Spectral Geometry Processing with Manifold Harmonics. Bruno Vallet Bruno Lévy

Transcription

1 Spectral Geometry Processing with Manifold Harmonics Bruno Vallet Bruno Lévy

2 Introduction Introduction II. Harmonics III. DEC formulation IV. Filtering V. Numerics Results and conclusion

3 Introduction Extend to meshes: Fourier transform Spectral filtering

4 Introduction Fourier transform =Σ f sin(kx)

5 Introduction Filtering x =

6 Introduction Filtering Filtering Convolution Fourier Transform Geometric space Frequency space Inverse Fourier Transform x

7 Introduction Filtering on a mesh Filtering [Taubin 95] Geometric space Frequency space? x?

8 Introduction Filtering on a mesh Filtering [Taubin 95] Geometric space Frequency space? x?

9 Introduction Filtering on a mesh Filtering [Taubin 95] Geometric space Frequency space? [Karni00] mesh compression [Zhang06] shape matching [Dong06] quadrangulation?

10 I Harmonics Introduction Harmonics DEC formulation Filtering Numerics Results and conclusion

11 I Harmonics Question on? sin(kx)

12 I Harmonics Harmonics and vibrations sin(kx) are the stationary vibrating modes = harmonics of a string

13 I Harmonics Manifold Harmonics Harmonics Harmonics?

14 I Harmonics Square Harmonics Harmonics Harmonics?

15 I Harmonics Chladni plates sand

16 I Harmonics Chladni plates

17 I Harmonics Chladni plates Discoveries concerning the theory of music, Chladni, 1787

18 I Harmonics Chladni plates and jpeg Chladni plates, 1787 Discrete cosine transform (jpeg)

19 I Harmonics Spherical Harmonics Harmonics Harmonics

20 I Harmonics Manifold Harmonics Harmonics Harmonics Stationary vibrating modes

21 I Harmonics Harmonics and vibrations Wave equation: T ²y/ x² = μ ²y/ t² T: stiffness μ: mass y x Stationary modes: y(x,t) = y(x)sin(ωt) ²y/ x² = -μω²/t y eigenfunctions of ²/ x²

22 I Harmonics I Harmonics : recap Harmonics are eigenfunctions of ²/ x² On a mesh, ²/ x² is the Laplacian Δ We need the eigenfunctions of Δ Let s use DEC

23 II DEC formulation Introduction Harmonics DEC formulation Filtering Numerics Results and conclusion

24 II DEC formulation Discrete Exterior Calculus (DEC) Discretize equations on a mesh Simple Rigorous [Mercat], [Hirani], [Arnold], [Desbrun] Based on k-forms

25 II DEC formulation k-forms mesh dual mesh

26 II DEC formulation 0-forms

27 II DEC formulation 1-forms

28 II DEC formulation 2-forms

29 II DEC formulation dual 0-forms

30 II DEC formulation dual 1-forms

31 II DEC formulation dual 2-forms

32 II DEC formulation Hodge star *0 from to 0-forms dual 2-forms i term *i mesh dual mesh

33 II DEC formulation Hodge star *1 from to term 1-forms dual 1-forms *ij / ij = cot(β)+cot(β ) β i *ij j mesh dual mesh β

34 II DEC formulation Exterior derivative d from to term 0-forms 1-forms df (ij) = fi - fj Oriented connectivity of the mesh: l i k j d ij jk ki il lj i j k l f fi fj fk fl

35 II DEC formulation DEC Laplacian In DEC the Laplacian is *0 dt *1 d -1 0-form (function) f

36 II DEC formulation DEC Laplacian In DEC the Laplacian is *0 dt *1 d -1 0-form (function) f 1-form (gradient) df d (fj-fi)

37 II DEC formulation DEC Laplacian In DEC the Laplacian is *0 dt *1 d -1 0-form (function) f 1-form (gradient) df d (cot(β)+cot(β ))(fj-fi) *1 dual 1-form (cogradient) df *

38 II DEC formulation DEC Laplacian In DEC the Laplacian is *0 dt *1 d -1 0-form (function) f 1-form (gradient) df d Σ (cot(β)+cot(β ))(fj-fi) *1 dt dual 0-form (integrated laplacian) d df * dual 1-form (cogradient) df *

39 II DEC formulation DEC Laplacian In DEC the Laplacian is *0 dt *1 d 0-form (function) f -1 0-form (pointwise laplacian) 1-form (gradient) df d * d*df -1 * -1 0 Σ*i (cot(β)+cot(β )) (fj-fi) *i *1 dt dual 0-form (integrated laplacian) d df * dual 1-form (cogradient) df *

40 II DEC formulation Manifold Harmonics Basis (MHB) Eigenfunctions of operator Δ +1 DEC Eigenvectors of -1 matrix *0-1dT*1d

41 II DEC formulation II DEC formulation : recap on = sin(kx)

42 III Filtering Introduction Harmonics DEC formulation Filtering Numerics Results and conclusion

43 III Filtering?

44 III Filtering Spectral Filtering The Manifold Harmonics Hk come with an eigenvalue λk The λk= ωk2 is a squared spatial frequency A filter is a transfer function F(ω) Geometric space Filtering fi MHT Frequency space f~k fifilt MHT-1 F(ωk) f~k

45 III Filtering Color Filtering Take f =(r,g,b)

46 III Filtering Geometry Filtering Take f =(x,y,z)

47 IV Numerics Introduction Harmonics DEC formulation Filtering Numerics Results and conclusion

48 IV Numerics Eigenvalues Compute the eigenpairs (Hk, λk) of L = *0-1dT*1d Solver returns eigenvectors of highest eigenvalue Compute x=lv x Problem: v Eigen Solver (Hk, λk) of L for 50 largest λk We want smallest λk We want more than 50

49 IV Numerics Shift Invert L (L λs Id)-1 same Hk shift λs 0 μk = 1/ (λk - λs) λk L

50 IV Numerics Eigen solver Compute a band of eigenpairs (Hk,λk) around λs x x= (L λs Id)-1 v Solve Lx - λsx = v v Sparse indefinite Cholesky factorization + backward substitution [Meshar & Toledo] Eigen Solver (Hk, μk) for 50 largest μk (Hk, λk) for 50 λk closest to λs

51 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk 0

52 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk λs0=0

53 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk λs0=0 λs1

54 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk λs0=0 λs1 λs2

55 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk λs0=0 λs1 λs2 λs3

56 IV Numerics Band by band algorithm Compute the eigenpairs (hk,λk) of L band by band λk λs0=0 λs1 λs2 λs3 λs4

57 Results and conclusion Introduction II. Harmonics III. DEC formulation IV. Filtering V. Numerics Results and conclusion

58 Results

59 Conclusion We make explicit Fourier Analysis and Filtering tractable Time to compute MHB ~ Time to compute a filter (5 minutes for 300k vertices) Time to update filter ~ real time

60 Acknowledgements Ramsay Dyer for personal communication Sivan Toledo for the sparse indefinite Cholesky factorization code

61 Questions?