Theory and Computation of LCS - Part II EFFICIENT ... · 22/05/2011 · OUTLINE • Computational...

EFFICIENT COMPUTATION OF LCS FOR INTERACTIVE VISUAL

ANALYSIS IN CFDXavier Tricoche

Purdue Computer Science

SIAM DS 2011, Snowbird, UTMay 22, 2011

Theory and Computation of LCS - Part II

Sunday, May 22, 2011

CLIMATELBNL, Utah


H. Werlé

AERONAUTICS


AERODYNAMICS

Pagani Automobili S.p.A.


Sartorius Bearing Tech

TURBOMACHINERY


COMBUSTION

small-scale phenomena.

are often empirical and

Jackie Chen, Sandia


OUTLINE• Computational challenge of LCS

• State of the art in FTLE / LCS computation

• Interactive visual exploration of LCS

• Extraction of high quality ridges from FTLE fields

• Beyond FTLE: Strong LCS in numerical data


LCS IN ACTIONNYTIMES 2009

C. GARTH, UC DAVIS


LCS IN 2D FLOWS


COMPUTATIONAL CHALLENGE•Excruciatingly slow to compute in 3D: compute particles trajectory “everywhere”• integrate for “long enough” in 4D domain: issues of I/O

bottleneck in very large datasets

• integrate forward and backward

• differentiate the flow map + spectral analysis

• restart the computation at each time step

• structures are fractal


OUTLINE• Computation challenge of LCS

•State of the art in FTLE / LCS computation





ADAPTIVE METHODS• Incremental refinement of flow map:

• Significant reduction in number of flow map evaluations


ADAPTIVE METHODS• Incremental refinement of flow map


Garth et al., Efficient computation and visualization of coherent structures in fluid flows, IEEE Visualization 2007.


ADAPTIVE METHODS• Incremental refinement of flow map



ADAPTIVE METHODS• Incremental refinement of FTLE field

• flow map sampling focused on regions containing ridges at current resolution


ADAPTIVE METHODS• Incremental refinement of FTLE field

• flow map sampling focused on regions containing ridges at current resolution

Sadlo, Peikert, Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction, IEEE Visualization 2007.


EXPLOIT TEMPORAL REDUNDANCY

• Advect sampling grid in vicinity of structures of interest

Sadlo et al.,Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection, TopoInVis 2009.


• Exploit the overlap of the space-time region traversed by FTLE computation from different initial time steps.

• cf. talk by Steve Brunton in this session

EXPLOIT TEMPORAL REDUNDANCY

Brunton and Rowley, Fast Computation of FTLE Fields for Unsteady Flows: A Comparison of Methods, Chaos, 2010.


COMPUTATIONAL CHALLENGE

None of these methods is interactive


PROPOSED SOLUTION•Intertwine adaptive FTLE computation and visualization on the GPU

•Strategy1. Prioritize computation based on visual impact2. Progressive refinement weighing responsiveness and image quality

3. Concurrent manipulation of data structures on the GPU4. Incorporate user feedback

4

sampling resolution in a setting that is responsive to

his interest and exploration behavior. Resolving FTLE

structures at a high uniform resolution is intractable

due to memory and time constraints. Instead, our

method restricts the computation to the visible portion

of the flow domain and further it adaptively focuses

on the regions contributes to the rendered image,

as determined by the user-defined transfer function.

More precisely, the basic idea of our method consists

in alternating partial FTLE sampling and ray casting

of the resulting field at each frame, following an

incremental approach. By intertwining computation

and rendering, we can leverage the rendering step to

inform the following computing step about missing

samples that were identified during the volume ren-

dering of the available data. Our method is built upon

a hierarchical data representation implemented in tex-

ture memory. In contrast to typical view-dependent

rendering techniques however, our method requires a

dynamic data representation, which disqualifies static

GPU data structures. We therefore propose a novel

data structure that extends the traditional texture

octree by allowing for the dynamic allocation of tex-

ture memory to the spatial region that is currently

visualized. The allocation policy is controlled by a

priority metric that quantifies the importance of any

spatial region for the current frame in terms of its

FTLE content, size, and visibility.

With these elements, our algorithm proceeds as an

incremental process that optimizes the use of avail-

able computational and memory resources at each

frame. Moreover, we adopt a progressive approach

that weighs responsiveness and image quality based

on the user interaction. Finally, the entire method

has been implemented and optimized on the GPU to

document its benefit in a practical setting. We present

the main building blocks of our method in Figure 1.

Fig. 1. Overview of the components of our method

4 DYNAMIC HIERARCHICAL DATA REPRE-SENTATIONWe describe in the following a novel data structure

that supports the dynamic and adaptive refinement

of both flow map and FTLE field. This data structure

is pointerless and can be implemented very efficiently

in texture memory on the graphics hardware.

4.1 Modified Texture OctreeOur data structure is based on the idea of texture

octree [16], whereby the octree nodes are saved and

interlinked in a texture called indirection pool. Our im-

plementation adopts a modified version of the texture

octree in which texture blocks rather than individual

values are assigned to the leaves [2], [26]. A direct

benefit of this approach is that texture caching and

texture interpolation can be directly exploited, thus

achieving high performance on the GPU. Note that

the boundaries of each block are replicated across

blocks to ensure the local memory footprint of both

interpolation and gradient computation. In this work

we improve upon this basic data structure in two

ways to support the dynamic modification of the

octree at runtime.

First, we introduce the concept of virtual leaf. A

virtual leaf, in contrast to a regular leaf, does not

have an assigned 3D texture block. In our terminol-

ogy, leaves that possess a texture block are called

sampled leaves since they contain sampled values of

the flow map and FTLE field. Virtual leaves allow

us to dynamically assign the limited number of 3D

blocks available in texture memory to the portion of

the tree that directly contributes to the visualization.

In particular, as the user zooms on a particular region

of the dataset, sampled leaves in coarser levels are

continuously evacuated when necessary to free blocks

for the finer levels, as explained in the algorithm

description in Section 6. Observe however that we

prevent the evacuation of nodes in the three coarsest

levels to ensure the availability of a basic data set

overview at all times (e.g., when the user is suddenly

zooming out). Refer to Figure 2.

Our second improvement consists in allowing con-

tinuous changes to the structure of the texture octree.

This necessitates the use of a map, hereafter referred

to as indirection pool map that keeps track of the free

tree nodes. This map associates each octree node in

the indirection pool with a value indicating whether

that entry is currently used or not. When an octree

node is deleted and an entry is freed in the indirection

pool, its corresponding entry in the indirection pool

map is marked as invalid. Upon sorting the indirec-

tion pool map with respect to the valid / invalid

value, free entries in the indirection pool are identified

and made available for subsequent processing. Since

the performance of the texture octree heavily relies

on the caching mechanism offered by the GPU, it

Barakat, Garth, Tricoche, submittedSunday, May 22, 2011

frame ratesteady case

Barakat, Garth, Tricoche, submitted


RESULTS



ACCURACY CHECK

CPU - 4th order solver GPU - adaptive method

ABC flow






•Extraction of high quality ridges from FTLE fields



What are crease surfaces? ”extremal” curves and surfaces

CREASE SURFACE EXTRACTION


RIDGE DEFINITION (EBERLY 1994)

• Constrained extremum• Gradient g

• Hessian eigensystem ei, λi

• Ridge: g orthogonal to one or more emin

• Eigenvalue gives ridge strength

g{ei}

Ridge surface: g . e3 = 0; λ3 < thresh

Valley surface g . e1 = 0; λ1 > thresh

Ridge surfaces in 3D


MARCHING CUBESNatural solution to crease extraction problem, but:• Slow: Nonlinear nature of the creases requires high sampling density

• Inflexible: Cannot filter on the fly to prevent spurious small scale structures


RAY-CASTING•Barakat & Tricoche, 2010:

• GPU-based ray casting of crease surfaces

• check for crease point criterion along each ray

• empty space skipping based on strength

• precompute scale-normalized quantities: gradient, hessian, strength, scalar field

• fast but no mesh


S. Barakat and X. Tricoche / Procedia Computer Science 00 (2010) 1–11 9

A B

C D

Figure 3: Left: Lagrangian coherent structures extracted as ridges of the FTLE field in the ABC flow (here A =√

3,B =

√2, C = 1). A: direct volume rendering of the same clipped FTLE field. B: corresponding ridge surfaces. Right:

LCS extracted in a CFD simulation of a turbulent jet. C: volume rendering of the LCS. D: Ridge surfaces formingunstable manifolds.

6. Conclusion and Future Work

We have presented a method for the interactive extraction and visualization of crease surfaceson the GPU. In significant contrast to existing techniques, our approach is fast and produces re-sults at several frames per second. As such, our algorithm allows the user to precisely controlparameters of the extraction based on their visual impact on the the resulting structures. It alsoprovides an exploratory tool through which the user can freely navigate across space and scales.An additional benefit of this method is its natural complementarity with volume rendering, thusenabling the creation of crease-centric volume visualizations. We have demonstrated the effec-tiveness of this approach on several examples corresponding to applications in science, medicine,and engineering.

References

[1] D. Eberly, R. Gardner, B. Morse, S. Pizer, Ridges for image analysis, Journal of Mathematical Imaging and Vision4 (1994) 351–371.

Barakat & Tricoche, ICCS 2010

FTLE LCS


Barakat & Tricoche, ICCS 2010

FTLE LCS


MESH EXTRACTIONChallenging task:

• Sampling is expensive

• Creases are non-orientable

• Presence of boundaries

• Strong curvature

Meshing theory: very dense sampling is necessary


Wish list:1. Bounded distance between surface and mesh2. Reasonable number of triangles3. Good triangle aspect ratio4. Smooth change of triangle size on the mesh5. Fast!

MESH EXTRACTION


Proposed Approach• Sampling and meshing as a single phase instead of two

separate phases.• Sample on the GPU, triangulate on the CPU•Benefits:• Filter out huge number of samples on GPU

based on accuracy and geometric criteria

• Steer sampling toward good triangles

• Limited memory footprint

• Parallel triangulation due to the small number of possible vertices

MESH EXTRACTION


CPU: Pipelined parallel triangulation using priority and locking

TEAM WORKGPU: Find new vertices using dense sampling + error tracking


High extraction cost Need to grow a large number of triangles at once to

exploit parallelismmeshing conflicts are very frequent

Conflicts happen between:• Neighboring triangles of the same front • Triangles of two encroaching fronts

Solution: •heuristic to minimize conflicts•discard conflicting triangles and re-insert active edges with different order. •Add active edge between encroaching fronts when necessary

ALGORITHM

Problematic cases


MESHING STRATEGYConsider various cases...


SAMPLING• Track error as sampling progresses along median axis

• Proceed as long as error threshold or triangle size are not exceeded


• A large number of triangles can be produced in parallel on the GPU and returned to CPU for meshing

• Space is divided into blocks where meshing is handled by different threads in parallel

• Meshing on the CPU is interleaved with triangle computation on the GPU using locking mechanism

PARALLEL IMPLEMENTATION


RESULTS: ABC FLOW

Comparison withSchultz’s method

Barakat & Tricoche, Eurovis 2011


RESULTS: QUALITY CHECKError comparison

Our method Marching Cubes

Barakat & Tricoche, Eurovis 2011Sunday, May 22, 2011

PERFORMANCE

Submission ID: paper360 / Fast Extraction of Crease Surfaces 9

Figure 11: Our extraction for the ridge in the IEEE Visu-alization 2010 Contest brain dataset colored by the scaleinformation (blue:coarse, red:fine).

Table 1: Results of our method compared to two variationsof [STS09]. "S1" uses the same resolution as our method,while "S2" is doubled along each axis. The percentage of dis-carded triangles for our method was 8.6%, 32.7%, 29.4%,and 27.5% for the Cube, ABC, Jet, and Brain data sets, re-spectively.

Time Edge- Edge- Error(sec) # Tri. Point Tri. (e-4)

Cub

e Our 4.73 55K 0.58 0.73 0.17

S1 4.2 86K 0.20 0.41 0.66

S2 24.14 350K 0.21 0.41 0.66

AB

C

Our 52.3 632K 0.46 0.61 2.57

S1 18.03 912K 0.37 0.54 3.33

S2 227.6 3866K 0.39 0.56 3.32

Jet

Our 143.3 1761K 0.47 0.67 1.68

S1 54.3 1790K 0.37 0.55 8.80

S2 996.1 6284K 0.39 0.57 8.65

Bra

in Our 32.75 405K 0.39 0.59 4.84

S1 149.1 3101K 0.36 0.50 -

We find the observed collaborative performance betweenCPU and GPU very promising and we would like to pur-sue this avenue. We believe that utilizing new technologiessuch as OpenCL will be useful in this endeavor. Additionallywe would like to see how our parallel approach performson even larger data sets and investigates its scalability onmany-core systems. This promises to be particularly inter-esting in the context of problems that far exceed the memoryof modern day desktop computers, such as in the process-

ing of high-performance computing simulations. Streamingstrategies will need to be considered in this case.

Though our method outperforms the current state of theart in ridge surface extraction and is fast compared to con-ventional front propagation techniques, we are confident thatsignificant performance gains can be achieved by using abetter spatial data structure for conflict detection. In partic-ular we found that the collision detection method we usedhad limited scalability, thus it can adversely impact the per-formance of our method in the context of very large meshes.

References[AB98] AMENTA N., BERN M.: Surface reconstruction by

voronoi filtering. Discrete and Computational Geometry 22(1998), 481–504. 5

[ABCO∗01] ALEXA M., BEHR J., COHEN-OR D., FLEISHMANS., LEVIN D., SILVA C. T.: Point set surfaces. In VIS ’01:Proceedings of the conference on Visualization ’01 (Washington,DC, USA, 2001), IEEE Computer Society, pp. 21–28. 2

[ABK98] AMENTA N., BERN M., KAMVYSSELIS M.: A newvoronoi-based surface reconstruction algorithm. In SIGGRAPH’98: Proceedings of the 25th annual conference on Computergraphics and interactive techniques (New York, NY, USA, 1998),ACM, pp. 415–421. 2

[BMR∗99] BERNARDINI F., MITTLEMAN J., RUSHMEIER H.,SILVA C., TAUBIN G.: The ball-pivoting algorithm for surfacereconstruction. IEEE Transactions on Visualization and Com-puter Graphics 5, 4 (1999), 349–359. 6

[BO05] BOISSONNAT J.-D., OUDOT S.: Provably good samplingand meshing of surfaces. Graph. Models 67, 5 (2005), 405–451.2

[BT10] BARAKAT S., TRICOCHE X.: An image-based approachto interactive crease extraction and rendering. Procedia Com-puter Science 1, 1 (2010), 1703 – 1712. ICCS 2010. 3, 5

[EGMP94] EBERLY D., GARDNER R., MORSE B., PIZER S.:Ridges for image analysis. Journal of Mathematical Imaging andVision 4 (1994), 351–371. 2

[Eve01] EVERITT C.: Interactive Order-Independent Trans-parency. NVIDIA Corporation, 2001. 3

[FP01] FURST J. D., PIZER S. M.: Marching ridges. In Proceed-ings of 2001 IASTED International Conference on Signal andImage Processing (2001), pp. 22–26. 2

[Hal01] HALLER G.: Distinguished material surfaces and coher-ent structures in three-dimensional flows. Physica D 149 (2001),248–277. 7

[KSSW09] KINDLMANN G., SAN JOSE ESTEPAR R., SMITHS. M., WESTIN C.-F.: Sampling and visualizing creases withscale-space particles. IEEE Transactions on Visualization andComputer Graphics 15, 6 (2009), 1415–1424. 2, 3, 8

[KTW06] KINDLMANN G., TRICOCHE X., WESTIN C.-F.:Anisotropy creases delineate white matter structure in diffusiontensor mri. In Proceedings of Medical Imaging Computing andComputer-Assisted Intervention, MICCAI ’06 (2006). 2

[KTW07a] KINDLMANN G., TRICOCHE X., WESTIN C.-F.: De-lineating white matter structure in diffusion tensor MRI withanisotropy creases. Medical Image Analysis 11, 5 (October2007), 492–502. 2, 3

submitted to Eurographics/ IEEE-VGTC Symposium on Visualization (2011)

Submission ID: paper360 / Fast Extraction of Crease Surfaces 9

Figure 11: Our extraction for the ridge in the IEEE Visu-alization 2010 Contest brain dataset colored by the scaleinformation (blue:coarse, red:fine).

Table 1: Results of our method compared to two variationsof [STS09]. "S1" uses the same resolution as our method,while "S2" is doubled along each axis. The percentage of dis-carded triangles for our method was 8.6%, 32.7%, 29.4%,and 27.5% for the Cube, ABC, Jet, and Brain data sets, re-spectively.

Time Edge- Edge- Error(sec) # Tri. Point Tri. (e-4)

Cub

e Our 4.73 55K 0.58 0.73 0.17

S1 4.2 86K 0.20 0.41 0.66

S2 24.14 350K 0.21 0.41 0.66

AB

C

Our 52.3 632K 0.46 0.61 2.57

S1 18.03 912K 0.37 0.54 3.33

S2 227.6 3866K 0.39 0.56 3.32

Jet

Our 143.3 1761K 0.47 0.67 1.68

S1 54.3 1790K 0.37 0.55 8.80

S2 996.1 6284K 0.39 0.57 8.65

Bra

in Our 32.75 405K 0.39 0.59 4.84

S1 149.1 3101K 0.36 0.50 -

We find the observed collaborative performance betweenCPU and GPU very promising and we would like to pur-sue this avenue. We believe that utilizing new technologiessuch as OpenCL will be useful in this endeavor. Additionallywe would like to see how our parallel approach performson even larger data sets and investigates its scalability onmany-core systems. This promises to be particularly inter-esting in the context of problems that far exceed the memoryof modern day desktop computers, such as in the process-

ing of high-performance computing simulations. Streamingstrategies will need to be considered in this case.

Though our method outperforms the current state of theart in ridge surface extraction and is fast compared to con-ventional front propagation techniques, we are confident thatsignificant performance gains can be achieved by using abetter spatial data structure for conflict detection. In partic-ular we found that the collision detection method we usedhad limited scalability, thus it can adversely impact the per-formance of our method in the context of very large meshes.

References[AB98] AMENTA N., BERN M.: Surface reconstruction by

voronoi filtering. Discrete and Computational Geometry 22(1998), 481–504. 5

[ABCO∗01] ALEXA M., BEHR J., COHEN-OR D., FLEISHMANS., LEVIN D., SILVA C. T.: Point set surfaces. In VIS ’01:Proceedings of the conference on Visualization ’01 (Washington,DC, USA, 2001), IEEE Computer Society, pp. 21–28. 2

[ABK98] AMENTA N., BERN M., KAMVYSSELIS M.: A newvoronoi-based surface reconstruction algorithm. In SIGGRAPH’98: Proceedings of the 25th annual conference on Computergraphics and interactive techniques (New York, NY, USA, 1998),ACM, pp. 415–421. 2

[BMR∗99] BERNARDINI F., MITTLEMAN J., RUSHMEIER H.,SILVA C., TAUBIN G.: The ball-pivoting algorithm for surfacereconstruction. IEEE Transactions on Visualization and Com-puter Graphics 5, 4 (1999), 349–359. 6

[BO05] BOISSONNAT J.-D., OUDOT S.: Provably good samplingand meshing of surfaces. Graph. Models 67, 5 (2005), 405–451.2

[BT10] BARAKAT S., TRICOCHE X.: An image-based approachto interactive crease extraction and rendering. Procedia Com-puter Science 1, 1 (2010), 1703 – 1712. ICCS 2010. 3, 5

[EGMP94] EBERLY D., GARDNER R., MORSE B., PIZER S.:Ridges for image analysis. Journal of Mathematical Imaging andVision 4 (1994), 351–371. 2

[Eve01] EVERITT C.: Interactive Order-Independent Trans-parency. NVIDIA Corporation, 2001. 3

[FP01] FURST J. D., PIZER S. M.: Marching ridges. In Proceed-ings of 2001 IASTED International Conference on Signal andImage Processing (2001), pp. 22–26. 2

[Hal01] HALLER G.: Distinguished material surfaces and coher-ent structures in three-dimensional flows. Physica D 149 (2001),248–277. 7

[KSSW09] KINDLMANN G., SAN JOSE ESTEPAR R., SMITHS. M., WESTIN C.-F.: Sampling and visualizing creases withscale-space particles. IEEE Transactions on Visualization andComputer Graphics 15, 6 (2009), 1415–1424. 2, 3, 8

[KTW06] KINDLMANN G., TRICOCHE X., WESTIN C.-F.:Anisotropy creases delineate white matter structure in diffusiontensor mri. In Proceedings of Medical Imaging Computing andComputer-Assisted Intervention, MICCAI ’06 (2006). 2

[KTW07a] KINDLMANN G., TRICOCHE X., WESTIN C.-F.: De-lineating white matter structure in diffusion tensor MRI withanisotropy creases. Medical Image Analysis 11, 5 (October2007), 492–502. 2, 3

submitted to Eurographics/ IEEE-VGTC Symposium on Visualization (2011)

Hybrid GPU/CPU method (our) vs. Marching Cubes at single (S1) and double (S2) resolution

Barakat & Tricoche, Eurovis 2011






•Beyond FTLE: Strong LCS in numerical data


VARIATIONAL CHARACTERIZATION• G. Haller ‘s Variational Theory of Lagrangian Coherent Structures

suggests following numerical characterization of LCS

i)Extract location set Z satisfying

ii) Identify WLCS as subset satisfying

iii) Identify repelling LCS as where is positive definite

(∇λn(x0, to, T ), ξn(x0, to, T )) = 0

ZWLCS ⊂ Z

λn−1 �= λn > 1

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2SZLCS ⊂ ZWLCS L(x0, t0, T )


IMPLEMENTATION• Gradients are computed over numerical volume of flow map

using smooth convolution kernels and analytical derivation of Cauchy-Green tensor’s eigenvectors and eigenvalues

• (W)LCS are visualized using a ray-casting approach

• Parallel computation on CPU


PRELIMINARY RESULTS• WLCS (step 2) in ABC flow


• Step 3 (LCS)

PRELIMINARY RESULTS


PRELIMINARY RESULTS• Testing the positive definiteness of L is tricky

• The robust numerical extraction of LCS remains work in progress.

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ACKNOWLEDGMENTS• Samer Barakat, Purdue

• Christoph Garth, UC Davis & University of Kaiserslautern

• Gordon Kindlmann, U Chicago


Theory and Computation of LCS - Part II EFFICIENT ... · 22/05/2011 · OUTLINE • Computational...

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Transcript of Theory and Computation of LCS - Part II EFFICIENT ... · 22/05/2011 · OUTLINE • Computational...