Registration of Point Cloud Data from a Geometric Optimization Perspective

1. Registration of Point Cloud Data from a Geometric Optimization Perspective Niloy J. Mitra1, Natasha Gelfand1, Helmut Pottmann2, Leonidas J. Guibas1 1 Stanford University 2 Vienna University of Technology

2. Q P data model Registration Problem Given Two point cloud data sets P (model) and Q (data) sampled from surfaces P and Q respectively. Assume Q is a part of P.

3. Q P data model Registration Problem Given Two point cloud data sets P and Q. Goal Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.

4. Registration Problem Given Two point cloud data sets P and Q. Goal Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms. Contributions • use of second order accurate approximation to the squared distance field. • no explicit closest point information needed. • proposed algorithm has good convergence properties.

5. Related Work • Iterated Closest Point (ICP) • point-point ICP [ Besl-McKay ] • point-plane ICP [ Chen-Medioni ] • Matching point clouds • based on flow complex [ Dey et al. ] • based on geodesic distance [ Sapiro and Memoli ] • MLS surface for PCD [ Levin ]

6. Notations

7. Notations

8. Squared Distance Function (F) x

9. Squared Distance Function (F) x

10. Rigid transform thattakes points Registration Problem Our goal is to solve for, An optimization problem in the squared distance field of P, the model PCD.

11. Registration Problem Our goal is to solve for, Optimize for R and t.

12. Overview of Our Approach • Construct approximate such that, to second order. • Linearize a. • Solve to get a linear system. • Apply a to data PCD (Q) and iterate.

13. Registration in 2D • Quadratic Approximant

14. Registration in 2D • Quadratic Approximant • Linearize rigid transform

15. Registration in 2D • Quadratic Approximant • Linearize rigid transform • Residual error

16. Registration in 2D • Minimize residual error Depends on F+ and data PCD (Q).

17. Registration in 2D • Minimize residual error • Solve for R and t. • Apply a fraction of the computed motion • F+ valid locally • Step size determined by Armijo condition • Fractional transforms [Alexa et al. 2002]

18. Registration in 3D • Quadratic Approximant • Linearize rigid transform • Residual error Minimize to get a linear system

19. valid in the neighborhood of x Approximate Squared Distance • Two methods for estimating F • d2Tree based computation • On-demand computation

20. F(x, P) using d2Tree • A kd-tree like data structure for storing approximants of the squared distance function. • Each cell (c) stores a quadratic approximant as a matrix Qc. • Efficient to query. [ Leopoldseder et al. 2003]

21. F(x, P) using d2Tree • A kd-tree like data structure for storing approximants of the squared distance function. • Each cell (c) stores a quadratic approximant as a matrix Qc. • Efficient to query. • Simple bottom-up construction • Pre-computed for a given PCD. Closest point information implicitly embedded in the squared distance function.

22. Example d2trees 2D 3D

23. Y Approximate Squared Distance For a curve Y, [ Pottmann and Hofer 2003 ]

24. Approximate Squared Distance For a curve Y, For a surface F, [ Pottmann and Hofer 2003 ]

25. On-demand Computation • Given a PCD, at each point p we pre-compute, • a local frame • normal • principal direction of curvatures • radii of principal curvature (r1and r2)

26. On-demand Computation • Given a PCD, at each point p we pre-compute, • a local frame • normal • principal direction of curvatures • radii of principal curvature (r1and r2) Estimated from a PCD using local analysis • covariance analysis for local frame • quadric fitting for principal curvatures

27. d/(d-j) if d < 0 0 otherwise. where j = On-demand Computation Given a point x, nearest neighbor (p) computed using approximate nearest neighbor (ANN) data structure

28. Iterated Closest Point (ICP) • Iterate • Find correspondence between P and Q. • closest point (point-to-point). • tangent plane of closest point (point-to-plane). • Solve for the best rigid transform given the correspondence.

29. ICP in Our Framework • Point-to-point ICP (good for large d) • Point-to-plane ICP (good for small d)

30. Convergence Properties • Gradient decent over the error landscapeGauss -Newton Iteration • Zero residue problem (model and data PCD-s match)Quadratic Convergence • For fractional steps, Armijo condition usedDamped Gauss-Newton IterationLinear convergence • can be improved by quadratic motion approximation (not currently used)

31. Convergence Funnel Set of all initial poses of the data PCD with respect to the model PCD that is successfully aligned using the algorithm. Desirable properties • broad • stable

32. Convergence Funnel Translation in x-z plane. Rotation about y-axis. Converges Does not converge

33. Convergence Funnel Plane-to-plane ICP Our algorithm

34. Convergence Rate I Bad Initial Alignment

35. Convergence Rate II Good Initial Alignment

36. Partial Alignment Starting Position

37. Partial Alignment After 6 iterations

38. Partial Alignment Different sampling density After 6 iterations

39. Future Work • Partial matching • Global registration • Non-rigid transforms

40. Questions?