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1 Mesh Smoothing 1

2 Mesh Processing Pipeline... Scan Reconstruct Clean Remesh 2

3 Mesh Quality Visual inspection of sensitive attributes Specular shading Flat Shading Gouraud Shading Phong Shading 3

4 Mesh Quality Visual inspection of sensitive attributes Specular shading 4

5 Mesh Quality Visual inspection of sensitive attributes Specular shading Reflection lines 5

6 Mesh Quality Visual inspection of sensitive attributes Specular shading Reflection lines differentiability one order lower than surface can be efficiently computed using graphics hardware 6

7 Mesh Quality Visual inspection of sensitive attributes Specular shading Reflection lines Curvature Mean curvature 7

8 Mesh Quality Visual inspection of sensitive attributes Specular shading Reflection lines Curvature Mean curvature Gaussian curvature 8

9 Motivation Filter out high frequency noise 9

10 Mesh Smoothing (aka Denoising, Filtering, Fairing) p Noisyy mesh ((scanned or other)) Input: Output: Smooth mesh How: Filter out high frequency noise 10

11 Smoothing by Filtering Fourier Transform f(x) =Σ sin(kx) Slides by Levy et al., SigAsia Course

12 Smoothing by Filtering Fourier Transform x = Slides by Levy et al., SigAsia Course

13 Smoothing by Filtering Fourier Transform Fourier Transform Filtering Convolution Geometric space Frequency space x Inverse Fourier Transform Slides by Levy et al., SigAsia Course

14 Filtering on a Mesh Filtering [Taubin 95] Geometric space Frequency space?? x Slides by Levy et al., SigAsia Course

15 Laplacian Smoothing An easier problem: How to smooth a curve? p i = (x i, y i ) p i-1 p i+1 (p i-1 + p i+1 )/2- p i 15

16 Laplacian Smoothing An easier problem: How to smooth a curve? p i = (x i, y i ) p i-1 p i+1 Finite difference discretization of second derivative = Laplace operator in one dimension 16

17 Laplacian Smoothing Algorithm: Repeat for m iterations (for non boundary points): For which λ? 0 < λ < 1 Closed curve converges to? Single point 17

18 Spectral Analysis Closed Curve Re-write in matrix notation: 18

19 The Eigenvectors of L L= VDV T 19

20 Spectral Analysis Then: After m iterations: Can be described d using eigendecomposition of L Filtering high frequencies 20

21 Spectral Analysis Laplacian Smoothing k i ( ) m k i 21

22 Laplacian Smoothing on Meshes Same as for curves: N i = {k,l,m,n} lmn} p i = (x i, y i, z i ) What is Δp i? p m p l p n p k 22

23 Laplacian Smoothing on Meshes 0 Iterations 5 Iterations 20 Iterations 23

24 Problem - Shrinkage Repeated iterations of Laplacian smoothing shrinks the mesh original 3 steps 6 steps 18 steps original 24

25 Taubin Smoothing Iterate: with λ > 0 and μ < 0 Shrink Inflate original 10 steps 50 steps 200 steps From Taubin, Siggraph

26 Spectral Analysis Taubin Smoothing f k = 1 λk 1 μk ( ) ( )( ) i i i From Taubin, Siggraph

27 Laplacian Smoothing Δp i = mean curvature normal mean curvature flow 27

28 Laplace Operator Discretization The Problem Sanity check what should happen if the mesh lies in the plane: p i = (x i, y i, 0)? 0 Iterations 5 Iterations 28

29 Laplace Operator Discretization The Problem Not good A flat mesh is smooth, should stay the same after smoothing 0 Iterations 5 Iterations 29

30 Laplace Operator Discretization The Problem Not good The result should not depend on triangle sizes From Desbrun et al., Siggraph

31 Laplace Operator Discretization What Went Wrong? Back to curves: p i-1 p i p i+1 Same weight for both neighbors, although one is closer 31

32 Laplace Operator Discretization The Solution Use a weighted average to define Δ Which h weights? p j l ij p i l ik p k Straight curves will be invariant to smoothing 32

33 Laplace Operator Discretiztion Cotangent Weights Use a weighted average to define Δ Which h weights?? N i = {k,l,m,n} p i p i l p i ij p j l ij p j h 2 h 1 p n p m p k p l 33

34 Laplace Operator Discretiztion Cotangent Weights Use a weighted average to define Δ Which weights? N i = {k,l,m,n} p i p i p j α ij β ij p n p m p l w h + h 1 2 ij ij 1 ij = = cot αij + cot βij lij 2 ( ) p k Planar meshes will be invariant to smoothing 34

35 Smoothing with the Cotangent t Laplacian normal and tangential movement normal movement original Uniform weights Cotangent weights From Desbrun et al., Siggraph

36 Filtering on a Mesh Filtering [Taubin 95] Geometric space Frequency space?? x Slides by Levy et al., SigAsia Course

37 The Eigenvectors of L L= VDV T 37

38 Spectral Analysis Cotangent Laplacian Demo v 2 v 50 From Vallet et al., Eurographics

39 Smoothing using the Laplacian Eigen-decomposition From Vallet et al., Eurographics

40 Geometry Filtering Demo: From Vallet et al., Eurographics

41 Surface Fairing Find surfaces which are as smooth as possible Applications Smooth blends Hole filling 41

42 Fairness Idea: Penalize unaesthetic behavior Measure fairness Principle of the simplest shape Physical interpretation Minimize some fairness functional Surface area, curvature Membrane energy, thin plate energy 42

43 Energy Functionals Membrane Surface Thin Plate Surface Minimum Variation Surface 43

44 Non-Linear Energies Membrane energy (surface area) Thin-plate energy (curvature) Too complex... simplify energies 44

45 Membrane Surfaces Linearized Energy Surface parameterization Membrane energy (surface area) Variational calculus 45

46 Thin-Plate Surfaces Linearized Energy Surface parameterization Thin-plate energy (curvature) Variational calculus 46

47 Fair Surfaces Membrane Thin Plate Demo Minimal Curvature Variation 47

48 Exercise Smoothing Uniform Cotangent formula Smoothness visualization Mean curvature (uniform, weighted) Gaussian curvature Triangle shape 48

49 Exercise Smoothing Uniform Laplace-Beltrami 49

50 Exercise Smoothing Cotangent Formula (simplified) 50

51 Exercise Smoothing Uniform Laplace-Beltrami Cotangent Formula Move vertices in parallel! 51

52 Exercise Curvature Mean Curvature Gaussian Curvature (simplified) 52

53 Exercise Triangle Shape circumradius vs. minimal edge length 53

54 Exercise Triangle Shape Smoothing 54

55 References A Signal Processing Approach to Fair Surface Design, Taubin, Siggraph 95 Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, Desbrun et al., Siggraph 99 An Intuitive Framework for Real-Time Freeform Modeling, Botsch et al., Siggraph 04 Spectral Geometry Processing with Manifold Harmonics, Vallet et al., Eurographics 08 55

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