1. A Diffusion Wavelet Approach for 3-D Model Matching Authors: K.P. Zhu, Y.S. Wong, W.F. Lu, J.Y.H. Fuh Presented by: Raphael Steinberg

2. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 2

3. Introduction Currently - A larger than ever number of 3D Models in CAD, computer games, multimedia, molecular biology, computer vision and more There is a need for 3D Retrieval 3

4. Introduction (2) Tagging are not always available or sufficient to describe the model we require Combine topological information with multi-scale properties 4

5. Model Reusability (CAD/Animation) Model Matching Video Retrieval (2.5D/Virtual environments) Ecommerce Correcting defects Efficient Representation Many other uses… Motivation for 3D Retrieval 5

6. Obstacles in Retrieval Partial retrieval - Non-transitive Functional description How to match text tags with vertices and texture? Orthonormal coordinate system 6

7. 7

8. 3D Model Matching – Prior Art Feature vectors using wavelets to mesh vertices – localized in both space & frequency – Paquet et. al. 2000 Random sampling for comparison – Osada et. al. 2001 8

9. Spherical harmonics (SH) Global method in Euclidean space lacks multi-scale analysis Legendre polynomials solve the Laplace equation in Spherical coordinates Vranic et. al. 2001 9

10. Spherical Wavelets (SW) Multi-scale in Euclidean space Lacks connectivity on the manifold Tannenbaum et. al. 2007 10

11. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 11

12. Diffusion Maps Introduction Originally suggested by Stephan Lafon and R.R. Coifman from Yale Math, circa 2005 Many other manifold learning techniques exist Data analysis based on geometric properties of the data set 12

13. Manifold Learning algorithms 13 MANI - Manifold learning Matlab tool

15. Contains descriptive information about the 3D model14

17. Smooth manifold (no fractals in our case)

18. Fixed boundary conditions

19. Enough points = feature vectors (N→∞)15

20. Use RBF Gaussian Kernel to choose ε Normalize W to create a Stochastic Matrix Diffusion Maps Algorithm 16 Lu et. al. 2009

21. Diffusion Maps algorithm (2) Diffuse by taking higher powers of t “The diffusion distance is equal to the Euclidean distance in the diffusion map space” , Nadler et. al. 2005 Cut manifold according to dominant eigenvalues 17

22. Diffusion Maps Code Example function checker(); close all; tetha=2*pi*rand(1,500); z=[cos(tetha);sin(tetha)]; figure(1);scatter(z(1,:),z(2,:),'b*');hold on; N=size(z,2); epsilon=linspace(0.01,.3,10); %epsilon=.3; W=nan(N); summer=nan(1,length(epsilon)); for k=1:length(epsilon) for i=1:N parfor j=1:N W(i,j)=exp(-sum((z(:,j)-z(:,i)).^2)/2/epsilon(k)); end end summer(k)=sum(sum(W)); end figure;scatter(log(epsilon),log(summer));title('Epsilon - linear region') p=polyfit(log(epsilon),log(summer),1); d=2*p(1);%manifold dimension M=W*diag(1./sum(W,2)); [U V]=svds(M); sync=max(U(:,2)); figure(1);scatter(U(:,2)./sync,U(:,3)./sync,'rd') title('Original manifold as stars and reconstructed manifold as diamonds') end 18

23. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 19

24. Problems with Mesh Simplification 20

25. Wavelets 21

26. Novelty – Diffusion Wavelets Combination of Diffusion Maps and Wavelets Used for non-linear dimensionality reduction Extension of wavelets to the unit circle (just as diffusion maps extends the Fourier transform) 22

27. Diffusion Wavelets Intuition 23

28. Example of Diffusion Wavelets 24 Wavelet basis ψ(2,2,3) Scaling basis φ(1,1,1) Wavelet basis ψ(4,2,5) Wavelet basis ψ(3,2,3)

29. Diffusion Wavelets Use an optimization scheme to construct the scaling functions Each scaling function should deal with a single dimension and be orthogonal to the other scaling functions Extension of wavelets to the sphere (or to any other manifold) 25

30. Diffusion Wavelets (2) Better than LOD (Level of Detail - simplifies meshes) Involved algorithm – very few implementations exist 26

31. Wavelet decomposition example 27

32. Wavelet coefficients 28 Scale 1 Scale 2 Scale 3 Scale 4

33. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 29

36. Fisher’s Discriminant Ratio 32

37. IRPR Curve Measure performance – use Princeton University 3D database IRPR – Information Retrieval Precision-Recall 33

38. IRPR Curve m = relevant matches r = # of retrieved models 1) Precision = 2) Recall = 34

39. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 35

40. 3D Model Retrieval Procedure Compute the diffusionwavelet for each 3D model Obtain the model representing vector X Compute the 2nd order statistics of X for each scale 36

41. 1) Start with a coarsest scale comparison 2)Advance up to the finest scale 3) Stop on threshold or when finest scale reached * Use a threshold to determine if a model is from a certain class Model Matching Procedure 37

42. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 38

43. Experimental Results 39 Differences in scaling levels DW gives better results than SH and SW

44. Visual Results 40

45. Schedule Introduction Diffusion Maps Wavelets and Diffusion Wavelets Fisher’s Discriminant Ratio (FDR) Retrieval Procedure Results Conclusions 41

46. Authors’ Conclusions Surfaces with sharp peaks, grooves or holes contain high-frequency information which is not addressed by the wavelet multi-resolution (use diffusion wavelet packets instead?) Possible to extend to partial matching DW presents better results than SH and SW 42

48. Less novel solutions:

49. IRPR is a common measure in database retrieval

50. Fischer Discriminant Ratio is a common statistical measure43

52. Seems like a reasonable solution to the problem of 3D object retrieval44

54. How are the wavelet functions affected when a new model is inserted?45

57. Can we have an extension of Diffusion Wavelets for non-rigid manifolds?47

58. “Would like to have” (Technical/2) How to automatically choose the level of decomposition 48

59. “Would like to have” (Technical/3) An intuitive explanation - why prefer Diffusion Wavelets over Diffusion Wavelet Packets? Wavelet Packets seem to give more information especially in high frequencies… 49

60. “Would like to have” (Technical/4) Numerical problems of overflow of the FDR - use logarithm instead of inverse? 50

61. “Would like to have” (Presentation/1) Block diagram of the algorithm 51

62. “Would like to have” (Presentation/2) Web-based Graphical User Interface 52

63. “Would like to have” (Presentation/3) Error analysis 53

64. “Would like to have” (Presentation/4) More explanations on Diffusion Wavelets 54

65. Conclusions Shape retrieval requires multi-scale analysis 3D models, like most real-life objects, are embedded in a low dimension manifold Results are robust to noise and to mesh simplifications 55

66. Conclusions (2) Diffusion Wavelets give good retrieval results for 3D objects Possible to extend the proposed method to include texture, sound, smell, elasticity and any other possibly given attribute of the 3D model 56

67. THE END 57

68. References [1]K.P. Zhu, Y.S. Wong, W.F. Lu, J.Y.H. Fuh. , Department of Mechanical Engineering, National University of Singapore “A diffusion wavelet approach for 3-D model matching” Computer Aided Design, Elsevier, Nov. 2008 [2] Presentation by R.R. Coifman et. al. [3] J. Lu et. al. “Dominant Texture and Diffusion Distance Manifolds“, Eurographics, Volume 28 , Issue 2, Pages 667 - 676, Mar. 2009 [4] Diffusion waveletsMatlab code: http://www.math.duke.edu/~mauro/diffusionwavelets.html#Code|outline [5] The Princeton Shape Benchmark: http://shape.cs.princeton.edu/benchmark/ [6] Nadler, B., Lafon, S., Coifman, R., Kevrekidis, I. “Diffusion maps, spectral clustering and eigenfunctions of Fokker-Planck operators”. [7] Ulrike von Luxburg, “A tutorial on spectral clustering”. Statistical Journal 2007 [8] Personal communications with K.P. Zhu [9] MANI - Manifold learning Matlab tool http://www.math.umn.edu/~wittman/mani/ [10] Vranic D, Saupe D, Richter J. Tools for 3D-object retrieval: Karhunen-Loeve transform and spherical harmonics. In: Proc. IEEE workshop on multimedia signal processing; 2001. p. 29398. 58

69. References [11] Osada R, Funkhouser T, Chazelle B, Dobkin D. Matching 3D models with shape distributions, In: Proc. shape modeling international. 2001. p. 15466. [12] Laga H, Nakajima M. Statistical spherical wavelet moments for content-based 3D model Retrieval. In: Computer graphics international 2007, CGI. 2007; 2007. p.1-8. [13] Nain D, Haker S, Bobick A, Tannenbaum A. Multiscale 3-D shape representation and segmentation using spherical wavelets. IEEE Transactions on Medical Imaging 2007;26(4), pages 598-618. 59