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deformable

Heat. diffusion. descriptors. for. deformable. shapes. Michael Bronstein. Institute of Computational Science Universita della Svizzera Italiana Lugano , Switzerland. Weizmann Institute of Science, 4 November 2010. Alex Bronstein TAU. Maks Ovsjanikov Stanford. Leo Guibas Stanford.

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deformable

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  1. Heat diffusion descriptors for deformable shapes Michael Bronstein Institute of Computational Science UniversitadellaSvizzeraItaliana Lugano, Switzerland Weizmann Institute of Science, 4 November 2010

  2. Alex Bronstein TAU MaksOvsjanikov Stanford Leo Guibas Stanford Iasonas Kokkinos ECP Paris Dan Raviv Technion Ron Kimmel Technion

  3. The next challenge Text Visual data Geometric data

  4. Shape retrieval today

  5. Bags of words Notre Dame de Paris is a Gothic cathedral in the fourth quarter of Paris, France. It was the first Gothic architecture cathedral, and its construction spanned the Gothic period. Notre Dame de Paris is a Gothic cathedral in the fourth quarter of Paris, France. It was the first Gothic architecture cathedral, and its construction spanned the Gothic period. construction architecture Italy France cathedral church basilica Paris Rome Gothic Roman St. Peter’s basilica is the largest church in world, located in Rome, Italy. As a work of architecture, it is regarded as the best building of its age in Italy. St. Peter’s basilica is the largest church in world, located in Rome, Italy. As a work of architecture, it is regarded as the best building of its age in Italy.

  6. Outline Scale invariance Geometric words Bag of geometric words “ ” Feature descriptor “ ” Geometric expressions Spatially-sensitive bag of words Volumetric descriptors

  7. Shape descriptors Rigid Scale Bending Topology Representation Any Curvature Integral volume1 Volume/Mesh Any Spin image2 Any Shape context3 Any HKS4 SI-HKS5 Any vHKS6 Volume/Mesh 1 Gelfandet al. 2005; 2 Johnson, Hebert 1999;3 Belongieet al. 2002; 4 Sun et al. 2009 5 B, Kokkinos 2010; 6 Raviv, BBK 2010

  8. Diffusion geometry Heat equation where - positive semidefinite Laplace-Beltrami operator - heat distribution Fundamental solution (heat kernel, ) – heat equation solution for initial conditions Amount of heat transferred from point x to point y in time t Spectral expression

  9. Heat kernel interpretation Geometric interpretation: “multiscale Gaussian curvature” Probabilistic interpretation: the probability of a random walk to remain at point x after time t. Sun, Ovsjanikov, Guibas, 2009

  10. Heat kernel signature Multiscale descriptor Time (scale) • Intrinsic, hence deformation-invariant • Provably informative • Efficiently computable on different shape representations • Multiscale Sun, Ovsjanikov, Guibas, 2009

  11. Shape Geometric vocabulary Bag of geometric words Ovsjanikov, BB, Guibas, 2009 BB. Ovsjanikov, Guibas 2010

  12. Bags of geometric words 1 64 Index in geometric vocabulary Ovsjanikov, BB, Guibas, 2009 BB. Ovsjanikov, Guibas 2010

  13. SHREC 2010: Robust shape retrieval benchmark Query set Transformation Database (>1K shapes) B et al. 2010

  14. Query Toldoet al. 2009 Shape B et al. 2010

  15. Performance results Toldoet al. 2009 Shape Bags of words using spin image descriptor Bags of words using HKS descriptor, vocabulary of size 48 Performance criterion: mean average precision (mAP) in % Toldoet al. 2009 B et al. 2010

  16. Scale invariance Original shape Scaled by Not scale invariant!

  17. 0 -5 4 0 -10 -0.01 3 -0.02 2 -15 0 100 200 300 -0.03 t 1 -0.04 0 0 100 200 300 0 2 4 6 8 10 12 14 16 18 20 t =2k/T Scale-invariant HKS Log scale-space log + d/d Fourier transform magnitude Scaling = shift and multiplicative constant Undo scaling Undo shift B, Kokkinos CVPR 2010

  18. Scale invariant HKS HKS SI-HKS B, Kokkinos 2010

  19. Query HKS SI-HKS B, Kokkinos 2010

  20. HKS vs SI-HKS HKS, vocabulary of size 48 SI-HKS, vocabulary of size 48 Performance criterion: mean average precision (mAP) in % B, Kokkinos 2010

  21. matrix decomposition is a the of in to by science form matrix decomposition matrix factorization science fiction canonical form Expressions In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem. In math science, matrix decomposition is a factorization of a matrix into some canonicalform. Each type of decomposition is used in a particular problem. In biological science, decomposition is the process of organisms to break down into simpler form of matter. Usually, decomposition occurs after death. Matrix is a science fiction movie released in 1999. Matrix refers to a simulated reality created by machines in order to subdue the human population. Matrix is a science fiction movie released in 1999. Matrix refers to a simulated reality created by machines in order to subdue the human population. In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem. In biological science, decomposition is the process of organisms to break down into simpler form of matter. Usually, decomposition occurs after death. Matrix is a science fiction movie released in 1999. Matrix refers to a simulated reality created by machines in order to subdue the human population. Ovsjanikov, BB & Guibas 2009

  22. Expressions In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem. In particular matrix used type a some science, decomposition form a factorization of is canonical. matrix math decomposition is in a Each problem. into of matrix decomposition is a the of in to by science form matrix decomposition matrix factorization science fiction canonical form Ovsjanikov, BB & Guibas 2009

  23. Geometric expressions Yellow “Yellow Yellow” No total order between points (only “far” and “near”) Geometric expression = a pair of spatially close geometric words Ovsjanikov, BB & Guibas 2009

  24. Spatially-sensitive bags of words Ovsjanikov, BB & Guibas 2009

  25. HKS vs SI-HKS HKS, vocabulary of size 48 Spatially-sensitive HKS, vocabulary of size 8x8 Performance criterion: mean average precision (mAP) in % B et al. 2010

  26. Is our shape model good? Boundary ∂X Interior X Raviv, BBK 2010

  27. Is our shape model good? Boundary isometry Volume isometry Preserves geodesic distances on the boundary surface Preserves geodesic distances inside the volume Camel illustration from Sumner et al. Raviv, BBK 2010

  28. Diffusion equation Boundary diffusion Volumetric diffusion where - Laplace-Beltrami operator - Euclidean Laplacian - normal to boundary surface Raviv, BBK 2010

  29. Heat kernels Boundary heat kernel Volumetric heat kernel where Geometric interpretation “Multiscale Gaussian curvature” Raviv, BBK 2010

  30. Heat kernel signatures HKS vHKS Boundary+volumeisometry Boundary+volumeisometry Boundary isometry Boundary isometry Raviv, BBK 2010

  31. HKS vHKS Raviv, BBK 2010

  32. HKS vsvHKS HKS, vocabulary of size 48 vHKS, vocabulary of size 48 Performance criterion: mean average precision (mAP) in % Raviv, BBK 2010

  33. Summary Scale invariance Geometric words Bag of geometric words “ ” Feature descriptor “ ” Geometric expressions Spatially-sensitive bag of words Volumetric descriptors

  34. Thank you

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