2D Triangulation Input 2D Boundary Output Triangle Mesh Physical - - PowerPoint PPT Presentation

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2D Triangulation Input 2D Boundary Output Triangle Mesh Physical - - PowerPoint PPT Presentation

Jan 09, 2023 224 likes •951 views

TriWild : Robust Triangulation with Curved Constraints 1 Yixin Hu, 1 1 Teseo Schneider, 2 2 Xifeng Gao, 3 Qingnan Zhou, 3 4 Alec Jacobson, 1 4 Denis Zorin, 1 Daniele Panozzo. 2D Triangulation Input 2D Boundary Output Triangle

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TriWild:

Robust Triangulation with Curved Constraints

Yixin Hu, Teseo Schneider, Xifeng Gao, Qingnan Zhou, Alec Jacobson, Denis Zorin, Daniele Panozzo.

1 1 1 1 1 2 3 4 2 3 4

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2D Triangulation

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Input 2D Boundary Output Triangle Mesh Physical Simulation

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Linear Triangulation

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(Generated by Triangle)

#V = 3013, #F = 4149

Want a smaller mesh?

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SLIDE 4

Linear Triangulation

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#V = 1503 #F = 304

#V = 3013, #F = 4149

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SLIDE 5

Linear Triangulation

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#V = 1503 #F = 304

#V = 3013, #F = 4149

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Curved Triangulation

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Curved mesh!

(Generated by TriWild)

#V = 1503 #F = 304

#V = 3013, #F = 4149

!

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SLIDE 7

2D Triangulation

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Faster Coarser Conforming Accurate

Input 2D Boundary Output Triangle Mesh Physical Simulation

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SLIDE 8

Existing Methods

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  • Curved triangle mesh is not a new concept.
  • Introduced in late 80s.
  • Haven’t been tested on large-scale dataset.
  • Not widely used because no robust tools.
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Tangled Curved Constraints

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Superposed Curves

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Tangled Curved Constraints

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Intersected Curves

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Tangled Curved Constraints

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Tangled Curved Constraints

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(Generated by TriWild)

Input:

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(Generated by TriWild)

Input:

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(Generated by TriWild)

Input:

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(Generated by TriWild)

Input:

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Why TriWild?

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Input High Curvature & Inflection Point Separation !-separation Linear Mesh Generation & Mesh Improvement Curved Mesh Improvement

“Cleanup” the input curves.

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SLIDE 18

Why TriWild?

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Input High Curvature & Inflection Point Separation !-separation Linear Mesh Generation & Mesh Improvement Curved Mesh Improvement

Linear mesh for easier curving.

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SLIDE 19

Pipeline - Input & Output

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  • Input:
  • Bezier curves. (SVG standard, degree 3)
  • Output:
  • Curved triangle mesh
  • Conforms to primary features.
  • Approximates secondary features up to ".

Primary features Secondary features

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Pipeline - Feature Preprocessing

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Input High Curvature & Inflection Point Separation

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Pipeline - Feature Preprocessing

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High Curvature & Inflection Point Separation

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Pipeline - Feature Preprocessing

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Input High Curvature & Inflection Point Separation !-separation

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SLIDE 23

Pipeline - Feature Preprocessing

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!-separation

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SLIDE 24

Pipeline - Curved Mesh Generation

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Input High Curvature & Inflection Point Separation !-separation Linear Mesh Generation & Mesh Improvement

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SLIDE 25

Recap - Linear Mesh Generation (TetWild)

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Stage I: Valid Mesh Generation (Rational)

Segment Soup Linear Triangle Mesh

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Recap - Linear Mesh Generation (TetWild)

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Stage I: Valid Mesh Generation (Rational) Stage II: Mesh Improvement (Mixed à Double)

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Recap - Linear Mesh Generation (TetWild)

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100% success rate on 10,000 real-world models. Stage I: Valid Mesh Generation (Rational) Stage II: Mesh Improvement (Mixed à Double)

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SLIDE 28

Pipeline - Linear Mesh Generation

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  • Feature Invariant:
  • Feature edges’ associated curve

has total curvature <#.

  • Feature vertices can only move

along feature curves.

Total curvature <% Move along feature

“feature” == “primary feature”

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SLIDE 29

Pipeline - Linear Mesh Generation

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  • Feature Invariant:
  • Feature edges’ associated curve

has total curvature <#.

  • Feature vertices can only move

along feature curves.

  • Vertex Projection
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Pipeline - Linear Mesh Generation

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  • Feature Invariant:
  • Feature edges’ associated curve

has total curvature <#.

  • Feature vertices can only move

along feature curves.

  • Vertex Projection
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SLIDE 31

Pipeline - Curved Mesh Generation

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Linear Mesh Generation & Mesh Improvement

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SLIDE 32

Pipeline - Curved Mesh Generation

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Input High Curvature & Inflection Point Separation !-separation Linear Mesh Generation & Mesh Improvement Curved Mesh Improvement

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Pipeline - Curving

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  • Curving:
  • All feature elements
  • With primary feature edges

(thickened in red)

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SLIDE 34

Pipeline - Curved Mesh Generation

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  • Inversion check for curved elements:
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Pipeline - Curved Mesh Generation

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Curved Mesh Improvement

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Pipeline - Curved Mesh Generation

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Input High Curvature & Inflection Point Separation !-separation Linear Mesh Generation & Mesh Improvement Curved Mesh Improvement

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Parameters – Envelope Size

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  • !
  • Used in !-separation for separating primary features and secondary

features.

  • " (same parameter used in TetWild)
  • Maximum distant that secondary feature edges can deviate from input.
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Parameters – Envelope Size

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! "

Red: primary feature edges Green: secondary feature edges

1e-3 1e-3

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SLIDE 40

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! "

Red: primary feature edges Green: secondary feature edges

1e-3 1e-3 1e-2

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SLIDE 41

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! "

Red: primary feature edges Green: secondary feature edges

1e-3 1e-3 1e-2 1e-2

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SLIDE 42

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! "

Red: primary feature edges Green: secondary feature edges

1e-3 1e-3 1e-2 1e-2

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Parameters – Targeted Edge Length

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Edge_length = diag_bbox/5

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Parameters – Targeted Edge Length

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Edge_length = diag_bbox/20

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Parameters – Targeted Edge Length

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Edge_length = diag_bbox/100

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MATLAB vs TriWild

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(Generated by TriWild) (Generated by MATLAB)

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MATLAB vs TriWild

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(Generated by TriWild) (Generated by MATLAB)

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SLIDE 48

MATLAB vs TriWild

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Failure Input Original Input

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GMsh vs TriWild

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(Generated by GMsh) (Generated by TriWild)

19 inverted elements

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SLIDE 50

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Large-scale Test

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  • Dataset: 19,686 real-world SVG images from openclipart.org.
  • Success rate: 19,685 output meshes
  • The only failure is due to large input size (1.5GB).

Success: <6 hours <64GB memory

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9/5/19

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Large-scale Test

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0% 2% 4% 6% 1 100 10000

Timing (in seconds)

  • Dataset: 19,686 real-world SVG images from openclipart.org.
  • Success rate: 19,685 output meshes
  • The only failure is due to large input size (1.5GB).

10s

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Application – Diffusion Curves

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Application – Diffusion Curve

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Application – Inflation

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Curved Mesh Using Curved Output Using Linear Output

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Application – Inflation

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Curved Mesh Using Curved Output Using Linear Output

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Application – Stokes

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Using Curved Mesh

9/5/19

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Application – Stokes

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Using Linear Mesh

9/5/19

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Application – Stokes

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Application – Elasticity

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Gravity

Using Dense Linear Mesh Using Curved Mesh Curved Mesh

0.09s 115s

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Linear Pipeline

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Dataset 1: Raw OpenClip20k

  • Piece-wise linear approximations of 19,686 SVG images.

Dataset 2: Cleaned OpenClip20k

  • Raw dataset with duplication and degeneracy removed.

Dataset 3: Snapped OpenClip20k

  • Raw dataset with iteratively snap rounding using " as pixel size.
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Linear Pipeline

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Compared with: 1) CGAL Conforming/Constrained Delaunay Triangulation 2) Triangle Conforming/Constrained Delaunay Triangulation

Success: <1 hour <16GB memory

CGAL conforming CGAL constrained Triangle conforming Triangle constrained Ours

50% 100% 0 50% 100% 0 50% 100% 0 50% 100% 0 50% 100% raw cleaned snapped 99.97% 99.97% 99.97% 99.97% 99.80% 99.66% 83.09% 99.84% 99.57% 81.92% 99.20% 9.56% 19.01 % 99.93% 34.35%

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Success: <1 hour <16GB memory

Linear Pipeline

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Compared with: 1) CGAL Conforming/Constrained Delaunay Triangulation 2) Triangle Conforming/Constrained Delaunay Triangulation

CGAL conforming CGAL constrained Triangle conforming Triangle constrained Ours

50% 100% 0 50% 100% 0 50% 100% 0 50% 100% 0 50% 100% raw cleaned snapped 99.97% 99.97% 99.97% 100.0% 99.80% 99.66% 83.09% 99.84% 100.0% 81.92% 99.20% 9.56% 19.01 % 100.0% 34.35%

64GB

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Thank You!

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ACKNOWLEDGMENTS

  • This work was supported in part through the NYU IT High

Performance Computing resources, services, and staff expertise.

  • This work was partially supported by the NSF CAREER

award under Grant No. 1652515, the NSF grant IIS- 1320635, the NSF grant DMS-1436591, the NSF grant 1835712, the SNSF grant P2TIP2_175859, NSERC Discovery Grants (RGPIN-2017-05235 & RGPAS-2017- 507938), Canada Research Chair award, Connaught Fund, a gift from Adobe Research, and a gift from nTopology. https://github.com/wildmeshing/TriWild

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Thank You!

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https://github.com/wildmeshing/TriWild

  • Please scan the QR code for reference

implementation in C++ and data.

  • Python wrapper via Conda:
  • Contact me if you have any questions!

yixin.hu@nyu.edu

conda config --add channels conda-forge conda install wildmeshing