Bounded Distortion Mapping and Shape Deformation
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1 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
2 Outline Planar Mapping & Applications Bounded Distortion Mapping Harmonic Shape Deformation Shape Interpolation 2
3 Mapping between planar shapes Source Target
4 Mapping between images 4
5 Applications keyframe animations Model key poses/frames Fill in between key poses 5
6 Applications image editing Aesthetic optimization Content-aware resizing Re-photography 6
7 Applications - virtual reality [Sun et al. 2016] [Dong et al. 2017] 7
8 Outline Planar Mapping & Applications Bounded Distortion Mapping Harmonic Shape Deformation Shape Interpolation 8
9 Mapping Distortions intuitive view Fold over 9
10 Planar map notations? 10
11 Mapping Distortions formal definitions
12 Mapping discretization Triangle mesh Piecewise linear mapping Polygonal cage Smooth barycentric mapping Polygonal mesh + barycentric mapping [Chen & Gotsman 2017] 12
13 Barycentric Mapping a polygon Affine Wachspress Cot/Laplace Mean-value Harmonic MLS 13
14 Image resizing using hexmesh
15 Bounded Distortion Mapping Distortion minimization Bounded Distortion
16 Outline Planar Mapping & Applications Bounded Distortion Mapping Harmonic Shape Deformation Shape Interpolation 16
17 Deformation desirable properties 17
![2012] Bounded distortion mapping spaces [Lipman 2012]](/docs-images/93/111174697/images/18-3.jpg "Locally injective mappings [Schüller et al.")
![2013] Locally injective parameterization [Weber &](/docs-images/93/111174697/images/18-4.jpg "Zorin 2014] Planar shape interpolation with bounded")
18 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] Meshless smooth Generalized barycentric coordinates not locally injective no distortion bounds Controllable conformal maps [Weber & Gotsman 2010] Provably good planar maps [Poranne & Lipman 2014] no positional constraints 18
19 Mapping Space for Deformation Piecewise linear map Non-smooth Pointwise (facewise) constraints Sparse (large) linear algebra Harmonic map Smooth Boundary constraints Dense (small) linear algebra 19
20 Harmonic Planar Mapping 20
21 Bounded Distortion Harmonic Mapping Bound the distortion at every point Harmonic - Boundary only? 21
22 Bounded Distortion Theorem [Chen & Weber 2015] 22
23 Bounded Distortion Theorem [Chen & Weber 2017] 23
24 Harmonic Shape Deformation Convexification [Lipman 2012] 24
25 Holomorphic Harmonic Mapping Space = + Reflection Harmonic Holomorphic (complex analytic) Anti- Holomorphic 25
26 Cauchy Complex Barycentric Coordinate cage 26
27 Harmonic Deformation Results Harmonic Source 27
28 An Interactive Session 28
29 Deformation faster/better optimization? [Chen & Weber 2015] Iterative convexification Conic optimization User-specified bounds Feasibility [Chen & Weber 2017] Newton s method GPU acceleration Smooth isometric energy Automatic distortion bounds Unconstrained optimization Convexification [Lipman 2012] 29
30 Newton s Method 30
31 Isometric Energy
32 Locally Injective Harmonic Deformation
33 Locally Injective Harmonic Deformation 33
34 Per-element SPD Hessian 34
35 Composite Majorization [Shtengel et al. 2017] April 4,
36 Newton iteration on GPU cublas cusolver 36
37 Results & Comparison 3.71s / 28 it 0.03s / 8 it Input [Chen & Weber 2015] [Chen & Weber 2017] 37
38 Results & Comparison s GPU 1.18s CPU s s 38
39 39
40 Results & Comparison 40
41 41
42 42
43 Outline Planar Mapping & Applications Bounded Distortion Mapping Harmonic Shape Deformation Shape Interpolation 43
44 Metric Interpolation [Chen et al 2013] bou Input Square edge length n o i t r o t s i d l a m r o f n o c d nde Output
45 Harmonic Interpolation [Chien et al 2016] = + Harmonic Holomorphic Anti- Holomorphic 45
46 Harmonic Interpolation 46
47 Interpolation comparison 47
48 Harmonic mapping animation 4/4/2018 Low distortion geometry modeling 48
49 Conclusion Planar Mapping Distortions Discretization Harmonic Mapping Bounded distortion theorem Deformation Interpolation 49
50 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
51 Code available 51
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