Mesh Processing Pipeline

Transcription

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) Input: Noisy mesh (scanned or other) Output: Smooth mesh How: Filter out high frequency noise 10

11 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

12 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

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

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

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

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

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

18 Laplacian Smoothing p i = mean curvature normal mean curvature flow 27

19 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

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

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

22 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

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

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

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

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

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

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

29 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

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

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

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

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

34 Fair Surfaces Membrane Thin Plate Demo Minimal Curvature Variation 47