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Structure Computation Scene Planes and Homographies

Structure Computation Scene Planes and Homographies. Slides modified from Marc Pollefeys’ slides. Problem Statement. Given P, P’ or F with great accuracy Given x, x’ Compute X. Invariant to Projective transformations. Point reconstruction. linear triangulation. homogeneous. invariance?.

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Structure Computation Scene Planes and Homographies

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  1. Structure ComputationScene Planes and Homographies Slides modified from Marc Pollefeys’ slides

  2. Problem Statement • Given P, P’ or F with great accuracy • Given x, x’ • Compute X Invariant toProjective transformations

  3. Point reconstruction

  4. linear triangulation homogeneous invariance? algebraic error yes, constraint no (except for affine) inhomogeneous

  5. geometric error possibility to compute using LM (for 2 or more points) or directly (for 2 points)

  6. Geometric error Reconstruct matches in projective frame by minimizing the reprojection error Non-iterative optimal solution (see Hartley&Sturm,CVIU´97)

  7. x1 l1 l1 l2 x1 x2 x2´ x1´ x2 l2 Optimal 3D point in epipolar plane Given an epipolar plane, find best 3D point for (x1,x2) Select closest points (x1´,x2´) on epipolar lines Obtain 3D point through exact triangulation Guarantees minimal reprojection error (given this epipolar plane)

  8. m1 l2(a) l1(a) m2 Optimal epipolar plane • Reconstruct matches in projective frame by minimizing the reprojection error • Non-iterative method Determine the epipolar plane for reconstruction Reconstruct optimal point from selected epipolar plane 3DOF (Hartley and Sturm, CVIU´97) 1DOF (polynomial of degree 6 check all minima, incl ∞)

  9. Reconstruction uncertainty consider angle between rays

  10. Line reconstruction doesn‘t work for epipolar plane

  11. Scenes and Homographies

  12. Homography given plane point on plane project in second view

  13. Calibrated stereo rig

  14. homographies and epipolar geometry points on plane also have to satisfy epipolar geometry! HTF has to be skew-symmetric l’

  15. Homography also maps epipole

  16. Homography also maps epipolar lines

  17. Compatibility constraint

  18. plane homography given F and 3 points correspondences Method 1: reconstruct explicitly, compute plane through 3 points derive homography Method 2: use epipoles as 4th correspondence to compute homography

  19. degenerate geometry for an implicit computation of the homography

  20. Estimastion from 3 noisy points (+F) Consistency constraint: points have to be in exact epipolar correspodence Determine MLE points given F and xi↔xi’ Use implicit 3D approach (no derivation here) M is a 3x3 matrix with rows xiT

  21. plane homography given F, a point and a line

  22. application: matching lines (Schmid and Zisserman, CVPR’97)

  23. epipolar geometry induces point homography on lines

  24. Degenerate homographies

  25. plane induced parallax

  26. 6-point algorithm x1,x2,x3,x4 in plane, x5,x6 out of plane Compute H from x1,x2,x3,x4

  27. Projective depth r=0 on plane sign of r determines on which side of plane

  28. Binary space partition

  29. Next class: The Trifocal Tensor

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