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Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors

Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors. Michael Kazhdan Th o mas Funkhouser Szymon Rusinkiewicz Princeton University. Motivation. Large databases of 3D models. Computer Graphics (Princeton 3D Search Engine). Mechanical CAD

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Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors

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  1. Rotation Invariant Spherical Harmonic Representation of3D Shape Descriptors Michael Kazhdan Thomas Funkhouser Szymon Rusinkiewicz Princeton University

  2. Motivation • Large databases of 3D models Computer Graphics (Princeton 3D Search Engine) Mechanical CAD (National Design Repository) Molecular Biology (Audrey Sanderson)

  3. Retrieval Approach 3D Model ShapeDescriptor Nearest Neighbor Model Database

  4. Shape Unchanged by Rotation =

  5. Problem • Many shape descriptors are functions that rotate with the shape Extended Gaussian Image [Horn ’84] Spherical Attribute Image [Ikeuchi ’95] Shape Histogram [Ankerst ’99] Spherical Extent Function [Vranic ’00] Reflective Symmetry Descriptor[Kazhdan ’02] Gaussian EDT [Funkhouser ’03]

  6. Goal Compute similarity of shape descriptors independent of rotation ? - =

  7. Brute Force Approach Impractical for databases - - min (rotation) - = - -

  8. Normalization • Use PCA to place models into a canonical coordinate frame Covariance Matrix Computation Principal Axis Alignment

  9. Normalization • Doesn’t always work • Only second order information

  10. Shape Descriptor Our Approach • Eliminate rotation dependence in spherical and 3D descriptors EGI [Horn ’84] SAI [Ikeuchi ’95] EXT [Vranic ’00] RSD [Kazhdan ’02] EDT [Funkhouser ’03] etc. Shape Descriptor

  11. Our Approach • Eliminate rotation dependence in spherical and 3D descriptors Shape Descriptor Rotation Invariant Representation

  12. Outline • Introduction • Background • Harmonic Representation • Properties • Experimental Results • Conclusion and Future Work

  13. Key Idea • Obtain rotation invariant representation by storing amplitude and eliminating phase … + + + + = [Lo 1989] [Burel 1995]

  14. Fourier Descriptors CircularFunction

  15. Fourier Descriptors … = + + + + CircularFunction Cosine/Sine Decomposition

  16. Fourier Descriptors … = + + + + CircularFunction = Constant Frequency Decomposition

  17. Fourier Descriptors … = + + + + + CircularFunction + = Constant 1st Order Frequency Decomposition

  18. Fourier Descriptors … = + + + + + CircularFunction + + = Constant 1st Order 2nd Order Frequency Decomposition

  19. Fourier Descriptors … = + + + + + CircularFunction … + + + + = Constant 1st Order 2nd Order 3rd Order Frequency Decomposition

  20. Fourier Descriptors Amplitudes invariantto rotation … = + + + + + CircularFunction … + + + + = Constant 1st Order 2nd Order 3rd Order Frequency Decomposition

  21. Harmonic Representation SphericalFunction

  22. Harmonic Representation … = + + + + SphericalFunction Harmonic Decomposition

  23. Harmonic Representation … = + + + + SphericalFunction Constant 1st Order 2nd Order 3rd Order … + + + + =

  24. Harmonic Representation Store “how much” (L2-norm) of the shape resides in each frequency Norms Invariantto Rotation … + + + + =

  25. 3D Function (Voxel Grid) Restrict to concentric spheres

  26. 3D Function (Voxel Grid) • Compute harmonic representation of each sphere independently + + + + = = + + + + = + + + +

  27. 3D Function (Voxel Grid) • Combine harmonic representations Radius Frequency

  28. Matching Harmonic Representation Harmonic Representation - 2 L2-difference of harmonic representations…

  29. Matching - min (rotations) - 2 2 … bounds proximity of descriptors over all rotations

  30. Outline • Introduction • Background • Harmonic Representation • Properties • Experimental Results • Conclusion and Future Work

  31. Advantages • The harmonic representations is: • Rotation invariant • Multi-resolution • Compact • Discriminating

  32. Compact … … …

  33. Compact … … … …

  34. Compact … … … …

  35. Compact … … … …

  36. Compact … … … …

  37. Information Loss • Intra-frequency information loss • Cross-frequency information loss • Cross-radial information loss

  38. Information Loss (Spherical Descriptor) • Intra-frequency information loss • Cross-frequency information loss

  39. Information Loss (Spherical Descriptor) • Intra-frequency information loss • Cross-frequency information loss + = 22.5o 90o = +

  40. Information Loss (3D Descriptor) • Cross-radial information loss

  41. Outline • Introduction • Background • Harmonic Representation • Properties • Experimental Results • Conclusion and Future Work

  42. Shape Descriptors Extended Gaussian Image Horn 1984 Shape Histogram Ankerst 1999 Spherical Extent Function Vranic 2000 Gaussian EDT Funkhouser 2003

  43. Experimental Database • Viewpoint “household” database1,890 models, 85 classes 153 dining chairs 25 livingroom chairs 16 beds 12 dining tables 8 chests 28 bottles 39 vases 36 end tables

  44. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 Gaussian EDT Results PCA-Normalized Results Harmonic Representation Results Query

  45. Gaussian EDT Results • Precision vs. Recall 100% Harmonics PCA Precision 50% 0% 0% 50% 100% Recall

  46. 100% 100% 100% 100% Harmonics Harmonics Harmonics Harmonics PCA PCA PCA PCA Precision Precision Precision Precision 50% 50% 50% 50% 0% 0% 0% 0% 0% 0% 0% 0% 50% 50% 50% 50% 100% 100% 100% 100% Recall Recall Recall Recall Retrieval Results SECT EGI • EGI: Extended Gaussian Image • SECT: Shape Histogram (Sectors) • EXT: Spherical Extent Function • EDT: Gaussian Euclidean Distance Transform EXT EDT

  47. SECT 100% 100% 100% 100% Harmonics Harmonics Harmonics Harmonics PCA PCA PCA PCA Precision Precision Precision Precision 50% 50% 50% 50% 0% 0% 0% 0% 0% 0% 0% 0% 50% 50% 50% 50% 100% 100% 100% 100% Recall Recall Recall Recall EXT EDT Retrieval Results EGI • EGI: Extended Gaussian Image • SECT: Shape Histogram (Sectors) • EXT: Spherical Extent Function • EDT: Gaussian Euclidean Distance Transform

  48. Exhaustive Gaussian EDT Results Gaussian EDT - 100% min L2 Harmonic - PCA min (rotation) Precision - 50% - 0% 0% 50% 100% Recall

  49. Summary and Conclusion • Provide a rotation invariant representation of shape descriptors that: • Eliminates PCA dependence • Gives better matching performance • Is more compact • Is a multi-resolution representation

  50. Future Work • Managing Information Loss • Obtain cross radial information for 3D descriptors • Obtain cross frequency information • Get finer resolution of rotation invariance within frequencies • More Generally • Consider new shape descriptors

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