1. Robust Global Registration Natasha Gelfand Niloy Mitra Leonidas Guibas Helmut Pottmann

2. Registration Problem Given: Two shapes P and Q which partially overlap. Goal: Using only rigid transforms, register Q against P by minimizing the squared distance between them.

3. Applications Shape analysis –similarity, symmetry detection, rigid decomposition Shape acquisition [Koller 05][Pauly 05] [ Anguelov 04]

4. Approaches Iterative minimization algorithms Voting methods Geometric descriptors

5. Approaches I Iterative minimization algorithms (ICP) Properties –Dense correspondence sets –Converges if starting positions are “close” 2. Align corresponding points 3. Iterate 1. Build a set of corresponding points [Besl 92, Chen 92]

6. Approaches II Voting methods –Geometric hashing, Hough transform, alignment method Rigid transform can be specified with small number of points –Try all possible transform bases –Find the one that aligns the most points Guaranteed to find the correct transform –But can be costly [Stockman 87, Hecker 94, Barequet 97]

7. Approaches III Use neighborhood geometric information Descriptors should be –Invariant under transform –Local –Cheap Improves correspondence search or used for feature extraction [Mokh 01, Huber 03, Li 05]

8. Registration Problem Why it’s hard –Unknown areas of overlap –Have to solve the correspondence problem Why it’s easy –Rigid transform is specified by small number of points –Prominent features are easy to identify We only need to align a few points correctly.

9. Method Overview 1.Use descriptors to identify features –Integral volume descriptor 2.Build correspondence search space –Few correspondences for each feature 3.Efficiently explore search space –Distance error metric –Pruning algorithms

10. Integral Descriptors Multiscale Inherent smoothing f(x) p B r (p) [Manay 04]

11. Integral Volume Descriptor Relation to mean curvature 0.20

12. Descriptor Properties Relation to curvature Influence of noise

13. Descriptor Computation Approximate using a voxel grid –Convolution of occupancy grid with ball

14. Feature Identification Pick as features points with rare descriptor values –Rare in the data  rare in the model  few correspondences Works for any descriptor Features

15. Multiscale Algorithm Features should be persistent over scale change r=0.5r=1r=2r=4r=8 YYYNN NNYYY NNYNN

16. Feature Properties Sparse Robust to noise Non-canonical

17. Search the whole model for correspondences –Range query for descriptor values –Cluster and pick representatives Correspondence Space P Q

18. Evaluating Correspondences Coordinate root mean squared distance –Requires best aligning transform –Looks at correspondences individually

19. Rigidity Constraint Pair-wise distances between features and correspondences should be the same P Q

20. Rigidity Constraint Pair-wise distances between features and correspondences should be the same P Q

21. Rigidity Constraint Pair-wise distances between features and correspondences should be the same P Q

22. Evaluating Correspondences Distance root mean squared distance –Depends only on internal distance matrix

23. Search Algorithm Few features, each with few potential correspondences –Minimize dRMS –Exhaustive search still too expensive P Q

24. Branch and bound –Initial bound using greedy assignment Discard partial correspondences that fail thresholding test – Prune if partial correspondence exceeds bound –Spaced out features make incorrect correspondences fail quickly Since we explore the entire search space, we are guaranteed to find optimal alignment –Up to cluster size Search Algorithm pipi pjpj qjqj qiqi RcRc

25. Branch and bound –Initial bound using greedy assignment Discard partial correspondences that fail thresholding test – Prune if partial correspondence exceeds bound –Spaced out features make incorrect correspondences fail quickly Since we explore the entire search space, we are guaranteed to find optimal alignment –Up to cluster size

26. Greedy Initialization Good initial bound is essential Build up correspondence set hierarchically …

27. Greedy Initialization Good initial bound is essential Build up correspondence set hierarchically … …

28. Greedy Initialization Good initial bound is essential Build up correspondence set hierarchically …

29. Partial Alignment Allow null correspondences, while maximizing the number of matches points

30. Alignment Results Input: 2 scansOur alignmentRefined by ICP

31. Alignment Results Input: 10 scans Our alignmentRefined by ICP

32. Symmetry Detection Symmetry detection –Match a shape to itself

33. Rigid Decomposition Repeated application of partial matching

34. Conclusions Simple feature identification algorithm –Used integral volume descriptor –Applicable for any low-dimensional descriptor Correspondence evaluation using dRMS error –Look at pairs of correspondences, instead of individually Efficient branch and bound correspondence search –Finds globally best alignment

35. Future Work Linear features

36. Future Work Linear features Fast rejection tests More descriptors

37. Acknowledgements Daniel Russel An Nguyen Doo Young Kwon Digital Michelangelo Project NSF CARGO-0138456, FRG-0454543, ARO DAAD 19-03-1-033, Max Plank Fellowship, Stanford Graduate Fellowship, Austrian Science Fund P16002- N05

38. Questions?