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Sub-sampling for Efficient Spectral Mesh Processing

Sub-sampling for Efficient Spectral Mesh Processing. Rong Liu, Varun Jain and Hao Zhang GrUVi lab, Simon Fraser University, Burnaby, Canada. Roadmap. Background Nyström Method Kernel PCA (KPCA) Measuring Nyström Quality using KPCA Sampling Schemes Applications

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Sub-sampling for Efficient Spectral Mesh Processing

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  1. Sub-sampling for Efficient Spectral Mesh Processing Rong Liu, Varun Jain and Hao Zhang GrUVi lab, Simon Fraser University, Burnaby, Canada CGI'06, Hangzhou China

  2. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  3. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  4. spectral mesh compression [Karni and Gotsman, 00] spectral clustering [Ng et. al., 02] spectral mesh segmentation [Liu and Zhang, 04] face recognition in eigenspace [Turk, 01] texture mapping using MDS [Zigelman et. al., 02] spectral mesh correspondence [Jain and Zhang, 06] watermarking [Ohbuchi et. al., 01] Spectral Applications “affinity matrix” W, its eigen-decomposition CGI'06, Hangzhou China

  5. j W= 0.56 i … j i E = embedding space, dimensionn row i Spectral Embedding j i npoints, dimension 2 W = EΛET CGI'06, Hangzhou China

  6. Bottlenecks • Computation of W, O(n2) . • Apply sub-sampling to compute partial W. • Eigenvalue decomposition of W, O(n3). • Apply Nyström method to approximate the eigenvectors of W. How to sample to make Nyström work better ? CGI'06, Hangzhou China

  7. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  8. affinities between XandY affinities within X W = O (n2) complexity: O (l . n) Sub-sampling • Compute partial affinities n points lsample points Z =XUY CGI'06, Hangzhou China

  9. A B BT C complexity: U O (n3) U = BTUΛ-1 approximate eigenvectors O (l2. n) Nyström Method [Williams and Seeger, 2001] • Approximate Eigenvectors W = , A = UΛUT CGI'06, Hangzhou China

  10. W = UΛUT = U U BTUΛ-1 BTUΛ-1 A B A B W = BT BTA-1B BT C F F Schur Complement T Λ = Schur Complement = C - BTA-1B SC = Practically, SC is not useful to measure the quality of a sample set. CGI'06, Hangzhou China

  11. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  12. feature space, high dimension (infinite) X is implicitly defined by a kernel matrix K, where Kij= PCA and KPCA [Schölkopf et al, 1998] dimension 2 X covariance matrix CX covariance matrix Cφ(X) CGI'06, Hangzhou China

  13. L M MT N E K = ˙Λ-1/2 E = MTEΛ-1 L = EΛET Training Set for KPCA CGI'06, Hangzhou China

  14. A B L M BT C MT N E K = ˙Λ-1/2 E = MTEΛ-1 Nyström Method and KPCA U Nyström W = U = BTUΛ-1 A = UΛUT KPCA w/ training set L = EΛET CGI'06, Hangzhou China

  15. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  16. subspace spanned by training points Training set should minimize: When Nyström Works Well ? • When the training set of KPCA works well ? CGI'06, Hangzhou China

  17. W = A B BT C Objective Function minimize: maximize: evaluation: CGI'06, Hangzhou China

  18. Test data are generated using Gaussian distribution; • Test is repeated for 100 times; • 4% inconsistency. Compare Γ and SC Given two samplingsets S1and S2 CGI'06, Hangzhou China

  19. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  20. W = A B BT C How to sample: Greedy Scheme • Maximize: Greedy Sampling Scheme: A B Best candidate sampling scheme: To find the best 1% with probability 95%, we only need to search for the best one from a random subset of size 90 (log(0.01)/log(0.95)) regardless of the problem size. CGI'06, Hangzhou China

  21. (0, m), m is the column size of B Properties of Γ • maximize 1T(A-11) • A is symmetric. • Diagonals of A are 1. • Off-diagonals of A are in (0, 1). • It can be shown that when A’s columns are • canonical basis of the Euclidean space, the • maxima is obtained. CGI'06, Hangzhou China

  22. How to Sample: Farthest Point Scheme In order for A’s columns to be close to canonical basis, the off-diagonals should be close to zero. 1 A = 1 … 1 This means the distances between each pair of samples should be as large as possible, namely Samples are mutually farthest away. CGI'06, Hangzhou China

  23. Farthest Sampling Scheme CGI'06, Hangzhou China

  24. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  25. D(1) W(1) EΛ-1/2 M(1) D(2) W(2) EΛ-1/2 M(2) Mesh Correspondence M(1) M(2) CGI'06, Hangzhou China

  26. without sampling farthest point sampling (vertices sampled: 10, total vertices: 250) random sampling CGI'06, Hangzhou China

  27. (vertices sampled: 10 total vertices: 2000) CGI'06, Hangzhou China

  28. correspondence error against mesh size • correspond a series a slimmed mesh with the original mesh • a correspondence error at a certain vertex is defined as the geodesic distance between the matched point and the ground-truth matching point. CGI'06, Hangzhou China

  29. D W EΛ-1/2 Mesh Segmentation M CGI'06, Hangzhou China

  30. (b, d) obtained using farthest point sampling • (a, c) obtained using random sampling • faces sampled: 10 • number in brackets: value of Γ CGI'06, Hangzhou China

  31. 2.2 GHz Processor 1GB RAM w/o sampling, it takes 30s to handle a mesh with 4000 faces. CGI'06, Hangzhou China

  32. Roadmap • Background • Nyström Method • Kernel PCA (KPCA) • Measuring Nyström Quality using KPCA • Sampling Schemes • Applications • Conclusion and Future Work CGI'06, Hangzhou China

  33. Conclusion • Nyström approximation can be considered as using training data in Kernel PCA. • Objective function Γ effectively quantifies the quality of a sample set. • Γ leads to two sampling schemes: greedy scheme and farthest point scheme. • Farthest point sampling scheme outperforms random sampling. CGI'06, Hangzhou China

  34. Future Work • Study the influence of kernel functions to Nyström method. • Further improve the sampling scheme. CGI'06, Hangzhou China

  35. Thank you !Questions ? CGI'06, Hangzhou China

  36. Mesh Correspondence • Given any two models, M(1) and M(2), build the geodesic distance matrices D(1) and D(2). Dijencodes the geodesic distance between vertices i and j; • D(1)  W(1) , D(2) W(2) , using Gaussian kernel. • Compute the eigenvalue decomposition of W(1) and W(2), and use the corresponding eigenvectors to define the spectral-embedded models M(1) and M(2). • handle bending, uniform scaling and rigid body transformation. • Compute the correspondence betweenM(1) and M(2). CGI'06, Hangzhou China

  37. Mesh Segmentation • Given a model M, somehow define the distances between each pair of faces; the distances are stored in matrix D; • D W ; • Compute the eigenvalue decomposition of W, and use the eigenvectors to spectral-embed the faces. • Cluster (K-means) the embedded faces. Each cluster corresponds to a segment of the original model. CGI'06, Hangzhou China

  38. SC = C - BTA-1B Γ and Schur Complement • Maximize: • Given any two sampling sets S1and S2 , S1 is superior to S2 iff • Efficient to compute. • Minimize: (schur complement) • S1 is superior to S2 iff • Very expensive to compute. CGI'06, Hangzhou China

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