Robust GPCA Algorithm with Applications in Video Segmentation Via Hybrid System Identification

1. MTNS 2004 Robust GPCA Algorithm with Applications in Video Segmentation Via Hybrid System Identification Kun Huang and Yi Ma Perception & Decision Laboratory Decision & Control Group, CSL University of Illinois at Urbana-Champaign http://black.csl.uiuc.edu/~kunh

2. GPCA – Problem Formulation • Problem Statement [Multiple Linear Model Fitting] Given a set ofNnoisy data points sampled froms different subspaces with possibly different dimensions in aK-dimensional ambient space: • Estimate the number of subspacessand the dimensionki • (i=1,2,…,s) of each subspace; identify a basis for each subspace; • Segment the given data points into the subspaces. • Difficulties • Are the two subspaces of dimension 2? • Is there one plane and two lines? • We can fit all points with zero fitting error • All points live in 1 subspace: R^3 • Each point lives in its own subspace: a line passing through the point

3. GPCA – Robust Algorithm • Contributions of the Robust GPCA Algorithm Research • Design a new model selection criterion for multiple linear subspaces. • Develop a Robust Recursive GPCA algorithm. • Recursively identify the correct number of subspaces and their dimensions and bases. • Robustly segment the data points based on specified maximum error tolerance.

4. GPCA – Effective Dimension • Model selection criteria • MML, MDL, AIC, G-AIC, Robust AIC Balance model complexity and data fidelity. • Effective dimension • Specifically developed for mixture of linear models (subspaces) Dimension of each subspace Number of subspaces Total number of points Number of points in each subspace

5. GPCA – Minimum Effective Dimension • Example Model selection criterion: Minimum Effective Dimension (MED)

6. GPCA - MED and Robust GPCA Algorithm MED of a data set is closely related to error tolerance t. • Extreme cases: If t is infinity, then MED=0; if t is 0, then MED=K. Robust approach for mixture linear model fitting: For a specified maximum error tolerance t, find the subspace that minimizes the Effective Dimension (ED) of the data set. t t Robust recursive GPCA algorithm • Recursively segment each group; • Automatically search for the number and the dimensions of the subspaces; • Assign points to the subspace based on the specified error tolerance t; • Accommodate outliers.

7. ED=3 ED=2.0067 ED=1.6717 A ROBUST RECURSIVE ALGORITHM – A Simulation Example

8. Mixture of LTI Systems HYBRID SYSTEM IDENTIFICTION –Problem Formulation Single LTI System

9. Hybrid LTI System Identification HYBRID SYSTEM IDENTIFICTION –Problem Formulation Hybrid LTI System Switching function

10. HYBRID SYSTEM IDENTIFICTION –Embedding • Embed the input ut and output yt in a high-dimensional space. • The embedded data point resides on a subspace defined by • the system. • Mixture of LTI systems generate a mixture of linear subspaces. • The mixture of linear subspaces can be identified using the • GPCA algorithms. Two ways of embedding: • Embedding via the oblique projection (Overschee et. al. ’96); • Direct input/output embedding.

11. InputBlock Hankel Matrix HYBRID SYSTEM IDENTIFICTION –Notations Past Input Future Input Future Output Past Output OutputBlock Hankel Matrix

12. Embedding via the oblique projection (Overschee et. al. ’96) • Subject to • The covariance matrix of the input block Hankel matrix is 2i; • The intersection of the row spaces of future input and past state is trivial. HYBRID SYSTEM IDENTIFICTION –Oblique Projection Orthogonal Projection Oblique Projection

13. Embedded data point HYBRID SYSTEM IDENTIFICTION –Direction Embedding • Direct input/output embedding • The embedded data points are on a subspace. • If the system is observable and , .

14. HYBRID LTI SYSTEM IDENTIFICATION - Simulations 4th order 3rd order 1st order Data points around the switching points do NOT belong to any of the three subspaces. They cause outliers!

15. HYBRID LTI SYSTEM IDENTIFICATION -Simulations 4th order 3rd order 1st order 4th order 1st order 3rd order 4th order 3rd order 1st order

16. Segmentation of the embedded data HYBRID LTI SYSTEM IDENTIFICATION -Simulations • Embedding via the oblique projection Average errors of the subspaces for 1000 trials • Direct input/output embedding

17. APPLICATIONS – Video Segmentation Experiment Setup: Testing sequence (150 image frames) Image Size: 352X240 Error Tolerance for the GPCA Algorithm: 0.05 rad Outlier Tolerance for the GPCA Algorithm: 15% m=3 Image Frames Grayscale Images PCA m-dimensional space

18. Camera Zooming Out APPLICATIONS – Segmentation of Embedded Output m=3, k=2

19. CONCLUSION • Present a new model selection criterion for multiple linear subspaces. • Develop a robust GPCA algorithm to recursively segment the data points into multiple subspaces with different dimensions. • The hybrid LTI system identification problem can be converted into a GPCA problem. Two ways of embedding are provided. • Video sequences can be modeled using hybrid linear systems. The robust GPCA algorithm can segment the video sequence based on the hidden dynamics via the embedding of the system output.