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slides courtesy of Frank Dellaert, adapted by Steve Seitz

Structure from Motion with Unknown Correspondence (aka: How to combine EM, MCMC, Nonlinear LS!) Frank Dellaert, Steve Seitz, Chuck Thorpe, and Sebastian Thrun. slides courtesy of Frank Dellaert, adapted by Steve Seitz. Traditionally: 2 Problems. Correspondence

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slides courtesy of Frank Dellaert, adapted by Steve Seitz

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  1. Structure from Motionwith Unknown Correspondence(aka: How to combine EM, MCMC, Nonlinear LS!)Frank Dellaert, Steve Seitz, Chuck Thorpe, and Sebastian Thrun slides courtesy of Frank Dellaert, adapted by Steve Seitz

  2. Traditionally: 2 Problems • Correspondence • Compute 3D positions, camera motion

  3. The Correspondence Problem

  4. The Structure from Motion Problem • Find the most likely structure and motion 

  5. Non-linear Least-Squares (Rick Szeliski’s lecture) = mi’ mi jik xj Structure From Motion Image i’ Image i uik

  6. Question How to recover structure and motion with unknown correspondence? Aside: related subproblems • known correspondence, unknown structure, motion • structure from motion • known motion, unknown correspondence, structure • stereo • known structure, unknown correspondence, motion • 3D registration, ICP

  7. Without Correspondence:Try Tracking Features Lucas-Kanade Tracker (figures from Tommasini et. al. 98)

  8. New Idea: try all correspondences Likelihood: P(; U, J) structure + motion image features correspondence Marginalize over J: P(; U) = J P(; U, J)

  9. Combinatorial Explosion • 3 images, 4 features: 4!3=13,824 • 5 images, 30 features: 30!5=1.3131e+162 • (number of stars:1e+20, atoms: 1e+79) • Computing P( ; U, J)is intractable !

  10. EM for Correspondence

  11. Expectation Maximization

  12. E-Step: Soft Correspondences P(J | U,t) View 1 View 2 View 3

  13. M-Step: SFM • Compute expected SFM solution • Problem: this requires solving an exponential number of SFM problems • If we assume correspondences jik are independent, we can simplify...

  14. vA=p1Au1+p2Au2+p3Au3+p4Au4 B A C D Virtual Measurements 1 2 B A C 4 D 3 View 2

  15. mi’ mi xj M-Step: SFM one SFM problem on virtual measurements Image i’ Image i vij jik

  16. Structure from Motion without Correspondence via EM: In each image, calculate the n2 “soft correspondences”

  17. Key Issue: Calculating Weights 1 2 B A C 4 D 3

  18. Weights = Marginal Probabilities Calculating Weights

  19. Mutual Exclusion

  20. Monte Carlo EM

  21. Monte Carlo Approximation

  22. P( ) = -------------- P( ) Sampling Assignments using the Metropolis Algorithm (1953)

  23. Swap Proposals (inefficient) a b c d e f

  24. ‘Chain Flipping’ Proposals

  25. Efficient Sampling (Example)

  26. MCMC to deal with mutex a b c d e f

  27. Input Images • Manual measurements • Random order

  28. Results: Cube

  29. Soft Correspondences ui scene feature xj EM iterations

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