This document summarizes a research paper on a multi-threaded algorithm for computing augmented contour trees from scalar field data. The key points are: 1) A simple parallel algorithm is proposed that partitions the scalar field domain, computes local contour trees in parallel on each partition, and stitches the local trees together. 2) The algorithm was implemented in C++ using VTK for generic input/output support. It achieves good parallel efficiency on workstations with up to 8 threads. 3) Experimental results show the parallel implementation outperforms sequential state-of-the-art methods for large data sets and provides interactive analysis capabilities for topological data exploration and visualization applications.

1. Contour Forests: Fast Multi-threaded
Augmented Contour Trees
Charles Gueunet, UPMC and Kitware
Pierre Fortin, UPMC
Julien Jomier, Kitware
Julien Tierny, UPMC

2. • Topological analysis: classical approach in Sci Viz
• Contour Trees, Reeb Graphs, Morse-Smale Complexes...
Introduction
[DoraiswamyTVCG13] [LandgeSC14] [MaadasamyHiPC12]

Context

Related Works

Challenges

Contributions

3. • Topological analysis: classical approach in Sci Viz
• Contour Trees, Reeb Graphs, Morse-Smale Complexes...
• Increasing data size and complexity
• Challenge for interactive exploration
• Multi-core architectures are common
Introduction
[DoraiswamyTVCG13] [LandgeSC14] [MaadasamyHiPC12]

Context

Related Works

Challenges

Contributions
Motivation for multi-threaded parallelism

4. Introduction

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]

5. Introduction
• [PascucciAlgorithmica04]
Parallel:

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗

6. Introduction
• [PascucciAlgorithmica04]
• [MaadasamyHiPC12]
Parallel:

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗
Maadasamy
✓ ✓ ✗ ✗

7. Introduction
• [PascucciAlgorithmica04]
• [MaadasamyHiPC12]
• [NatarajanPV15]
Parallel:

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗
Maadasamy
✓ ✓ ✗ ✗
Natarajan
✓ ✗ ✗ ✗

8. Introduction
• [PascucciAlgorithmica04]
• [MaadasamyHiPC12]
• [NatarajanPV15]
Parallel:
Distributed:
• [MorozovPPoPP13]

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗
Maadasamy
✓ ✓ ✗ ✗
Natarajan
✓ ✗ ✗ ✗
Morozov MT
✗ ~ ✓ -

9. Introduction
• [PascucciAlgorithmica04]
• [MaadasamyHiPC12]
• [NatarajanPV15]
Parallel:
Distributed:
• [MorozovPPoPP13]
• [MorozovTopoInVis13]

Context

Related Works

Challenges

Contributions
Sequential:
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗
Maadasamy
✓ ✓ ✗ ✗
Natarajan
✓ ✗ ✗ ✗
Morozov MT
✗ ~ ✓ -
Morozov CT
✓ ~ ✓ ✗

10. Introduction
• [PascucciAlgorithmica04]
• [MaadasamyHiPC12]
• [NatarajanPV15]
Sequential:
Parallel:
Distributed:
• [MorozovPPoPP13]
• [MorozovTopoInVis13]
• [LandgeTopoInVis15]

Context

Related Works

Challenges

Contributions
• [CarrSODA00,ChiangCG05]
Ref CT Tet. mesh Augmented Combination
Pascucci
✓ ~ ✓ ✗
Maadasamy
✓ ✓ ✗ ✗
Natarajan
✓ ✗ ✗ ✗
Morozov MT
✗ ~ ✓ -
Morozov CT
✓ ~ ✓ ✗
Landge
✗ ✗ ✓ -

11. • Topological data analysis algorithms
• Intrinsically sequential approaches
• Challenging parallelization
Introduction

Context

Related Works

Challenges

Contributions

12. • Topological data analysis algorithms
• Intrinsically sequential approaches
• Challenging parallelization
• Contour Tree:
• No complete parallelization (only subroutines)
• No efficient parallel algorithm for augmented trees
Introduction

Context

Related Works

Challenges

Contributions

13. • Efficient multi-threaded algorithm for contour tree computation
• Simple approach
• Good parallel efficiency on workstations
Introduction

Context

Related Works

Challenges

Contributions

14. • Efficient multi-threaded algorithm for contour tree computation
• Simple approach
• Good parallel efficiency on workstations
• Ready-to-use VTK-based C++ implementation
• Generic input (VTU/VTI, 2D/3D)
• Generic output (augmented trees)
Introduction

Context

Related Works

Challenges

Contributions

15. 1) Introduction
2) Preliminaries

Background

Overview
3) Algorithm

Domain partitioning

Local computation

Contour forest stitching
Summary
4) Experimental results

Scalability

Efficiency

Limitations
5) Application
6) Conclusion

Recall

Perspective

16. Preliminaries

17. Input data:
Preliminaries

Background

Overview
• Piecewise linear scalar field
2D Example 3D Example

18. Level set:
Preliminaries

Background

Overview
• Preimage of a scalar value
• Isovalue: onto
2D Example 3D Example

19. Contour tree:
Preliminaries

Background

Overview
• Simply connected input domain
• 1-dimensional simplicial complex
Regular vertices
in augmented tree
2D Example 3D Example

20. Contour tree:
Preliminaries

Background

Overview
• Quotient space:
• Equivalence relation:
2D Example 3D Example

21. 1) Sort vertices
2) Create partitions using level sets
3) Compute local trees
4) Stitch local trees
Main steps of our approach:
Preliminaries

Background

Overview

22. 1) Sort vertices
2) Create partitions using level sets
3) Compute local trees
4) Stitch local trees
• Simple approach
• Local computation of CT
• Augmented trees
• Simple stitching of the forests
Main steps of our approach:
Preliminaries

Background

Overview Pros:

23. 1) Sort vertices
2) Create partitions using level sets
3) Compute local trees
4) Stitch local trees
• Simple approach
• Local computation of CT
• Augmented trees
• Simple stitching of the forests
Main steps of our approach:
Preliminaries

Background

Overview Pros:
• Depends on interface level sets
Cons:

24. Algorithm

25. • Range driven partitions
• Sorted scalars ⇒ balanced partitions
• Use partitions
Algorithm

Domain partitioning

Local computation

Contour forest stitching

26. • Range driven partitions
• Sorted scalars ⇒ balanced partitions
• Use partitions
Algorithm

Domain partitioning

Local computation

Contour forest stitching

27. • Range driven partitions
• Sorted scalars ⇒ balanced partitions
• Use partitions
Algorithm

Domain partitioning

Local computation

Contour forest stitching
3D Example

28. • Range driven partitions
• Sorted scalars ⇒ balanced partitions
• Use partitions
Algorithm

Domain partitioning

Local computation

Contour forest stitching
3D Example

29. • Range driven partitions
• Sorted scalars ⇒ balanced partitions
• Use partitions
Algorithm

Domain partitioning

Local computation

Contour forest stitching
noise
2D Example 3D Example

30. • Extended partition:
initial partition + boundary vertices
• Correct tree in initial partition
• Visit all edges in parallel
Algorithm

Domain partitioning

Local computation

Contour forest stitching
extended
green + red
Redundant computations:
2D Example 3D Example

31. • In extended partitions
• Using Carr et al. Algorithm:
• Join Tree
• Split Tree
• Combination
• Union-Find [CormenItA01]
Algorithm

Domain partitioning

Local computation

Contour forest stitching
Independent
2D Example 3D Example

32. • Join and Split trees: 2 threads
• Combine: 1 thread
• O(|σi
|×α(|σi
|)) + O(|S|)
• Keep arcs on the boundary (n-h)
Algorithm

Domain partitioning

Local computation

Contour forest stitching
Each partition:
2D Example 3D Example

33. • Simple step
• Visit crossing arcs
• Cut interface arcs
• Segmentation: fast lookup
Algorithm

Domain partitioning

Local computation

Contour forest stitching
2D Example 3D Example

34. • New super-arc:
• Union of the two facing halves
• Once per connected component
of level set (∼ crossing arc)
Algorithm

Domain partitioning

Local computation

Contour forest stitching
2D Example 3D Example

35. • Repeat for each interface
• Fast in practice
Algorithm

Domain partitioning

Local computation

Contour forest stitching
2D Example 3D Example

36. Experimental results
Intel Xeon CPU E5-2630 v3 (2.4 Ghz, 8 cores)
64GB of RAM
C++ (GCC-4.9), VTK, OpenMP (additional material)

37. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• Uniform sampling

38. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• Foot: fast computation

39. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• Overlap and Stitching: efficient

40. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• High range of value

41. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• About ten seconds

42. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
8 threads, 4 partitions
• Parallel efficiency max: 67% (speedup / )
• Parallel efficiency: 40 ~ 55 % (except extrema)

43. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
• Open-source reference
implementation
• pTourtre: naive parallel
version
Libtourtre:

44. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
• Open-source reference
implementation
• pTourtre: naive parallel
version
• Always better than
sequential libtourtre for
big data sets
• Better than parallel except
for the foot data set
Libtourtre:

45. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
• Different slopes
• Few arithmetic operations per
memory access

46. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
Computational speed: vertices/second
• Different slopes
• Few arithmetic operations per
memory access
• Load imbalance

47. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
Computational speed: vertices/second
• Different slopes
• Few arithmetic operations per
memory access
• Load imbalance
• Memory congestion

48. Exp. Results

Efficiency

Comparison

Speed

Pros & cons
Partitions sizes in vertices
• Different slopes
• Few arithmetic operations per
memory access
• Load imbalance
• Memory congestion
• Redundant computations

49. • Redundant computation
• Load imbalance
• Memory congestion (augmented
tree)
Pros: Cons:

Efficiency

Comparison

Speed

Pros & cons
Exp. Results
• Good parallel efficiency
• Faster than reference
implementation
• Large spectrum of data

50. Application

51. Application
• Highlight features
• Allows features grouping
Exploration:

52. Application
• Remove noise
• Keep the most important features
Occlusion reduction:

53. Conclusion

54. Take home message:

Recall

PerspectiveConclusion
• Efficient algorithm:
• Multi-threaded
• Augmented
• Simple approach, subtle details

55. Take home message:

Recall

PerspectiveConclusion
• Efficient algorithm:
• Multi-threaded
• Augmented
• Simple approach, subtle details
• VTK-based implementation:
• Generic input
– VTU,VTI
– 2D/3D
• Generic output (augmented trees)
• Ready-to-use

56. Take home message:

Recall

PerspectiveConclusion
• Efficient algorithm:
• Multi-threaded
• Augmented
• Simple approach, subtle details
• VTK-based implementation:
• Generic input
– VTU,VTI
– 2D/3D
• Generic output (augmented trees)
• Ready-to-use
• Lesson learned
• Memory bound
• Memory congestion: price to pay
for augmented trees
• Efficient implementation: hard

57. 
Recall

PerspectiveConclusion
• Improve partitioning:
●
Better cutting isovalue selection
●
Contour Spectrum
• Distributed systems

58. The end
Thank you for your attention,
Do you have any question ?
This work is partially supported by the Bpifrance grant “AVIDO” (Programme d’Investissements d’Avenir, reference
P112017-2661376/DOS0021427) and by the French National Association for Research and Technology (ANRT), in the
framework of the LIP6 - Kitware SAS CIFRE partnership reference 2015/1039.
Paper and Code at: http://www-pequan.lip6.fr/~tierny/

59. Appendice

60. Seq. Approaches
Monotone paths:Union find:

61. Critical points