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2D/3D Shape Manipulation, 3D Printing

2D/3D Shape Manipulation, 3D Printing

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2D/3D Shape Manipulation, 3D Printing

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  1. CS 6501 2D/3D Shape Manipulation,3D Printing Digital Geometry Processing -- Mesh Smoothing

  2. Outlook – Topics • Smoothing • Parameterization • Remeshing Olga Sorkine-Hornung

  3. Surface Smoothing – Motivation • Scanned surfaces can be noisy Olga Sorkine-Hornung

  4. Surface Smoothing – Motivation • Scanned surfaces can be noisy Olga Sorkine-Hornung

  5. Surface Fairing – Motivation • Marching Cubes meshes can be ugly • Why is the left mesh ugly? • Why is the right mesh ugly? • What is the problem with such triangles? Olga Sorkine-Hornung

  6. Filtering Curves • Discrete second derivative: • In matrix-vector form for the whole curve Olga Sorkine-Hornung

  7. Filtering Curves • Gaussian filtering • Scale factor 0 <  < 1 • Matrix-vector form: Olga Sorkine-Hornung

  8. Filtering Curves • Gaussian filtering • Scale factor 0 <  < 1 • Drawbacks • ? Olga Sorkine-Hornung

  9. Filtering Curves • Gaussian filtering • Scale factor 0 <  < 1 • Drawbacks • Causes the curve/mesh to shrink; slow Olga Sorkine-Hornung

  10. Filtering Curves Original curve Olga Sorkine-Hornung

  11. Filtering Curves 1st iteration; =0.5 Olga Sorkine-Hornung

  12. Filtering Curves 2nd iteration; =0.5 Olga Sorkine-Hornung

  13. Filtering Curves 8th iteration; =0.5 Olga Sorkine-Hornung

  14. Filtering Curves 27th iteration; =0.5 Olga Sorkine-Hornung

  15. Filtering Curves 50th iteration; =0.5 Olga Sorkine-Hornung

  16. Filtering Curves 500th iteration; =0.5 Olga Sorkine-Hornung

  17. Filtering Curves 1000th iteration; =0.5 Olga Sorkine-Hornung

  18. Filtering Curves 5000th iteration; =0.5 Olga Sorkine-Hornung

  19. Filtering Curves 10000th iteration; =0.5 Olga Sorkine-Hornung

  20. Filtering Curves 50000th iteration; =0.5 Olga Sorkine-Hornung

  21. Recap: Laplace-Beltrami • High-pass filter: extracts local surface detail • Detail = smooth(surface) – surface • Smoothing = averaging Olga Sorkine-Hornung

  22. Recap: Laplace-Beltrami • The direction of iapproximates the normal • The size approximates the mean curvature  Olga Sorkine-Hornung

  23. L-B: Weighting Schemes • Ignore geometry • Integrate over Voronoiregion of the vertex duniform: Wi = 1, wij= 1/di  n dcotan: wij = 0.5(cotaij+ cotbij) aij bij Wi =Ai Olga Sorkine-Hornung

  24. LaplacianMatrix • The transition between xyzand  is linear: L x x = L y y = z z L = Olga Sorkine-Hornung

  25. LaplacianMatrix • Breaking down the Laplace matrix: • M = mass matrix; Lw = stiffness matrix L M–1 Lw = Olga Sorkine-Hornung

  26. Taubin Smoothing: Explicit Steps • Iterate: •  > 0to smooth;  < 0 to inflate • Originally proposed with uniform Laplacianweights 10 iterations original 50 iterations 200 iterations A Signal Processing Approach to Fair Surface DesignGabriel Taubin ACM SIGGRAPH 95 Olga Sorkine-Hornung

  27. Taubin Smoothing: Explicit Steps • Per-vertex iterations • Simple to implement • Requires many iterations • Need to tweak μand λ 10 iterations original 50 iterations 200 iterations A Signal Processing Approach to Fair Surface DesignGabriel Taubin ACM SIGGRAPH 95 Olga Sorkine-Hornung

  28. Example 20 iterations 5 iterations 0 iterations Olga Sorkine-Hornung

  29. Mesh Independence • Result of smoothing with uniform Laplacian depends on triangle density and shape • Why? • Asymmetric results although underlying geometry is symmetric original uniform cotan Olga Sorkine-Hornung

  30. Mesh Independence • Result of smoothing with uniform Laplacian depends on triangle density and shape • Why? • Asymmetric results although underlying geometry is symmetric original uniform cotan Olga Sorkine-Hornung

  31. Implicit Fairing: Implicit Euler Steps • In each iteration, solve for the smoothed x: Implicit fairing of irregular meshes using diffusion and curvature flowM. Desbrun, M. Meyer, P. Schroeder, A. Barr ACM SIGGRAPH 99 Olga Sorkine-Hornung

  32. Smoothing as (mean curvature) Flow • Model smoothing as a diffusion process • Discretize in time, forward differences: Olga Sorkine-Hornung

  33. Smoothing as (mean curvature) Flow • Model smoothing as a diffusion process • Discretize in time, forward differences: Explicit integration Unstable unless time step dt is small Olga Sorkine-Hornung

  34. Smoothing as (mean curvature) Flow • Model smoothing as a diffusion process • Backward Euler for unconditional stability Olga Sorkine-Hornung

  35. Implicit Fairing Implicit fairing of irregular meshes using diffusion and curvature flowM. Desbrun, M. Meyer, P. Schroeder, A. Barr ACM SIGGRAPH 99 Olga Sorkine-Hornung

  36. Implicit Fairing • The importance of using the right weights Mean curvature Mean curvature Uniform(umbrella) cotan Olga Sorkine-Hornung

  37. Laplacian Mesh Optimization • Smoothing, improving of triangle shapes • Basic idea: formulate a “shopping list”, solve with least squares Laplacian Mesh Optimization ACM GRAPHITE 2006 A. Nealen, T. Igarashi, O. Sorkine, M. Alexa Olga Sorkine-Hornung

  38. Laplacian Mesh Optimization • Smoothing, improving of triangle shapes • Basic idea: formulate a “shopping list”, solve with least squares • Smooth mesh: Lx = 0 • Passes close to input data set: x = x • Well-shaped triangles: Lunix = Lcotx • The terms are weighted according to importance and geometry Laplacian Mesh Optimization ACM GRAPHITE 2006 A. Nealen, T. Igarashi, O. Sorkine, M. Alexa Olga Sorkine-Hornung

  39. WP WP WL WL X Smoothing • Meshsmoothing L = Lcot (outer fairness) or L = Luni(outer and inner fairness) • Controlled by WP and WL(Intensity, Features) L x 0 Lx = 0 = x = x n aij bij Olga Sorkine-Hornung

  40. Using WP original w = 0.2 w = 0.02 Olga Sorkine-Hornung

  41. Using WP and WL test model with added noise without WL with WL Olga Sorkine-Hornung

  42. WP WP x Triangle Shape Optimization Luni dcot x = n n Lunix = Lcotx Olga Sorkine-Hornung

  43. Positional Weights Olga Sorkine-Hornung

  44. Constant Weights Olga Sorkine-Hornung

  45. Linear Weights Olga Sorkine-Hornung

  46. CDF Weights Olga Sorkine-Hornung

  47. Original Olga Sorkine-Hornung

  48. Triangle Shape Optimization Olga Sorkine-Hornung

  49. Thank You