A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation

1. 2006. 6. 9 (Fri) Young Ki Baik, Computer Vision Lab. A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation

2. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • References • A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation • Rene Vidal and Yi Ma (ECCV 2004) • Generalized Principal Component Analysis (GPCA) • Rene Vidal, Yi Ma, et. al. (PAMI 2005)

3. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Contents • Introduction • GPCA • 2-D motion segmentation • 3-D motion segmentation • Experimental results • Summary

4. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Introduction (Motion segmentation) • Target • 2D, 3D motion segmentation • Problem statement • Most previous work • Iterative approach (EM, RANSAC, etc.) • Manual or random initial value. • Cause of divergence or bad results • Proposed algorithm • Good initial value using non-iterative algebraic method

5. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Introduction (This paper) • Contribution • Applying GPCA to 2D, 3D motion segmentation problem • Condition • Subspaces are all linear. • Known correspondences • Known number of subspace (class)

6. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • GPCA (Generalized PCA) • GPCA treats heterogeneous data and multiple subset with different linear model.

7. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • GPCA (Generalized PCA) • Find the basis of each subspace which orthogonal to data x

8. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • GPCA (Generalized PCA) • A homogeneous polynomial of degree n • The mixture of subspaces can be linearly fitting general polynomial to the given data. • Example • Number of subspace n = 2 • Vector size k = 3

9. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • GPCA (Generalized PCA) • Find the basis from derivatives of the polynomials with y on the S

10. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • GPCA (Algorithm) • Fitting polynomials to data lying in multiple subspaces • Obtaining basis of each subspace by polynomial differentiation • Choosing data per subspace by using basis

11. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Motion segmentation • Notation • Let be a vector in or .

12. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 2D Motion segmentation • Translation case • Under 2-d translation motion model, the two images are related by on out of n possible 2-d translation .

13. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 2D Motion segmentation • Translation case • Finding coefficient c (fitting polynomial)

14. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 2D Motion segmentation • Translation case • Finding basis (using polynomial differentiation)

15. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 2D Motion segmentation • Translation case • Segmentation

16. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 2D Motion segmentation • Affine motion case • In this case, we assume that the images are related by a collection on n 2-D affine motion models .

17. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • 3D Motion segmentation • Epipolar constraint • Homography

18. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Experimental results

19. A Unified Algebraic Approach to 2D and 3D Motion Segmentation • Summary • Contribution • Applying GPCA to 2D, 3D motion segmentation problem • Good initial value using non-iterative algebraic method • Limitation • Linear subspace • Known correspondences and number of subspace • If subspace is increased, then computational complexity will be exponentially increased.