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Harmonic 3D Shape Matching

Harmonic 3D Shape Matching. Michael Kazhdan Thomas Funkhouser Princeton University. Motivation. Large databases of 3D models. Computer Graphics (Princeton 3D Search Engine). Mechanical CAD (National Design Repository). Molecular Biology (Audrey Sanderson). Goal.

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Harmonic 3D Shape Matching

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  1. Harmonic3D Shape Matching Michael KazhdanThomas Funkhouser 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. Goal • Find 3D models with similar shapes 3D Model ShapeDescriptor Nearest Neighbor Model Database

  4. Research Challenge • Need shape descriptor that is: • Discriminating • Concise to store • Quick to compute • Efficient to match Nearest Neighbor 3D Model ShapeDescriptor Model Database

  5. Research Challenge • Finding a 3D shape descriptor that is: • Discriminating • Concise to store • Quick to compute • Efficient to match Nearest Neighbor 3D Model ShapeDescriptor Model Database

  6. Research Challenge • Finding a 3D shape descriptor that is: • Discriminating • Concise to store • Quick to compute • Efficient to match Nearest Neighbor 3D Model ShapeDescriptor Model Database

  7. Research Challenge • Finding a 3D shape descriptor that is: • Discriminating • Concise to store • Quick to compute • Efficient to match Nearest Neighbor 3D Model ShapeDescriptor Model Database

  8. Research Challenge • Finding a 3D shape descriptor that is: • Discriminating • Concise to store • Quick to compute • Efficient to match Many possible alignments Nearest Neighbor 3D Model ShapeDescriptor Model Database

  9. 3D Model Matching Approaches • Search over all possible alignments • Too slow for large database - - - - min - -

  10. 3D Model Matching Approaches • Search over all possible alignments • Too slow for large database • Normalize alignment (e.g., with moments) • OK for translation and scale, not for rotation PCA Aligned Models

  11. 3D Model Matching Approaches • Search over all possible alignments • Too slow for large database • Normalize alignment (e.g., with moments) • OK for translation and scale, not for rotation • Build alignment invariance into descriptor • Previous methods not very discriminating Shape Histograms [Ankerst et al., 1999]

  12. Outline • Introduction • Approach • Implementation • Experimental Results • Conclusion and Future Work

  13. Our Approach • Harmonic 3D shape descriptor • Decompose 3D shapes into irreducible set of rotation independent components • Store “how much” of the model resides in each component 3D Model Rotation IndependentComponents ShapeDescriptor

  14. Our Approach • Harmonic 3D shape descriptor • Decompose 3D shapes into irreducible set of rotation independent components • Store “how much” of the model resides in each component Concentric Spheres 3D Model Rotation IndependentComponents ShapeDescriptor

  15. Our Approach • Harmonic 3D shape descriptor • Decompose 3D shapes into irreducible set of rotation independent components • Store “how much” of the model resides in each component Frequency Decomposition 3D Model Rotation IndependentComponents ShapeDescriptor

  16. Our Approach • Harmonic 3D shape descriptor • Decompose 3D shapes into irreducible set of rotation independent components • Store “how much” of the model resides in each component Amplitudes 3D Model Rotation IndependentComponents ShapeDescriptor

  17. Outline • Introduction • Approach • Implementation • Experimental Results • Conclusion and Future Work

  18. Voxelization • Convert polygonal model to 3D voxel grid • Rasterize surfaces (no solid reconstruction) • Normalize for translation and scale 3D Model 3D Voxel Grid

  19. Spherical Decomposition • Intersect with concentric spheres

  20. Frequency Decomposition • Represent each spherical function as a sum of different frequencies Frequency Components Spherical Functions

  21. Fourier Analysis CircularFunction

  22. Fourier Analysis … = + + + + CircularFunction Cosine/Sine Decomposition

  23. Fourier Analysis … = + + + + CircularFunction = Constant Frequency Decomposition

  24. Fourier Analysis … = + + + + + CircularFunction + = Constant 1st Order Frequency Decomposition

  25. Fourier Analysis … = + + + + + CircularFunction + + = Constant 1st Order 2nd Order Frequency Decomposition

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

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

  28. Harmonic Analysis SphericalFunction

  29. Harmonic Analysis … = + + + + SphericalFunction Harmonic Decomposition

  30. Harmonic Analysis … = + + + + SphericalFunction … = + + + + Constant 1st Order 2nd Order 3rd Order

  31. Building Shape Descriptor • Store “how much” (L2-norm) of the shape resides in each frequency of each sphere HarmonicShapeDescriptor Amplitudes Frequency Decomposition

  32. Matching • Model similarity defined as L2-distance between their descriptors • Bounds proximity of voxel gridsover all rotations - - - Sim = , - -

  33. Outline • Introduction • Approach • Implementation • Experimental Results • Conclusion and Future Work

  34. Query Princeton 3D Search Engine

  35. Retrieval Experiment • 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

  36. Query Retrieval Results • Precision-recall curve (mean for all queries) 1 0.8 0.6 Precision 0.4 3D Harmonics 0.2 Random 0 0 0.2 0.4 0.6 0.8 1 Recall

  37. Retrieval Results • Precision versus recall (mean for all queries) 1 3D Harmonics (Our Method) D2 Shape Distributions [Osada et al., 2001] Shape Histograms [Ankerst, 1999] EGI [Horn, 1984] Moments [Elad et al., 2001] Random 0.8 0.6 Precision 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Recall

  38. Retrieval Results • Precision versus recall (mean for all queries) 1 3D Harmonics (Our Method) D2 Shape Distributions [Osada et al., 2001] Shape Histograms [Ankerst, 1999] EGI [Horn, 1984] Moments [Elad et al., 2001] Random 0.8 0.6 Precision 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Recall

  39. Retrieval Results • Precision versus recall (mean for all queries) 1 3D Harmonics (Our Method) D2 Shape Distributions [Osada et al., 2001] Shape Histograms [Ankerst, 1999] EGI [Horn, 1984] Moments [Elad et al., 2001] Random 0.8 0.6 Precision 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Recall

  40. Retrieval Results • Precision versus recall (mean for all queries) 1 3D Harmonics (Our Method) D2 Shape Distributions [Osada et al., 2001] Shape Histograms [Ankerst, 1999] EGI [Horn, 1984] Moments [Elad et al., 2001] Random 0.8 0.6 Precision 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Recall

  41. Summary and Conclusion • Harmonic shape descriptor is a rotation invariant representation that is: • Discriminating (46%-245% better than others tested) • Concise to store (2048 bytes) • Quick to compute (1-5 seconds) • Efficient to match (0.45 seconds: 20,000 model DB)

  42. Query Future Work • Extensions • Partial object matching? • Other 3D shape functions? • Other applications • Molecular biology • Medicine • Paleontology • Forensics http://shape.cs.princeton.edu/

  43. Thank You • Funding • National Science Foundation • Sloan Foundation • People • Bernard Chazelle, David Dobkin, David Jacobs, David Kazhdan, Allison Klein, Patrick Min, Szymon Rusinkiewicz, Peter Sarnak, Julianna Tymoczko http://shape.cs.princeton.edu/

  44. Analysis • The Harmonic Descriptor is not invertible • Different spheres rotate independently • Different orders rotate independently • Rotations are not transitive within an order

  45. Analysis • The Harmonic Descriptor is not invertible • Different spheres rotate independently • Different orders rotate independently • Rotations are not transitive within an order l=0 l=1 l=2 l=3

  46. Inter-Radial Coherence • Force same orders on different spheres to rotate together by setting up the rotation invariant matrix Mk with: • where fk,j is the k-th order component of the restriction of f to the j-th radius. • (The diagonal is precisely the collection of L2-norms.)

  47. Polygon Rasterization • Rasterize using the Euclidean Distance Transform to measure how much models miss by: Polygonal Model Voxel Model

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