1. PG 2011 Pacific Graphics 2011 The 19th Pacific Conference on Computer Graphics and Applications (Pacific Graphics 2011) will be held on September 21 to 23, 2011 in Kaohsiung, Taiwan. Improving Performance and Accuracy of Local PCA V. Gassenbauer, J. Křivánek, K. Bouatouch, C. Bouville, M. Ribardière University of Rennes 1, France, and Charles University, Czech Republic

2. Pacific Graphics 2011 2 | 25 Kaohsiung, Taiwan Need to compress large data set Objective Precomputed Radiance Transfer (PRT) Bidirectional Texture Function (BTF) compression Image courtesy of Hongzhi Wu

3. Pacific Graphics 2011 3 | 25 Kaohsiung, Taiwan Relighting as a matrix-vector multiply Matrix T(x, ω i ) - Billions of elements Infeasible multiplication Compression (wavelet, Local PCA) Precomputed Radiance Transfer

4. Pacific Graphics 2011 4 | 25 Kaohsiung, Taiwan Precomputation-based rendering Related Work [Sloan et al. 02,03] Low-freq shadows, inter-reflections [Xu et al. 08] Dynamic scene, BRDF editing [Liu et al. 04], [Huang et al. 10] High-freq shadows, inter-reflections Still use slow and inaccurate LPCA !

5. Pacific Graphics 2011 5 | 25 Kaohsiung, Taiwan BTF compression [Müller et al. 03], [Filip et al. 09] Related Work Simpler LPCA: k-means problem Better performance [Phillips 02], [Elkan 03], [Hamerly10] Better accuracy [Arthur et al. 07], [Kanugo et al. 02]

6. Pacific Graphics 2011 6 | 25 Kaohsiung, Taiwan Input High-dimensional points (rows of T) Number of clusters k Compression Problem Output Find k clusters – i.e. low-dimensional subspaces x 13 x 14 x 104 x 960

7. Pacific Graphics 2011 7 | 25 Kaohsiung, Taiwan Guess k clusters (i.e. subs.) Initialize by randomly selected centers LPCA – The Algorithm Assign each point to the nearest cluster Update PCA in the cluster Repeat until convergence Generate seeds Classification Update Until converge

8. Pacific Graphics 2011 8 | 25 Kaohsiung, Taiwan Motivation Inaccurate Inefficient For each point compute distance to all clusters a i Prone to get stuck in a local optimum x a1a1 a2a2 akak Generate seeds Classification Update Until converge

9. Pacific Graphics 2011 9 | 25 Kaohsiung, Taiwan Our Contribution Inaccurate Inefficient For each point compute distance to all clusters a i Prone to get stuck in a local optimum SortMeans++ x a1a1 a2a2 akak SortCluster LPCA (SC-LPCA) Generate seeds Classification Update Until converge

10. Pacific Graphics 2011 10 | 25 Kaohsiung, Taiwan Improving Performance SortCluster-LPCA (SC-LPCA) Improving Accuracy SortMeans++ Results Outline

11. Pacific Graphics 2011 11 | 25 Kaohsiung, Taiwan If d(c a,c b ) ≥ 2d(x,c a ) ∆- inequality [Phillips 02] Accelerated k-means x d(c a,c b ) d(x,c b ) d(x,c a ) cbcb caca → d(x,c b ) ≥ d(x,c a ) → d(x,c b ) not necessary to compute !

12. Pacific Graphics 2011 12 | 25 Kaohsiung, Taiwan How does it work ? We know: potentially nearest cluster to x We know: Distances of other cluster w.r.t. c a Check d(c a,c b ) ≥ 2d(x,c a ) Compute d(x,c a ) Compute d(x,c b ) caca cbcbc x Check d(c a,c c ) ≥ d(x,c a )+d(x,c b ) c b becomes a new nearest cluster

13. Pacific Graphics 2011 13 | 25 Kaohsiung, Taiwan Our Contribution: From k-means to LPCA x caca cbcb d(c a,c b ) d(x,c b ) x d(x,ab)d(x,ab) d(x,ab)d(x,ab) d(aa,ab)d(aa,ab) abab a k-meansLPCA Piecewise reconstruction Piecewise reconstruction in low-dimensional subspaces

14. Pacific Graphics 2011 14 | 25 Kaohsiung, Taiwan Distance between subspaces d(a a,a b ) = inf { ║p - q║; p in a a, q in a b } ∆-inequality for LPCA a abab x d(x,ab)d(x,ab) d(x,ab)d(x,ab) d(aa,ab)d(aa,ab) ∆-inequality for subspaces If d(a a,a b ) ≥ 2d(x,a b ) → d(x,a b ) ≥ d(x,a b ) → d(x,a b ) not necessary to compute !

15. Pacific Graphics 2011 15 | 25 Kaohsiung, Taiwan Our SortCluster-LPCA When assigning x Start from a j Proceed a i in increasing order of distances w.r.t. a j Check ∆-inequality For all a i precompute Distances to each others Ordering Precompute distances Generate seeds Classification Update Until converge using ∆-ineq.

16. Pacific Graphics 2011 16 | 25 Kaohsiung, Taiwan Improving Performance SortCluster-LPCA (SC-LPCA) Improving Accuracy SortMeans++ Results Outline

17. Pacific Graphics 2011 17 | 25 Kaohsiung, Taiwan Observation Error comes from poor selections of cluster centers LPCA Accuracy LPCA prone to stuck in local optimum

18. Pacific Graphics 2011 18 | 25 Kaohsiung, Taiwan Some heuristic Farthest first [Hochbaum et al. 85] Sum based [Hašan et al. 06] k-means++ [Arthur and Vassil. 07] … Our approach: SortMeans++ Based on k-means++ Faster Generation of Seeds Generate seeds Classification Update Until converge Precompute distances using ∆-ineq.

19. Pacific Graphics 2011 19 | 25 Kaohsiung, Taiwan Select initial seeds Our SortMeans++ Select 1 st seed at random Take 2 nd seed with probability equal distances Recluster using ∆-inequality Take 3 rd seed with probability equal distances Recluster …

20. Pacific Graphics 2011 20 | 25 Kaohsiung, Taiwan Improving Performance SortCluster-LPCA (SC-LPCA) Improving Accuracy SortMeans++ Results Outline

21. Pacific Graphics 2011 21 | 25 Kaohsiung, Taiwan Evaluate method with different parameters Scalability with the subspace dimension Results More than 5x speed-up

22. Pacific Graphics 2011 22 | 25 Kaohsiung, Taiwan Did several iterations of (SC)LPCA up to 24 basis Overall Performance ModelVertices [#]LPCASC-LPCASpeed UpPRT Horse 67.6k3h 48m11m20.3x22.5 s Dragon 57.5k3h 1m28m6.4x25.5 s Buddha 85.2k4h 38m50m5.6x31.6 s Disney 106.3k5h 50m1h 8m5.1x46.5 s

23. Pacific Graphics 2011 23 | 25 Kaohsiung, Taiwan Latency and accuracy of seeding algs: Improving Accuracy Our SortMeans++ Generally lowest error & fast Improve performance of SC-LPCA !

24. Pacific Graphics 2011 24 | 25 Kaohsiung, Taiwan SC-LPCA (accelerated LPCA) Avoid unnecessary distance comp. Speed-up of 5 to 20 on our PRT data Without changing output ! Conclusion Improve accuracy (SortMeans++) More accurate data approximation Future work Test on other CG data GPU acceleration Generate seeds Classification Update Until converge Precompute distances using ∆-ineq. SortMeans++

25. Pacific Graphics 2011 25 | 25 Kaohsiung, Taiwan SC-LPCA speed-up of about only 1.5x Reason: Small number of subspaces BTF Compression Try several data sets

26. Pacific Graphics 2011 26 | 25 Kaohsiung, Taiwan Acknowledgement European Community Marie Curie Fellowship PIOF-GA-2008- 221716 Ministry of Education, Czech Republic Research program LC-06008. Anonymous reviewers The End Thank You for your attention