Subspace Clustering Algorithms and Applications for Computer
Vision Amir Adler
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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) 2
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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
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The Subspace Clustering Problem 4 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
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Subspace Clustering Challenges 5 Independent subspaces:
Disjoint subspaces: Independent Disjoint However, disjoint
subspaces are not necessarily independent, and considered more
challenging to cluster.
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) 7
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Video Motion Segmentation 8 Input: video frames of a scene with
multiple motions Output: Segmentation of tracked feature points
into motions.
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Video Motion Segmentation 9
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Affine Camera Model 10
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Video Motion Segmentation 11
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Video Motion Segmentation 12
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Temporal Video Segmentation 13 R. Vidal, Applications of GPCA
for Computer Vision, CVPR 2008.
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Face Clustering 14 Moghaddam & Pentland, Probabalistic
Visual Learning for Object Recognition, IEEE PAMI 1997.
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Face Clustering 15
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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) 16
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The Spectral Clustering Approach 17
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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) 18
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The Data Model 19
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Sparse Subspace Clustering (SSC) 20
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Self Expressive Data Single Subspace 21
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Self Expressive Data Multiple Subspaces 22
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Extension to Noisy Data 24
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Performance Evaluation 25 Applied to the motion segmentation
problem. Utilized the Hopkins-155 database:
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Performance Evaluation 26
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Paper Evaluation 27 Novelty Clarity Experiments Code
availability Limitations High complexity: O(L^2)+O(L^3) Sensitivity
to noise (data represented by itself)
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28 Low Rank Representation (LRR)
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29 Why Low Rank Representation(1/3)?
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30 Why Low Rank Representation(2/3)?
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31 Why Low Rank Representation(3/3)?
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32 Summary of the Algorithm
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33 Performance Face Clustering
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Paper Evaluation 34 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
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Closed Form Solutions 35 Favaro, Vidal & Ravichandran (CVPR
2011) Separation between clean and noisy data. Provides several
relaxations to:
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Case 1:Noiseless Data & Relaxed Constraint 36
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Noiseless Data & Relaxed Constraint 37
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Case 2: Noisy Data & Relaxed Constraints 38
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Polynomial Shrinkage Operator 39
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Performance Evaluation 40 The motion segmentation problem
(Hopkins-155). Case 1 algorithm. Comparable to SSC, LRR. Processing
time of 0.4 sec/sequence.
