1. Bootstrapping a Heteroscedastic Regression Model with Application to 3D Rigid Motion Evaluation Bogdan Matei Peter Meer Electrical and Computer Engineering Department Rutgers University

2. Rigorous method based on resampling the data Data must be independent and identically distributed (i.i.d.) Statistical measures computed from one data set Bootstrap Principle [Efron, 1979] Data Bootstrap samples Sampling with replacement Bootstrap replicates Example: covariance of the estimate

3. B = 50 TRUE ESTIMATE X Y B = 200 TRUE ESTIMATE X Y Bootstrap for Regression Resample from residuals Obtain bootstrap samples as Model Measurements

4. – pseudoinverse of the bootstrapped covariance matrix – the percentile of the distribution Building Confidence Regions Relation to error propagation –does not imply linearization –provides more accurate coverage –trades computation time for analytical derivations The ellipsoid contains the true estimate with probability

5. Heteroscedasticity Point dependent errors Appears in many 3D vision problems –due to linearization –multi-stage tasks e.g. estimating the 3D rigid motion of a stereo head

6. Total least squares (TLS) algorithm assumes i.i.d. data. Under heteroscedasticity yields biased solutions. Non-linear methods, like Levenberg-Marquard –may converge to local minima –are computationally intensive Proposed methods –renormalization [Kanatani, 1996] –HEIV algorithm [Leedan & Meer, ICCV’ 98; Matei & Meer, CVPR’ 99] Heteroscedastic Regression

7. Iterative method Can start from random initial solution Central module solves the generalized eigenvalue problem Provides consistent estimate Converges in less than 5 iterations It is the Maximum Likelihood solution for normal noise Multivariate HEIV Algorithm semi-positive definite matrices

8. Multivariate HEIV Algorithm The true values satisfy the linear constraint The true values are corrupted by heteroscedastic noise

9. Multivariate HEIV Algorithm Start with an initial solution Compute Find the scatter Update the solution as the smallest eigenvalue of

10. Error Analysis for Heteroscedastic Problems To analyze any algorithm applied to heteroscedastic data the bootstrap samples must be based on the HEIV residuals First order approximation of the HEIV estimate covariance

11. Bootstrap for Heteroscedastic Regression The measurements are not i.i.d. Need a consistent estimator for the residuals Use a whiten-color cycle to generate bootstrap samples Outliers must be eliminated with robust preprocessing Data Data correction Residuals Whitening Sampling with replacement Coloring B. samples B. replicates

12. 3D Rigid Motion of a Stereo Head 3D points recovered from stereo have heteroscedastic noise [Blostein et al., 1987] In quaternion representation the rigid motion constraint is True values are related Rigid motion estimation of a stereo head is a multivariate heteroscedastic regression problem

13. The corrected measurements are The covariance of the residuals The covariance matrices of the 3D points, are obtained through bootstrap Error Analysis of 3D Rigid Motion

14. Evaluation of 3D Rigid Motion Methods Methods –quaternion [Horn et al., 1988] and SVD [Arun et al., 1987] algorithms give identical results. Both are TLS type (biased). –HEIV algorithm B = 200 bootstrap replicates were used for the covariances (confidence regions) of the motion parameters Angle-axis representation for the rotation matrix Using error propagation is very difficult [Pennec & Thirion, 1997]

15. Bootstrap compared with Monte Carlo analysis –Monte Carlo uses the true data and the true noise distribution –bootstrap uses only the available measurements Synthetic Data bootstrap: ‘o’ HEIV ‘x’ quaternion/SVD bootstrap ‘ ’ HEIV ‘+’ quaternion/SVD Translation errorRotation error

16. Real Data Four images, planar texture sequence (CIL-CMU) –ground truth about the relative position of the frames available Frame 1 Points were matched using Z. Zhang’s program 3D data recovered by triangulation [Hartley, 1997] Frame 4

17. Real Data Translation estimate quaternion/SVD Translation estimate HEIV Bootstrap confidence regions with 0.95 probability of coverage

18. Real Data Rotation estimate quaternion/SVD Rotation estimate HEIV Bootstrap confidence regions with 0.95 probability of coverage

19. Conclusions The HEIV algorithm is a general tool for 3D vision Bootstrap can supplement the execution of a vision task with statistical information which –captures the actual operating conditions –reduces the dependence on simplifying assumptions Confidence regions in the input domain can provide uncertainty information about the true locations of features