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Bounded Distortion Mapping and Shape Deformation - - PowerPoint PPT Presentation

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Bounded Distortion Mapping and Shape Deformation GAMES Web Seminar, 29 March 2018 Outline Planar Mapping & Applications Bounded Distortion Mapping Harmonic Shape Deformation

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Bounded Distortion Mapping and Shape Deformation

陈仁杰 德国马克斯普朗克计算机研究所

GAMES Web Seminar, 29 March 2018

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Outline

  • Planar Mapping & Applications
  • Bounded Distortion Mapping
  • Harmonic Shape Deformation
  • Shape Interpolation

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Source Target

Mapping – between planar shapes

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Mapping – between images

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Applications – keyframe animations

  • Model key poses/frames
  • Fill in between key poses

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Applications – image editing

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Content-aware resizing Aesthetic optimization Re-photography

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Applications - virtual reality

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[Sun et al. 2016] [Dong et al. 2017]

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Outline

  • Planar Mapping & Applications
  • Bounded Distortion Mapping
  • Harmonic Shape Deformation
  • Shape Interpolation

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Mapping Distortions – intuitive view

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Fold over

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Planar map – notations

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？

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Mapping Distortions – formal definitions

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1 1

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Mapping – discretization

  • Triangle mesh
  • Piecewise linear mapping
  • Polygonal cage
  • Smooth barycentric mapping
  • Polygonal mesh + barycentric mapping

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[Chen & Gotsman 2017]

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Affine

  • Wachspress
  • Cot/Laplace
  • Mean-value
  • Harmonic
  • MLS
  • …

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Barycentric Mapping a polygon

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Image resizing using hexmesh

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Distortion minimization Bounded Distortion

Bounded Distortion Mapping

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Outline

  • Planar Mapping & Applications
  • Bounded Distortion Mapping
  • Harmonic Shape Deformation
  • Shape Interpolation

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Deformation – desirable properties

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  • Mesh-based
  • Extremal quasiconformal maps [Weber et al. 2012]
  • Bounded distortion mapping spaces [Lipman 2012]
  • Locally injective mappings [Schüller et al. 2013]
  • Locally injective parameterization [Weber & Zorin 2014]
  • Planar shape interpolation with bounded distortion [Chen et al. 2013]
  • Meshless
  • Generalized barycentric coordinates
  • Controllable conformal maps [Weber & Gotsman 2010]
  • Provably good planar maps [Poranne & Lipman 2014]

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✘ not locally injective ✘ no distortion bounds ✘ no positional constraints

✔ smooth

Deformation – previous work

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Piecewise linear map

  • Non-smooth
  • Pointwise (facewise) constraints
  • Sparse (large) linear algebra

Harmonic map

  • Smooth
  • Boundary constraints
  • Dense (small) linear algebra

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Mapping Space for Deformation

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Harmonic Planar Mapping

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  • Bound the distortion at every point
  • Harmonic - Boundary only?

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Bounded Distortion Harmonic Mapping

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Bounded Distortion Theorem

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[Chen & Weber 2015]

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Bounded Distortion Theorem

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[Chen & Weber 2017]

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Convexification [Lipman 2012]

Harmonic Shape Deformation

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Harmonic Mapping Space

= +

Holomorphic

(complex analytic)

Anti- Holomorphic Harmonic

Reflection

Holomorphic

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Cauchy Complex Barycentric Coordinate

cage

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Harmonic Deformation Results

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Source Harmonic

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An Interactive Session

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Convexification [Lipman 2012]

  • Iterative convexification
  • Conic optimization
  • User-specified bounds
  • Feasibility

[Chen & Weber 2017]

  • Newton’s method
  • GPU acceleration
  • Smooth isometric energy
  • Automatic distortion bounds
  • Unconstrained optimization

[Chen & Weber 2015]

Deformation – faster/better optimization?

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Newton’s Method

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1 1

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Isometric Energy

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1 1

Locally Injective Harmonic Deformation

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Locally Injective Harmonic Deformation

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Per-element SPD Hessian

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Composite Majorization

April 4, 2018 35

[Shtengel et al. 2017]

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Newton iteration on GPU

cuBLAS cuSolver

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[Chen & Weber 2015] [Chen & Weber 2017] Input

3.71s / 28 it 0.03s / 8 it

Results & Comparison

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5 10 15 0 200 400 5 50 100

0.11s GPU

1.18s CPU

4.32s 5.49s

Results & Comparison

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Results & Comparison

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Outline

  • Planar Mapping & Applications
  • Bounded Distortion Mapping
  • Harmonic Shape Deformation
  • Shape Interpolation

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Input Output

bounded conformal distortion Square edge length [Chen et al 2013]

Metric Interpolation

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= +

Holomorphic Anti- Holomorphic Harmonic

Harmonic Interpolation

[Chien et al 2016]

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Harmonic Interpolation

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Interpolation comparison

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4/4/2018 Low distortion geometry modeling 48

Harmonic mapping animation

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Conclusion

  • Planar Mapping
  • Distortions
  • Discretization
  • Harmonic Mapping
  • Bounded distortion theorem
  • Deformation
  • Interpolation

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[1] R.Chen, O. Weber. GPU-Accelerated Locally Injective Shape Deformation SIGGRAPH Asia, 2017 [2] R. Chen, C. Gotsman. Approximating Planar Conformal Maps Using Regular Polygonal Meshes CGF 2017 [3] E. Chien, R. Chen*, O. Weber. Bounded Distortion Harmonic Shape Interpolation SIGGRAPH 2016 [4] R. Chen, C. Gotsman. Generalized As-Similar-As-Possible Warping with Applications in Digital Photography Eurographics 2016 [5] R. Chen, O. Weber. Bounded Distortion Harmonic Mappings in the Plane SIGGRAPH 2015 [6] R. Chen, O. Weber, D. Keren, M. Ben-Chen. Planar Shape Interpolation with Bounded Distortion SIGGRAPH 2013

References

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Code available http://people.mpi-inf.mpg.de/~chen/