Geometrically Stable Sampling for the ICP Algorithm

1. Geometrically Stable Sampling for the ICP Algorithm

2. Point-point error Point-plane error ni qi qi pi pi Two Flavors of ICP • faster to converge

3. Source of Failures • Complex error landscape • Many local minima • Usually have to start from a good guess • Point selection and error metric in minimization are key to convergence • Some shapes are particularly “difficult” • Noise

4. Requirements for Error Metric • Planar and spherical areas with no features should not contribute to distance metric • “Lock and key” features should pull the surfaces to correct relative pose • Robust even as features get small and featureless regions noisy

5. Solution • Detect when parts of the input patch are prone to sliding • Use covariance analysis • Concentrate samples in “lock and key” areas to improve convergence • Use geometrically stable point-selection strategy • Similar to normal-space sampling [Rusinkiewicz & Levoy] • Detect when input has no “lock and key” features • Do not align that scan pair at all • Improves convergence of global registration [Ikemoto et.al]

6. ni Minimization Equation • Find (R, t) that minimize • Linearizing rotations: find r=[rx ry rz]T, t=[tx ty tz]T that minimize qi pi

7. “torque” “force” P P Q Q Analyzing the Metric (1) • Change in error from each point-pair: • For some transformations e=0: translation perpendicular to “force”

8. “torque” “force” P P Q Q Analyzing the Metric (2) • Change in error from each point-pair: • For some transformations e=0: rotation perpendicular to “torque”

9. Covariance Matrix • Aligning transform is given by Cx=b, where • C encodes the change in error when surfaces are moved from optimal alignment

10. Sliding Directions • Eigenvectors of C with small eigenvalues correspond to sliding transformations 3 small eigenvalues 2 translation 1 rotation 2 small eigenvalues 1 translation 1 rotation 3 small eigenvalues 3 rotation 1 small eigenvalue 1 translation 1 small eigenvalue 1 rotation

11. Sample Selection • Goal of sampling is to produce covariance matrix C with a good condition number • Equally constrain all eigenvectors • Find points that well sample the small features on P • Since P and Q are similar in the overlap region, use both points and normals from P to compute C

12. Implementation (1) • Estimate C from sparse uniform sampling of input mesh • C = XLXT • 6x6 matrix • 6 eigenvectors • form basis for the space of transforms

13. x1 x2 x3 x4 x5 x6 p5 p5 p3 p8 p1 p1 p3 p6 p7 p7 p2 p2 p1 p7 p8 p5 p3 p3 p7 p8 p9 p9 p4 p4 … … … … … … Implementation (2) • For each point form • For each eigenvector x j compute • Magnitude of the constraint piexerts on x j • Sort points in decreasing “order of influence” for each eigenvector

14. Implementation (3) • Let be an estimate of how well eigenvector x j is constrained by already chosen points • Choose next pi from from list with the smallest corresponding t j • Most unconstrained eigenvector • Compute closest point qi in Q • Once enough points are chosen, solve for alignment and iterate

15. Sample Selection • Small eigenvalues

16. Sample Selection • Large eigenvalues

17. Without noise, any sampling works fine Similar to normal-space sampling for planar regions Results (Planar Patches)

18. uniform sampling stable sampling Results (Planar Patches) • 25 iterations to converge • Naïve implementation about 3 times slower per iteration than uniform sampling

19. Results (Spherical Patches) • Normal-space sampling cannot align this

20. Conclusions • Point-plane error metric can have shallow error landscape when registering shapes with small features • Noise adds many small local minima • These conditions can be detected by analyzing the covariance matrix of the minimization equation • Eigenvectors with small eigenvalues give sliding directions • Stable sampling improves these landscapes by equally constraining all eigenvectors • Selects point samples inside the features • Allows ICP to align “difficult” input

21. Limitations: Noise • The algorithm can fail if input data is particularly noisy • Smoothing can help

22. Limitations: Noise • Success depends on relative size of features

23. Extensions and Future Work • Use stability analysis to weigh mesh pairs in global registration • A hierarchical method for aligning warped meshes [Ikemoto et al] • Stable sampling for global registration • Simultaneous covariance analysis of the entire set of scans • Weighing eigenvectors based on the area outside the overlap • Large vs. small leverage on the entire scan

24. Extensions and Future Work • Use of stability analysis as shape descriptor • Segmentation into areas with similar sliding transformations