Smoothing & Fairing. Mario Botsch

Transcription

1 Smoothing & Fairing Mario Botsch

2 Motivation Filter out high frequency noise Desbrun, Meyer, Schroeder, Barr: Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 99 2

3 Motivation Filter out high frequency noise 3

4 Motivation Advanced Filtering Original Low-Pass Exaggerate Kim, Rosignac: Geofilter: Geometric Selection of Mesh Filter Parameters, Eurographics 05 4

5 Motivation Fair Surface Design Schneider, Kobbelt: Geometric fairing of irregular meshes for free-form surface design, CAGD 18(4),

6 Motivation Hole-filling with energy-minimizing patches Liepa: Filling Holes in Meshses, SGP

7 Motivation Demo 7

8 Smoothing & Fairing

9 Smoothing How would you smooth a 1D/2D signal? 9

10 Diffusion Flow diffusion constant f t = λ f Laplace operator 10

11 Diffusion Flow 2nd order elliptic y, 2 f(x, y, 2 f(x, y, 2 Solve numerically Discretize in space & time Discretize time derivative Discretize spatial derivatives 11

12 Discretize in Space & Time Sample function f(x,y,t) on a regular grid Grid spacing h, time step δt f[i, j, t] = f (i h, j h, t t), y h i =1,...,n, j =1,..., m, t =0, 1, 2,... f[i,j,t] h x 12

13 Finite Differences Approximate f(x+h) from Taylor series f(x + h) = f(x)+hf 0 (x)+ h2 f(x)+hf 0 (x) 2! f 00 (x)+... Brook Taylor ( ) Approximate of f (x) f 0 (x) f(x + h) f(x) h 13

14 Finite Differences Approximate f(x-h) from Taylor series f(x h) = f(x) hf 0 (x)+ h2 2! f 00 (x)+... f(x) hf 0 (x) Brook Taylor ( ) Approximate of f (x) f 0 (x) f(x) f(x h) h 14

15 Finite Difference Method Approximation of spatial derivatives f x [i, j, t] f[i +1,j,t] f[i, j, t] h forward differences f x [i, j, t] f[i, j, t] f[i 1,j,t] h backward differences f x [i, j, t] f[i +1,j,t] f[i 1,j,t] 2h central differences 15

16 Finite Difference Method Approximation of higher-order derivatives f xx [i, j, t] f x[i, j, t] f x [i 1,j,t] h = f[i +1,j,t] 2f[i, j, t]+f[i 1,j,t] h 2 Approximation of Laplacian f[i, j, t] f xx [i, j, t]+f yy [i, j, t] = f[i +1,j,t]+f[i 1,j,t]+f[i, j +1,t]+f[i, j 1,t] 4f[i, j, t] h 2 16

17 Finite Differences Approximate f(t+δt) from Taylor series f(t + t) = f(t)+ t f(t)+ f(t)+ t f(t) t2 2! f(t)+... Brook Taylor ( ) Explicit Euler time integration f(t + t) f(t)+ t f(t) Leonhard Euler ( ) 17

18 Finite Difference Method Approximation of temporal derivative f t [i, j, t] f[i, j, t + 1] f[i, j, t] t 18

19 Diffusion Flow Continuous y, 2 f(x, y, 2 f(x, y, 2 Finite difference discretization f[i, j, t + 1] f[i, j, t] t = f[i +1,j,t]+f[i 1,j,t]+f[i, j +1,t]+f[i, j 1,t] 4f[i, j, t] h 2 f[i, j, t + 1] = f[i, j, t]+ t f[i +1,j,t]+f[i 1,j,t]+f[i, j +1,t]+f[i, j 1,t] 4f[i, j, t] h 2 δtλ has to be small (<=1 in this case) 19

20 Diffusion in 2D Demo 20

21 Diffusion Flow on = p δλ has to be small (~0.5 in exercises) Discretization p i p i + t p i 0 Iterations 10 Iterations 100 Iterations 21

22 Laplace Discretization Laplace discretization p i = P 1 v j 2N 1 (v i ) w ij X v j 2N 1 (v i ) w ij p j p i Uniform Laplace w ij =1 Note: No Voronoi area Ai for explicit smoothing Cotangent Laplace w ij = cot ij + cot ij ij v i ij v j 22

23 Comparison Original Uniform Laplace Cotan Laplace 23

24 Further Topics Implicit time integration p i (t + 1) = p i (t)+ t p i (t + 1) = ( 1) k+1 k p Boundary constraints Nonlinear flows, anisotropic flows 24

25 Smoothing & Fairing

26 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 26

27 Nonlinear Energies Membrane energy (surface area) S da min with S = c Thin-plate surface (curvature) S da min with S = c, n( S) =d Too complex... simplify energies 27

28 Membrane Surfaces Membrane energy (surface area) p u 2 + p v 2 dudv min 28

29 Variational Calculus in 1D 1D membrane energy L(f) = a b f 2 (x) dx min Add test function u with u(a) = u(b) = 0 L(f + u) = a b (f + u ) 2 = a b f f u + 2 u 2 If f minizes L, the following has to vanish L(f + u) =0 = b a 2f u! = 0 29

30 Variational Calculus in 1D Has to vanish for any u with u(a) = u(b) = 0 a b f u = [f u] b a =0 a b f u! = 0 u 0 1 f g = [fg] fg Only possible if f = f = 0 Euler-Lagrange equation 30

31 Membrane Surfaces Membrane energy (surface area) p u 2 + p v 2 dudv min Variational calculus p = 0 31

32 Thin-Plate Surfaces Thin-plate energy (curvature) p uu p uv 2 + p vv 2 dudv min Variational calculus 2 p = 0 32

33 Laplace Discretization Original Uniform Laplace Cotan Laplace 33

34 Smoothing Fairing Fair minimizer surfaces satisfy k p =0 Hence they are stationary points of = ( 1) k+1 k p Diffusion flow converges to fair surfaces k k 2 k k 2 k k k 2 dudv! min Explicit time integration corresponds to an iterative solution of the Euler-Lagrange equation 34

35 Literature Laplacian flow, curvature flow The Book: Chapter 4 Taubin, A Signal Processing Approach to Fair Surface Design, SIGGRAPH 1995 Desbrun, Meyer, Schröder, Barr: Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH

36 Literature Nonlinear smoothing Bobenko & Schröder: Discrete Willmore Flow, SGP 2005 Anisotropic smoothing Bajaj & Xu: Anisotropic Diffusion of Surfaces and Functions of Surfaces, ACM Trans. on Graphics 22(1), 2003 Hildebrandt & Polthier, Anisotropic Filtering of Nonlinear Surface Feature, Eurographics 2004 Bilaterial smoothing Fleishman et al, Bilateral Mesh Denoising, SIGGRAPH 2003 Jones et al, Non-iterative Feature-Preserving Mesh Smoothing, SIGGRAPH