Subspace Clustering Algorithms and Applications for Computer Vision Amir Adler

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)

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)

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

Subspace Clustering Challenges Independent subspaces: Disjoint subspaces: Independent  Disjoint However, disjoint subspaces are not necessarily independent, and considered more challenging to cluster.

Subspace Clustering Challenges Intersecting subspaces: Corrupted data: Noise Outliers

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)

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

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

Affine Camera Model

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

Video Motion Segmentation

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

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

Face Clustering

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)

The Spectral Clustering Approach

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)

The Data Model

Sparse Subspace Clustering (SSC)

Self Expressive Data – Single Subspace

Self Expressive Data –Multiple Subspaces

Extension to Noisy Data

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

Performance Evaluation

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

Low Rank Representation (LRR)

Why Low Rank Representation(1/3)?

Why Low Rank Representation(2/3)?

Why Low Rank Representation(3/3)?

Summary of the Algorithm

Performance – Face Clustering

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

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

Case 1:Noiseless Data & Relaxed Constraint 𝛬 1 V 1 𝑇 U 1 I 1 = 𝑖: 𝜆 𝑖 > 1 𝜏

Noiseless Data & Relaxed Constraint

Case 2: Noisy Data & Relaxed Constraints ⇓

Polynomial Shrinkage Operator

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

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

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