Subspace Clustering Algorithms and Applications for Computer Vision Amir Adler

1. Subspace ClusteringAlgorithms and Applications for Computer VisionAmir Adler

2. Agenda • The Subspace Clustering Problem • Computer Vision Applications • A Short Introduction to Spectral Clustering • Algorithms • Sparse Subspace Clustering (CVPR 2009) • Low Rank Representation (ICML 2010) • Closed Form Solutions (CVPR 2011)

3. Agenda • The Subspace Clustering Problem • Computer Vision Applications • A Short Introduction to Spectral Clustering • Algorithms • Sparse Subspace Clustering (CVPR 2009) • Low Rank Representation (ICML 2010) • Closed Form Solutions (CVPR 2011)

4. The Subspace Clustering Problem • Given a set of points drawn from a union-of-subspaces, obtain the following: • 1) Clustering of the points • 2) Number of subspaces • 3) Bases of all subspaces • Challenges: • 1) Subspaces layout • 2) Corrupted data

5. Subspace Clustering Challenges • Independent subspaces: • Disjoint subspaces: • Independent  Disjoint • However, disjoint subspaces arenot necessarily independent, and considered more challenging to cluster.

6. Subspace Clustering Challenges • Intersecting subspaces: • Corrupted data: • Noise • Outliers

7. Agenda • The Subspace Clustering Problem • Computer Vision Applications • A Short Introduction to Spectral Clustering • Algorithms • Sparse Subspace Clustering (CVPR 2009) • Low Rank Representation (ICML 2010) • Closed Form Solutions (CVPR 2011)

8. Video Motion Segmentation • Input: video frames of a scene with multiple motions • Output: Segmentation of tracked feature points into motions.

9. Video Motion Segmentation

10. Affine Camera Model

11. Video Motion Segmentation

12. Video Motion Segmentation

13. Temporal Video Segmentation R. Vidal, “Applications of GPCA for Computer Vision”, CVPR 2008.

14. Face Clustering Moghaddam & Pentland, “Probabalistic Visual Learning for Object Recognition”, IEEE PAMI 1997.

15. Face Clustering

16. Agenda • The Subspace Clustering Problem • Computer Vision Applications • A Short Introduction to Spectral Clustering • Algorithms • Sparse Subspace Clustering (CVPR 2009) • Low Rank Representation (ICML 2010) • Closed Form Solutions (CVPR 2011)

17. The Spectral Clustering Approach

18. Agenda • The Subspace Clustering Problem • Computer Vision Applications • A Short Introduction to Spectral Clustering • Algorithms • Sparse Subspace Clustering (CVPR 2009) • Low Rank Representation (ICML 2010) • Closed Form Solutions (CVPR 2011)

19. The Data Model

20. Sparse Subspace Clustering (SSC)

21. Self Expressive Data – Single Subspace

22. Self Expressive Data –Multiple Subspaces

24. Extension to Noisy Data

25. Performance Evaluation • Applied to the motion segmentation problem. • Utilized the Hopkins-155 database:

26. Performance Evaluation

27. Paper Evaluation • Novelty • Clarity • Experiments • Code availability • Limitations • High complexity: O(L^2)+O(L^3) • Sensitivity to noise (data represented by itself)

28. Low Rank Representation (LRR)

29. Why Low Rank Representation(1/3)?

30. Why Low Rank Representation(2/3)?

31. Why Low Rank Representation(3/3)?

32. Summary of the Algorithm

33. Performance – Face Clustering

34. Paper Evaluation • Novelty • Clarity • Experiments • Code availability • Limitations • High complexity: kO(L^3), k=200~300 • Sensitivity to noise (data represented by itself) • Parameter setting not discussed

35. Closed Form Solutions • Favaro, Vidal & Ravichandran (CVPR 2011) • Separation between clean and noisy data. • Provides several relaxations to:

36. Case 1:Noiseless Data & Relaxed Constraint =

37. Noiseless Data & Relaxed Constraint

38. Case 2: Noisy Data & Relaxed Constraints

39. Polynomial Shrinkage Operator

40. Performance Evaluation • The motion segmentation problem (Hopkins-155). • Case 1 algorithm. • Comparable to SSC, LRR. • Processing time of 0.4 sec/sequence.

41. Paper Evaluation • Novelty • Clarity • Experiments • Partial Complexity Analysis • Spectral clustering remains O(L^3) • Parameter setting unclear

42. Thank You!