1 Point registration via efficient convex relaxation Haggai
![6 Previous work: Low dimensional PM • RANSAC [Fischler and Bolles 1981] Exponential in](https://slidetodoc.com/presentation_image/2a398e891507751dec499576d6865ee4/image-6.jpg "6 Previous work: Low dimensional PM • RANSAC [Fischler and Bolles 1981] Exponential in")
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- Slides: 37
1 Point registration via efficient convex relaxation Haggai Maron, Nadav Dym, Itay Kezurer, Shahar Kovalsky, Yaron Lipman Weizmann Institute of Science
2 Orthogonal Procrustes Problem Orthogonal
3 Procrustes matching (PM) Permutation Orthogonal
4 Procrustes matching (PM) Orthogonal Given Orthogonal Point clouds Permutation
5 Motivation embedding 3 D Non Rigid Alignment High dimensional Rigid Alignment [Jain et al. 2006, Ovsjanikov et al. 2008]
6 Previous work: Low dimensional PM • RANSAC [Fischler and Bolles 1981] Exponential in the dimension • Combinatorial optimization [Gelfand et al. 2005, Yang et al. 2013] Worst case: exponential • ICP [Besl and Mckey 1992] Only local minimum guaranteed
7 Previous work: Shape matching • Functional maps [Ovsjanikov et al. 2012] Depends on functional correspondences • SDP relaxations [Kezurer et al. 2015] Works for small-scale problems • LP relaxations [Chen and Koltun, 2015] Relies on extrinsic alignment
9 Convex Relaxation Orthogonal Permutation
10 Convex Relaxation PM Convex Superset
11 Convex Relaxation Slow & Tight LP SDP PM Fast & Loose
12 Why SDP relaxation? Theorem: If No noise, No symmetries Then SDP relaxation is exact! SDP
13 Why SDP relaxation? Theorem: No symmetries? If No noise, No symmetries Then SDP relaxation is exact!
14 Why SDP relaxation? Theorem: If No noise, bilateral symmetries Then SDP relaxation is exact!
15 Our Approach Use SDP relaxation for real problems Idea: exploit PM structure SDP PM
16 1
17 1 Huge! # points dimension
18 1
19 1
20 1
21 1
22 1
23 1
24 1 1
25 1 1
26 1 1 1
27 Slow Fast Tight?
28 Theorem: Equivalent Relaxations!
29 Theorem: Equivalent Relaxations!
30 Scalability of PM-SDP
31 Results: Isometric matching SCAPE dataset Source [Anguelov at al. 05] Target
32 Isometric matching SCAPE Raw Scans Source [Anguelov at al. 05] Target
33 Non-Isometric matching % Correspondences FAUST Source [Bogo et al. 14] Target Intra subject Inter subject
34 Non-Isometric matching [Giorgi et al. 07] SHREC 07’ Failure
35 Limitations •
36 Summary •
37 Acknowledgments: - European Research Council (ERC Starting Grant) - Israel Science Foundation Thank you! • Code Available online! http: //www. wisdom. weizmann. ac. il/~haggaim/ • More results in the paper! Anatomical classification Collection alignment