A Diffusion Wavelet Approach For 3 D Model Matching
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A Diffusion Wavelet Approach For 3 D Model Matching

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  • Google has their own format –SketchUp (SU) and has been investing a lot of effort in 3D technologies. More examples from google include – google earth, google sketch-up and O3D platform for browser 3D display
  • Other manifold learning techniques – PCA (+variants), Kernel PCA, LLE (locally linear embedding), ISOMAP, Hessian LLE, LaplacianEigenMaps and many more…
  • Reference[9]
  • Reference [2]
  • See [1,6] for diffusion maps equations and [3] for graph
  • Ignore λmax=1 since it is a trivial eigenvalue of the markov transition matrix and corresponds to the stationary distribution of the markov chain at t=∞Cut is promised to conform with min-cut max-flow algorithmSee reference [6] for a thorough description of diffusion maps
  • We see that it is possible to reconstruct the manifold using just a single eigenvalue. Cuts can be made on the eigenvectors that represent the manifold (in this case, the second or the third eigenvectors corresponding to the second or third largest eigenvalue) – these cuts are meaningful since they are taking into account the geometric distribution of the original points. We see that diffusion maps approximate the Fourier series over the circle as the sine and cosine functions are the solution of the differential equation f’’=-f
  • Animated objects can be more sensitive to mesh simplification algorithms than CAD models.
  • Haar wavelet with 3 levels of decomposition to the Stanford Bunny image. By applying a threshold in the wavelet domain we can efficiently find similar images. The threshold is a very efficient way to remove noise. The high value coefficients correspond to edges at various scales. Collecting the high value coefficients to create a feature vector would ensure a good representation of the image. For example, check the lossy compression algorithm JPEG-2000.
  • We take 4 levels of decomposition since there is no real advantage in taking more decomposition levels and the computational burden is heavy. This result is specific to Princeton’s database and can change when dealing with different databases.
  • Take inverse of FDR to avoid numeric problems of overflow. Select model with minimum within cluster scattering and maximum within cluster scattering
  • Can do training on the entire database…See [5] for database
  • Chair – sparse structurePlane – smooth surface with local singularityKangaroo – smooth surfaceFlower – combines smooth surface with local singularity

Transcript

  • 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
  • 14. Diffusion Maps
    Z
    Y
    X
    Coifman - 2005
    • vi is a feature vector
    • 15. Contains descriptive information about the 3D model
    14
  • 16. Diffusion Maps
    Assumptions
    • Points are sampled uniformly on the manifold
    • 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
  • 34. Finding Shape Feature Vectors (X)
    30
    • Take 1,450 coefficients (wavelet + scaling) at each decomposition level
    • 35. Increase wavelet coefficient number from 0 to 450 and decreasescaling coefficient number from 1,450 to 1,000
  • Fisher’s Discriminant Ratio
    b – between classes
    w – within class (after wavelet decomposition)
    j – scale
    31
  • 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
  • 47. My Conclusions
    Paper presents a novel solution
    • Diffusion Wavelets was never used before for 3D Retrieval
    • 48. Less novel solutions:
    • 49. IRPR is a common measure in database retrieval
    • 50. Fischer Discriminant Ratio is a common statistical measure
    43
  • 51. My Conclusions (2)
    Technically sound, feasible
    • Taking available code it seems possible to reconstruct the results
    • 52. Seems like a reasonable solution to the problem of 3D object retrieval
    44
  • 53. My Conclusions (3)
    The diffusion wavelet part could be explained in more detail
    • Not clear in which way the wavelets are constructed
    • 54. How are the wavelet functions affected when a new model is inserted?
    45
  • 55. My Conclusions (4)
    Not self-containing but reference papers are exceptionally good
    • Missing explanations about diffusion wavelets
    46
  • 56. “Would like to have” (Technical/1)
    Non-rigid extensions
    • How would retrieval change if we know the 3D model is non-rigid?
    • 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