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Scene Planes and Homographies class 16

Scene Planes and Homographies class 16. Multiple View Geometry Comp 290-089 Marc Pollefeys. Multiple View Geometry course schedule (subject to change). Two-view geometry. Epipolar geometry 3D reconstruction F-matrix comp. Structure comp. Planar rectification. (standard approach).

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Scene Planes and Homographies class 16

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  1. Scene Planes and Homographiesclass 16 Multiple View Geometry Comp 290-089 Marc Pollefeys

  2. Multiple View Geometry course schedule(subject to change)

  3. Two-view geometry Epipolar geometry 3D reconstruction F-matrix comp. Structure comp.

  4. Planar rectification (standard approach) Bring two views to standard stereo setup (moves epipole to ) (not possible when in/close to image)

  5. Polar rectification (Pollefeys et al. ICCV’99) Polar re-parameterization around epipoles Requires only (oriented) epipolar geometry Preserve length of epipolar lines Choose  so that no pixels are compressed original image rectified image Works for all relative motions Guarantees minimal image size

  6. polar rectification: example

  7. polar rectification: example

  8. Example: Béguinage of Leuven Does not work with standard Homography-based approaches

  9. Stereo matching • attempt to match every pixel • use additional constraints

  10. Similarity measure (SSD or NCC) Optimal path (dynamic programming ) Stereo matching • Constraints • epipolar • ordering • uniqueness • disparity limit • disparity gradient limit • Trade-off • Matching cost (data) • Discontinuities (prior) (Cox et al. CVGIP’96; Koch’96; Falkenhagen´97; Van Meerbergen,Vergauwen,Pollefeys,VanGool IJCV‘02)

  11. Disparity map image I´(x´,y´) image I(x,y) Disparity map D(x,y) (x´,y´)=(x+D(x,y),y)

  12. Point reconstruction

  13. Line reconstruction doesn‘t work for epipolar plane

  14. Scene planes and homographies plane induces homography between two views

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

  16. Homography given plane and vice-versa

  17. Calibrated stereo rig

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

  19. homographies and epipolar geometry (pick lp =e’, since e’Te’≠0)

  20. Homography also maps epipole

  21. Homography also maps epipolar lines

  22. Compatibility constraint

  23. 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

  24. degenerate geometry for an implicit computation of the homography

  25. Estimastion from 3 noisy points (+F) Consistency constraint: points have to be in exact epipolar correspodence Determine MLE points given F and x↔x’ Use implicit 3D approach (no derivation here)

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

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

  28. epipolar geometry induces point homography on lines

  29. Degenerate homographies

  30. plane induced parallax

  31. 6-point algorithm x1,x2,x3,x4 in plane, x5,x6 out of plane Compute H from x1,x2,x3,x4

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

  33. Binary space partition

  34. Two planes H has fixed point and fixed line

  35. Next class: The Trifocal Tensor

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