Hierarchical Line IntegrationTVCG Papers

Marcel Hlawatsch, Filip Sadlo, Daniel WeiskopfUniversity of Stuttgart, Germany

2

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

4

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

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

• Perform operations inside hierarchical scheme• Combine quantities• , min, max, etc.• LIC: convolution of Gaussian with Gaussian is Gaussian

Computation of Quantities: LIC

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

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

10

Results: FTLE in Time-Dependent 3D Flow

straightforward hierarchical FTLE error FTLE ridge error

speedup 22• 1283 resol.• 64 time frames

11

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 ;-) )

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

IBFVstraightforward 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