Non-Rigid Shape and Motion Recovery: Degenerate Deformations

1. Non-Rigid Shape and Motion Recovery: Degenerate Deformations Jing Xiao and Takeo Kanade CVPR 2004

2. Problem Addressed • Ambiguity in non-rigid SFM if only Rotation Constraints used (ECCV 2004) • SFM recovery in Degenerate Deformations

3. Basis Formulation for Non-Rigid Shape • Shape at any time instant:

4. With Degenerate Deformation • Of the K bases • K1 are rank 1 • K2 are rank 2 • K3 are rank 3 • Kd = K1 + 2xK2 + 3xK3 • W=MB is not a unique decomposition • Essentially the problem is to find G

5. Degenerate cont’d • First K3 triple columns of M correspond to non-degenerate basis and the rest to degenerate basis • rj is a 3x1 eigen vector that corresponds to the degenerate basis shape • Arises cause Bi (degenerate) can be factored as:

6. Rotation Constraints • Denote by Qi, we have: • Qi has unknowns, so given enough frames we can find a solution? • This is not true in general • Qi can be written as where H satisfies:

7. Rotation cont • is an arbitrary scalar and is a skew symmetric matrix • The solution has degrees of freedom

8. Basis Constraints • Ambiguity cause any non-singular transformation on the bases gives another valid set of bases. • To handle this choose the set of K3 frames that have smallest condition number (ECCV paper) • Denote the chosen frames as the first K3 frames • Coefficients will be:

9. Basis contd • New constraints: • Finally combining Rotation and Basis constraints:

10. Constraints cont’d • degrees of freedom and linear constraints. • Therefore solution space has degrees of freedom. • When ND is 0, there is a unique solution.

11. Finding Qi • If K2>0 then • If K2=0, a unique solution exists using the constraints.

12. Results • Synthetic data:

13. Results • Real Data: