Local Barycentric Coordinates

Transcription

1 Local Barycentric Coordinates Juyong Zhang Bailin Deng Zishun Liu Giuseppe Patanè Sofien Bouaziz Kai Hormann Ligang Liu USTC EPFL USTC CNR-IMATI EPFL USI USTC

2 Introduction Given a point p inside a polygon with vertices {c i } c 1 c 2 p c 5 c 3 c 4 2

3 Introduction Given a point p inside a polygon with vertices {c i } {w i } Barycentric coordinates of : X p p = X i w i c i, i w i =1 c 1 c 2 p c 5 c 3 c 4 3

4 Introduction Given a point p inside a polygon with vertices {c i } {w i } Barycentric coordinates of : X p p = X i w i c i, i w i =1 c 1 functions inside the polygon c 2 p c 5 c 3 c 4 4

5 Introduction Application: interpolation c 1 c 2 c 5 c 3 c 4 5

6 Introduction Application: interpolation c 1 c 2 c 5 c 3 c 4 6

7 Introduction Application: interpolation f(x) = X i w i (x) f(c i ) interpolated value values at vertices c 1 c 2 x c 5 c 3 c 4 7

8 Introduction Application: interpolation f(x) = X i w i (x) f(c i ) barycentric coordinates c 2 c 1 x c 5 c 3 c 4 8

9 Introduction Cage based deformation cage vertex deformation d(x) = X i w i (x) d(c i ) deformation field inside the cage 9

10 Global Deformation Global influence Mean Value Coordinates (MVC) 10

11 Our Goal: Local Control Control points influence nearby regions only 11

12 Problem Formulation Input: control cage with vertices {c i } c 1 c 2 c 5 c 3 c 4 12

13 Problem Formulation Input: control cage with vertices {c i } Output: barycentric coordinate functions {w i (x)} with local influence c 1 c 2 c 5 c 3 c 4 13

14 Previous Work Poisson-based Weight Reduction [Landreneau & Schaefer 2009] 14

15 Previous Work Poisson-based Weight Reduction [Landreneau & Schaefer 2009] *Cages [García et al. 2013] 15

16 Previous Work Poisson-based Weight Reduction [Landreneau & Schaefer 2009] *Cages [García et al. 2013] Bounded Biharmonic Weights (BBW) [Jacobson et al. 2011] 16

17 Optimization Approach min F (w 1,...,w n ) subject to some constraints: w 1,...,w n X n i=1 w i(x) c i = x, w i 0 X n i=1 w i(x) =1, 8 x w i (c j )= 1, if i = j 0, otherwise w i linear on cage edges 17

18 Optimization Approach min F (w 1,...,w n ) subject to some constraints: w 1,...,w n X n i=1 w i(x) c i = x, w i 0 X n i=1 w i(x) =1, 8 x w i (c j )= 1, if i = j 0, otherwise w i linear on cage edges 18

19 Optimization Approach min F (w 1,...,w n ) subject to some constraints: w 1,...,w n X n i=1 w i(x) c i = x, w i 0 X n i=1 w i(x) =1, 8 x w i (c j )= 1, if i = j 0, otherwise w i linear on cage edges 19

20 Optimization Approach min F (w 1,...,w n ) subject to some constraints: w 1,...,w n X n i=1 w i(x) c i = x, w i 0 X n i=1 w i(x) =1, 8 x w i (c j )= 1, if i = j 0, otherwise w i linear on cage edges 20

21 Optimization Approach min F (w 1,...,w n ) subject to some constraints: w 1,...,w n Convex functional inducing locality 21

22 Local Influence Function w i for control vertex c i w i =0 w i > 0 c i 22

23 Condition for the Gradient Function w i for control vertex c i rw i = 0 23

24 Condition for the Gradient Necessary condition: large region with zero gradient rw i = 0 24

25 Condition for the Gradient Necessary condition: large region with zero gradient Z min rw i (x) dx rw i = 0 25

26 Condition for the Gradient Necessary condition: large region with zero gradient Z min rw i (x) dx Total variation of : convex functional w i rw i = 0 26

27 Condition for the Gradient Target functional: nx Z F = rw i (x) dx i=1 rw i = 0 27

28 Comparison MVC HBC Ours 28

29 Weighted Total Variation Extension: more locality using weighted total variation 1 O 1 c i unweighted weighted 29

30 Controlling Locality Local influence: w i decreases to zero quickly w i =0 w i > 0 c i 30

31 Controlling Locality Local influence: w i decreases to zero quickly Requires large gradients! w i =0 w i > 0 c i 31

32 Controlling Locality Total variation: Z rw i (x) dx w i =0 w i > 0 c i 32

33 Controlling Locality Total variation: Z rw i (x) dx Same penalty everywhere w i =0 w i > 0 c i 33

34 Controlling Locality Weighted total variation: Z i(x) rw i (x) dx w i =0 w i > 0 c i 34

35 Controlling Locality Weighted total variation: Z i(x) rw i (x) dx Monotonically increasing w.r.t. geodesic distance to cage vertex w i =0 w i > 0 c i 35

36 Comparison 1 O 1 c i i(x) =1 i(x) =[d i (x)] 2 36

37 Geometry of Total Variation 37

38 Geometry of Total Variation Scalar function w defined on domain 38

39 Geometry of Total Variation Scalar function w defined on domain Superlevel set of s w>s 39

40 Geometry of Total Variation Scalar function w defined on domain Perimeter: P (w >s) w>s 40

41 Geometry of Total Variation Coarea formula: Z Z +1 rw i (x) dx = 1 P (w >s) ds Perimeter: P (w >s) w>s 41

42 Geometry of Total Variation c i 42

43 Geometry of Total Variation Superlevel set of w i for s 2 [0, 1) : c i 43

44 Geometry of Total Variation Superlevel set of w i for s 2 [0, 1) : w i (a) =w i (b) =s boundary curve connects c i a b 44

45 Geometry of Total Variation Superlevel set of w i for s 2 [0, 1) : w i (a) =w i (b) =s boundary curve connects a, b c i a w i >s b 45

46 Geometry of Total Variation Penalizing the superlevel set area 46

47 Geometry of Total Variation Penalizing the superlevel set area larger perimeter 47

48 Geometry of Total Variation Regularizing the boundary curve 48

49 Geometry of Total Variation Regularizing the boundary curve larger perimeter 49

50 Geometry of Total Variation Total variation penalize superlevel set size regularize level set curves 50

51 Numerical Optimization 51

52 Numerical Optimization Discretization: triangulated domain 52

53 Numerical Optimization Discretization: triangulated domain Piecewise linear functions 53

54 Numerical Optimization Discretization: triangulated domain Piecewise linear functions determined by values at vertices 54

55 Numerical Optimization Discretization: triangulated domain Piecewise linear functions determined by values at vertices Convex optimization for values at interior vertices 55

56 Comparison 56

57 MVC HBC BBW LBC 57

58 MVC LBC 58

59 MVC LBC 59

60 3D Example 60

61 61

62 3 Superlevel set of 10 3 CNR - Istituto di Matemat /n Figure 1: Using LBC for 3D cage-based manipulated control points are deformed, a Abstract Barycentric coordinates yield a powerful a to interpolate data values on polyhedral d interior points of the domain as an affine control points, defining an interpolation s defined on a set of control points. Numerou schemes have been proposed satisfying a la However, they typically define interpolation control points. Thus a local change in the point will create a global change by prop domain. In this context, we present a fam coordinates (LBC), which select for each 62 of control points and satisfy common requ

63 Weight Reduction Cage based deformation: matrix multiplication WC = P 63

64 Weight Reduction new positions of control points WC = P barycentric coordinates of sample points new positions of sample points 64

65 Weight Reduction Global influence: dense matrix WC : = P 65

66 Weight Reduction Local influence: sparse matrix WC : = P 66

67 Weight Reduction Local influence: sparse matrix lower memory footprint faster multiplication WC : = P 67

68 Weight Reduction Store LBC values 68

69 Memory Storage Deformation Time 100% MVC 80% 60% 40% 20% 0% 69

70 Limitation Less smoothness: C 1 almost everywhere LBC BBW 70

71 Limitation Less smoothness: C 1 almost everywhere LBC 71

72 Conclusion Local barycentric coordinates by convex optimization Total variation induces locality via superlevel set perimeters 72

73 Future Work Higher order continuity Fundamental question: how local can smooth barycentric coordinates become? 73

74 Acknowledgements NSF of China ( , ) NSF of Anhui Province, China ( QF119) Specialized Research Fund for the Doctoral Program of Higher Education ( ) EU FP7 Integrated Project IQmulus (FP7-ICT ) Swiss National Science Foundation (200021_137626) The 100 Talents Program of the Chinese Academy of Sciences 74

75 Thank You!