1 Point registration via efficient convex relaxation Haggai

  • Slides: 37
Download presentation
1 Point registration via efficient convex relaxation Haggai Maron, Nadav Dym, Itay Kezurer, Shahar

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

2 Orthogonal Procrustes Problem Orthogonal

3 Procrustes matching (PM) Permutation Orthogonal

3 Procrustes matching (PM) Permutation Orthogonal

4 Procrustes matching (PM) Orthogonal Given Orthogonal Point clouds Permutation

4 Procrustes matching (PM) Orthogonal Given Orthogonal Point clouds Permutation

5 Motivation embedding 3 D Non Rigid Alignment High dimensional Rigid Alignment [Jain et

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

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

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

9 Convex Relaxation Orthogonal Permutation

10 Convex Relaxation PM Convex Superset

10 Convex Relaxation PM Convex Superset

11 Convex Relaxation Slow & Tight LP SDP PM Fast & Loose

11 Convex Relaxation Slow & Tight LP SDP PM Fast & Loose

12 Why SDP relaxation? Theorem: If No noise, No symmetries Then SDP relaxation is

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

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

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

15 Our Approach Use SDP relaxation for real problems Idea: exploit PM structure SDP PM

16 1

16 1

17 1 Huge! # points dimension

17 1 Huge! # points dimension

18 1

18 1

19 1

19 1

20 1

20 1

21 1

21 1

22 1

22 1

23 1

23 1

24 1 1

24 1 1

25 1 1

25 1 1

26 1 1 1

26 1 1 1

27 Slow Fast Tight?

27 Slow Fast Tight?

28 Theorem: Equivalent Relaxations!

28 Theorem: Equivalent Relaxations!

29 Theorem: Equivalent Relaxations!

29 Theorem: Equivalent Relaxations!

30 Scalability of PM-SDP

30 Scalability of PM-SDP

31 Results: Isometric matching SCAPE dataset Source [Anguelov at al. 05] Target

31 Results: Isometric matching SCAPE dataset Source [Anguelov at al. 05] Target

32 Isometric matching SCAPE Raw Scans 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

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

34 Non-Isometric matching [Giorgi et al. 07] SHREC 07’ Failure

35 Limitations •

35 Limitations •

36 Summary •

36 Summary •

37 Acknowledgments: - European Research Council (ERC Starting Grant) - Israel Science Foundation Thank

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

CONSIDER YOUR WAYS HAGGAI Haggai Haggai was aCONSIDER YOUR WAYS HAGGAI Haggai Haggai was a
Convex Mirrors Convex Mirrors Convex Mirror F andConvex Mirrors Convex Mirrors Convex Mirror F and
Harmonization exercises III Relaxation techniques Relaxation techniques RelaxationHarmonization exercises III Relaxation techniques Relaxation techniques Relaxation
RELAXATION MUSCULAIRE RELAXATION INTERNE RELAXATION MENTALE EXPRIENCE DERELAXATION MUSCULAIRE RELAXATION INTERNE RELAXATION MENTALE EXPRIENCE DE
Efficient Convex Relaxation for Transductive Support Vector MachineEfficient Convex Relaxation for Transductive Support Vector Machine
Efficient Convex Relaxation for Transductive Support Vector MachineEfficient Convex Relaxation for Transductive Support Vector Machine
Prioritized Living Haggai 1 1 15 Haggai 1Prioritized Living Haggai 1 1 15 Haggai 1
Das Buch Haggai Der Prophet Haggai und seineDas Buch Haggai Der Prophet Haggai und seine
HAGGAI an introduction HOW DO YOU PRONOUNCE HAGGAIHAGGAI an introduction HOW DO YOU PRONOUNCE HAGGAI
Scripture Reading Haggai 2 1 9 Haggai 2Scripture Reading Haggai 2 1 9 Haggai 2
Haggai Consider your ways Background Haggai was aHaggai Consider your ways Background Haggai was a
HAGGAI Priorities Excuses Haggai 1 2 4 2HAGGAI Priorities Excuses Haggai 1 2 4 2
HAGGAI ZECHARIAH MALACHI LESSON 2 INTRODUCTION TO HAGGAIHAGGAI ZECHARIAH MALACHI LESSON 2 INTRODUCTION TO HAGGAI
Haggai Lesson 4 Text Haggai 2 20 23Haggai Lesson 4 Text Haggai 2 20 23
convex concave convex concave Eigenfaces PhotobookEigenfaces MIT Mediaconvex concave convex concave Eigenfaces PhotobookEigenfaces MIT Media
Convex Hull obstacle start end Convex Hull 1Convex Hull obstacle start end Convex Hull 1
Convex hull smallest convex set containing all theConvex hull smallest convex set containing all the
computing CONVEX HULLS Presentation Outline 2 D Convexcomputing CONVEX HULLS Presentation Outline 2 D Convex
Convex Hull 20121023 Convex vs Concave A polygonConvex Hull 20121023 Convex vs Concave A polygon
CONVEX LENSES 18 1 Lenses n A convexCONVEX LENSES 18 1 Lenses n A convex
Convex Lens in Water Convex Lens in WaterConvex Lens in Water Convex Lens in Water
Convex Hull for Dynamic Data Convex Hull andConvex Hull for Dynamic Data Convex Hull and
CONVEX LENS Focus Convex lens is thicker inCONVEX LENS Focus Convex lens is thicker in
Convex Hull obstacle start 362021 end Convex HullConvex Hull obstacle start 362021 end Convex Hull