Bounded Distortion Mapping and Shape Deformation - PowerPoint PPT Presentation

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

Outline • Planar Mapping & Applications • Bounded Distortion Mapping • Harmonic Shape Deformation • Shape Interpolation 2

Mapping – between planar shapes Source Target

Mapping – between images 4

Applications – keyframe animations • Model key poses/frames • Fill in between key poses 5

Applications – image editing Aesthetic optimization Content-aware resizing Re-photography 6

Applications - virtual reality [Sun et al. 2016] [Dong et al. 2017] 7

Outline • Planar Mapping & Applications • Bounded Distortion Mapping • Harmonic Shape Deformation • Shape Interpolation 8

Mapping Distortions – intuitive view Fold over 9

Planar map – notations ？ 10

Mapping Distortions – formal definitions 1 1 11

Mapping – discretization • Triangle mesh • Piecewise linear mapping • Polygonal cage • Smooth barycentric mapping • Polygonal mesh + barycentric mapping 12 [Chen & Gotsman 2017]

Barycentric Mapping a polygon Affine • Wachspress • Cot/Laplace • Mean-value • Harmonic • MLS • … 13

Image resizing using hexmesh

Bounded Distortion Mapping Distortion minimization Bounded Distortion

Outline • Planar Mapping & Applications • Bounded Distortion Mapping • Harmonic Shape Deformation • Shape Interpolation 16

Deformation – desirable properties 17

Deformation – previous work • 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] ✔ smooth • Meshless ✘ not locally injective ✘ no distortion bounds • Generalized barycentric coordinates ✘ no positional constraints • Controllable conformal maps [Weber & Gotsman 2010] • Provably good planar maps [Poranne & Lipman 2014] 18

Mapping Space for Deformation Piecewise linear map Harmonic map • Non-smooth • Smooth • Pointwise (facewise) constraints • Boundary constraints • Sparse ( large ) linear algebra • Dense ( small ) linear algebra 19

Harmonic Planar Mapping 20

Bounded Distortion Harmonic Mapping • Bound the distortion at every point • Harmonic - Boundary only? 21

[Chen & Weber 2015] Bounded Distortion Theorem 22

[Chen & Weber 2017] Bounded Distortion Theorem 23

Harmonic Shape Deformation Convexification [Lipman 2012] 24

Holomorphic Harmonic Mapping Space Reflection = + Anti- Harmonic Holomorphic Holomorphic (complex analytic) 25

Cauchy Complex Barycentric Coordinate cage 26

Harmonic Deformation Results Source Harmonic 27

An Interactive Session 28

Deformation – faster/better optimization? [Chen & Weber 2015] [Chen & Weber 2017] • Iterative convexification • Newton’s method • Conic optimization • GPU acceleration • User-specified bounds • Smooth isometric energy • Feasibility • Automatic distortion bounds • Unconstrained optimization Convexification [Lipman 2012] 29

Newton’s Method 30

Isometric Energy 1 1 31

Locally Injective Harmonic Deformation 1 1 32

Locally Injective Harmonic Deformation 33

Per-element SPD Hessian 34

[Shtengel et al. 2017] Composite Majorization April 4, 2018 35

Newton iteration on GPU cuBLAS cuSolver 36

Results & Comparison 3.71s / 28 it 0.03s / 8 it Input [Chen & Weber 2015] [Chen & Weber 2017] 37

Results & Comparison 0.11s GPU 5 10 15 1.18s CPU 0 200 400 4.32s 5.49s 5 50 100 38

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Results & Comparison 40

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Outline • Planar Mapping & Applications • Bounded Distortion Mapping • Harmonic Shape Deformation • Shape Interpolation 43

Metric Interpolation [Chen et al 2013] bounded conformal distortion Input Output Square edge length

Harmonic Interpolation [Chien et al 2016] = + Anti- Holomorphic Harmonic Holomorphic 45

Harmonic Interpolation 46

Interpolation comparison 47

Harmonic mapping animation 4/4/2018 Low distortion geometry modeling 48

Conclusion • Planar Mapping • Distortions • Discretization • Harmonic Mapping • Bounded distortion theorem • Deformation • Interpolation 49

References [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 50

Code available http://people.mpi-inf.mpg.de/~chen/ 51