1.7.1 Laplacian Smoothing
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1 1.7.1 Laplacian Smoothing : Advanced Graphics - Chapter 1 434
2 Theory Minimize energy functional total curvature estimate by polynomial-fitting non-linear (very slow!) : Advanced Graphics - Chapter 1 435
3 Theory Minimize energy functional membrane or thin-plate energy derivative is Laplacian : Advanced Graphics - Chapter 1 436
4 Discrete Laplacian for w ij =1/n: umbrella : Advanced Graphics - Chapter 1 437
5 Laplacian smoothing flow P new P old + λl ( Pold) L Pold Average of the vectors to neighboring vertices Move each vertex Pold Pnew λl Q k 1 Q k Median direction Q k : Advanced Graphics - Chapter 1 438
6 Laplacian smoothing Move each vertex by the given formula Iterate the process for further smoothing : Advanced Graphics - Chapter 1 439
7 Regular diffusion Problem with umbrella operator does not distinguish these cases: P Q j umbrella assumes regular parameterization : Advanced Graphics - Chapter 1 440
8 Umbrella problem Initial mesh Umbrella smoothing : Advanced Graphics - Chapter 1 441
9 Volume preservation Diffusion induces shrinkage! Enforce exact volume preservation Compute volume Rescale after integration step(s) Incorporate other invariants Volume, surface area,... Build directly into PDE : Advanced Graphics - Chapter 1 442
10 1.7.2 Non-shrinking Laplacian : Advanced Graphics - Chapter 1 443
11 Fourier analysis : Advanced Graphics - Chapter 1 444
12 Fourier analysis : Advanced Graphics - Chapter 1 445
13 Fourier analysis : Advanced Graphics - Chapter 1 446
14 Fourier analysis : Advanced Graphics - Chapter 1 447
15 Fourier analysis : Advanced Graphics - Chapter 1 448
16 Fourier analysis : Advanced Graphics - Chapter 1 449
17 Non-shrinking Laplacian : Advanced Graphics - Chapter 1 450
18 Alternative: Length-scaled Laplacian Length-scale weighted operator: L P 1 ( M ) = l scale-dependent operator But: stability issues j P tiny explicit time step: dt < l min2 / λ implicit integration a must! j Q j l j (Fujiwara operator) : Advanced Graphics - Chapter 1 451
19 Non-shrinking Umbrella smoothing Scale-dependent smoothing : Advanced Graphics - Chapter 1 452
20 Solver Explicit Euler iterations: (Diffusion Eqn) M& = λ L(M ) M = M + λ L( M ) dt t + dt t t Strict stability requirements Small time steps : Advanced Graphics - Chapter 1 453
21 Stability Explicit Euler scheme vs. Implicit Euler scheme y ( t + dt) = y( t) + y& ( t) dt y ( t + dt) = y( t) + y& ( t + dt) dt : Advanced Graphics - Chapter 1 454
22 Explicit solver Explicit integration: M n+ = M + λdt L( M 1 n M ( ) = I + λdtl M n+ 1 n n ) Small time steps for large meshes : Advanced Graphics - Chapter 1 455
23 Implicit solver Implicit integration: M = M + λdt L( M 1 n n+ n+ 1 ( I λdtl) M = M n +1 n ) Conjugate Gradient for efficiency (Solve Ax = b by iterating with ) Large time steps : Advanced Graphics - Chapter 1 456
24 Signal analysis Benefit of implicit scheme evident in transfer functions explicit implicit : Advanced Graphics - Chapter 1 457
25 Results Initial mesh : Advanced Graphics - Chapter 1 458
26 Results smoothed mesh : Advanced Graphics - Chapter 1 459
27 Shape preservation Preventing tangential shift : Advanced Graphics - Chapter 1 460
28 1.7.3 Shape-preserving Smoothing : Advanced Graphics - Chapter 1 461
29 Noise components Shape denoising Normal think shape Tangent think parameterization Alternative to diffusion: curvature flow equation (Laplace-Beltrami) : Advanced Graphics - Chapter 1 462
30 Curvature flow Replace Laplacian operator with Laplace-Beltrami operator. Proceed as before : Advanced Graphics - Chapter 1 463
31 Comparison Initial Mesh Regular Diffusion Improved Diffusion Curvature Flow : Advanced Graphics - Chapter 1 464
32 Results on 3D Scanned Data Initial mesh After one fairing : Advanced Graphics - Chapter 1 465
33 Results on 3D Scanned Data : Advanced Graphics - Chapter 1 466
34 Curvature operator Curvature visualization using false colors Low curvature High curvature : Advanced Graphics - Chapter 1 467
35 Constraint enforcement Fixed points/fixed regions: Set the Laplacian to zero Soft constraints: Locally adjust smoothing amount (λ) : Advanced Graphics - Chapter 1 468
36 1.8 Summary : Advanced Graphics - Chapter 1 469
37 Modeling Object representation: Meshes Subdivision Surfaces - Arbitrary Meshes Mesh data structures - Half-edge structure etc. Multiresolution modeling: (LOD, local coefficients, filtering, editing) Wavelets - Simplification - Progressive Meshes -Normal Meshes Mesh Parametrization -Planar mappings - Non-planar mappings -MAPS Curves on Surfaces Smoothing : Advanced Graphics - Chapter 1 470
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