Hierarchical Line Integration TVCG Papers Marcel Hlawatsch, Filip Sadlo, Daniel Weiskopf University of Stuttgart, Germany
Feb 23, 2016
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]
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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