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Stanford CS223B Computer Vision, Winter 2005 Lecture 11: Structure From Motion 2

Stanford CS223B Computer Vision, Winter 2005 Lecture 11: Structure From Motion 2. Sebastian Thrun, Stanford Rick Szeliski, Microsoft Hendrik Dahlkamp and Dan Morris, Stanford. Overall Distribution. Question 1: Calibration. Question 1: Calibration. Calibration with planar unknown target

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Stanford CS223B Computer Vision, Winter 2005 Lecture 11: Structure From Motion 2

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  1. Stanford CS223B Computer Vision, Winter 2005Lecture 11: Structure From Motion 2 Sebastian Thrun, Stanford Rick Szeliski, Microsoft Hendrik Dahlkamp and Dan Morris, Stanford

  2. Overall Distribution

  3. Question 1: Calibration

  4. Question 1: Calibration • Calibration with planar unknown target • Unknown parameters • 4 intrinsics • 6K extrinsics (K = #images) • 2M calibration target parameters (but can’t recover 3) • 2KM constraints

  5. Question 2: Perspective Geometry

  6. Question 2: Perspective Geometry • Collinearity in 3D  2D (but not converse) • Order in 3D  2D (but not converse) • Equidistance: Not preserved! • Proof (collinearity in 2D):

  7. Question 3: Stereopsis

  8. Question 3: Stereopsis

  9. Question 3: Stereopsis • How does DZ scale with Z? – in approximation!!!

  10. Question 4: True or False

  11. Question 5: Build A System!

  12. Question 5: Build A System! • Range: stereo or laser • Classification : template, optical flow?, SIFT? • Alternatively: segmentation, range discontinuities • Prediction: person and car • Robustness: normalize image, bring light source • (many other possibilities)

  13. Stanford CS223B Computer Vision, Winter 2005Lecture 11: Structure From Motion 2 Sebastian Thrun, Stanford Rick Szeliski, Microsoft Hendrik Dahlkamp and Dan Morris, Stanford

  14. Structure From Motion (1) [Tomasi & Kanade 92]

  15. Structure From Motion (2) [Tomasi & Kanade 92]

  16. Structure From Motion (3) [Tomasi & Kanade 92]

  17. Structure From Motion • Problem 1: • Given n points pij =(xij, yij) in m images • Reconstruct structure: 3-D locations Pj =(xj, yj, zj) • Reconstruct camera positions (extrinsics) Mi=(Aj, bj) • Problem 2: • Establish correspondence: c(pij)

  18. The “Trick Of The Day” • Replace Euclidean Geometry by Affine Geometry • Solve SFM linearly (“closed” form) • Post-Process to make Euclidean • By Tomasi and Kanade, 1992

  19. Orthographic Camera Model Extrinsic Parameters Rotation Orthographic Projection Limit of Pinhole Model:

  20. Orthographic Projection Limit of Pinhole Model: Orthographic Projection

  21. The Affine SFM Problem

  22. Count # Constraints vs #Unknowns • m camera poses • n points • 2mn point constraints • 8m+3n unknowns • Suggests: need 2mn  8m + 3n • But: Can we really recover all parameters???

  23. How Many Parameters Can’t We Recover? We can recover all but… Place Your Bet!

  24. The Answer is (at least): 12

  25. Points for Solving Affine SFM Problem • m camera poses • n points • Need to have: 2mn  8m + 3n-12

  26. Affine SFM Fix coordinate system by making p0=origin Rank Theorem: Q has rank 3 Proof:

  27. The Rank Theorem 2m elements n elements

  28. Tomasi/Kanade 1992 Singular Value Decomposition

  29. Tomasi/Kanade 1992 Gives also the optimal affine reconstruction under noise

  30. Back To Orthographic Projection Find C and d for which constraints are met

  31. Back To Projective Geometry Orthographic (in the limit) Projective

  32. The “Trick Of The Day” • Replace Euclidean Geometry by Affine Geometry • Solve SFM linearly (“closed” form) • Post-Process to make Euclidean • By Tomasi and Kanade, 1992

  33. SFM With Projective Camera: See Rick Szeliski’s Lecture! Non-Linear Optimization Problem: Bundle Adjustment!

  34. Structure From Motion • Problem 1: • Given n points pij =(xij, yij) in m images • Reconstruct structure: 3-D locations Pj =(xj, yj, zj) • Reconstruct camera positions (extrinsics) Mi=(Aj, bj) • Problem 2: • Establish correspondence: c(pij)

  35. The Correspondence Problem View 1 View 2 View 3

  36. Correspondence: Solution 1 • Track features (e.g., optical flow) • …but fails when images taken from widely different poses

  37. Correspondence: Solution 2 • Start with random solution A, b, P • Compute soft correspondence: p(c|A,b,P) • Plug soft correspondence into SFM • Reiterate • See Dellaert et al 2003, Machine Learning Journal

  38. Example

  39. Results: Cube

  40. Animation

  41. Tomasi’s Benchmark Problem

  42. Reconstruction with EM

  43. 3-D Structure

  44. Summary SFM • Problem • Determine feature locations (=structure) • Determine camera extrinsic (=motion) • The name SFM is somewhat of a misdemeanor • Two Principal Solutions • Nonlinear optimization (local minima) • Linear (affine geometry) • Correspondence • RANSAC • Expectation Maximization

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