1. Subspace Clustering Algorithms and Applications for Computer Vision Amir 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 are not 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.
Input: video with several motions
Output: Video with feature points clustered according to their motions

9. Video Motion Segmentation
Input: video with several motions
Output: Video with feature points clustered according to their motions

10. Affine Camera Model

11. Video Motion Segmentation
Objective: cluster the trajectories such that each cluster belongs to the motion (subspace) of a single object.

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
𝛬 1
V 1 𝑇
U 1
I 1 = 𝑖: 𝜆 𝑖 > 1 𝜏

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!