error (max. = 0.1%)• max. error = 1.23%• longer advection time than LIC
straightforward hierarchical
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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 …)
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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