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Computing F and rectification class 14

Computing F and rectification class 14. 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. Epipolar geometry: direct computation.

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Computing F and rectification class 14

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  1. Computing F and rectificationclass 14 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. Epipolar geometry: direct computation Basic equation 8-point algorithm (normalize!) 7-point algorithm (impose rank 2)

  5. Epipolar geometry: iterative computation Maximum Likelihood Estimation (= least-squares for Gaussian noise) Sampson error (first order approx. to MLE) Symmetric epipolar error

  6. Automatic computation of F • Interest points • Putative correspondences • RANSAC • (iv) Non-linear re-estimation of F • Guided matching • (repeat (iv) and (v) until stable)

  7. Feature points • Extract feature points to relate images • Required properties: • Well-defined (i.e. neigboring points should all be different) • Stable across views (i.e. same 3D point should be extracted as feature for neighboring viewpoints)

  8. Feature points (e.g.Harris&Stephens´88; Shi&Tomasi´94) Find points that differ as much as possible from all neighboring points homogeneous edge corner Mshould have large eigenvalues Feature = local maxima (subpixel) of F(1,2)

  9. Feature points Select strongest features (e.g. 1000/image)

  10. ? Feature matching Evaluate NCC for all features with similar coordinates Keep mutual best matches Still many wrong matches!

  11. 3 3 2 2 4 4 1 5 1 5 Feature example Gives satisfying results for small image motions

  12. Wide-baseline matching… • Requirement to cope with larger variations between images • Translation, rotation, scaling • Foreshortening • Non-diffuse reflections • Illumination geometric transformations photometric changes

  13. Wide-baseline matching… (Tuytelaars and Van Gool BMVC 2000) Wide baseline matching for two different region types

  14. (generate hypothesis) (verify hypothesis) RANSAC Step 1. Extract features Step 2. Compute a set of potential matches Step 3. do Step 3.1 select minimal sample (i.e. 7 matches) Step 3.2 compute solution(s) for F Step 3.3 determine inliers until (#inliers,#samples)<95% Step 4. Compute F based on all inliers Step 5. Look for additional matches Step 6. Refine F based on all correct matches

  15. Finding more matches restrict search range to neighborhood of epipolar line (1.5 pixels) relax disparity restriction (along epipolar line)

  16. Degenerate cases: • Degenerate cases • Planar scene • Pure rotation • No unique solution • Remaining DOF filled by noise • Use simpler model (e.g. homography) • Model selection (Torr et al., ICCV´98, Kanatani, Akaike) • Compare H and F according to expected residual error (compensate for model complexity)

  17. motion structure Model selection Geometric Robust Information Criterion MLE n = number of measurements (inliers+outliers) r = dimension of data k = motion model parameters d = dimension of structure

  18. Model selection F Video tracking H Dominant planes

  19. More problems: • Absence of sufficient features (no texture) • Repeated structure ambiguity • Robust matcher also finds • support for wrong hypothesis • solution: detect repetition (Schaffalitzky and Zisserman, BMVC‘98)

  20. geometric relations between two views is fully described by recovered 3x3 matrix F two-view geometry

  21. e Image pair rectification simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e map epipole e to (1,0,0) try to minimize image distortion problem when epipole in (or close to) the image

  22. ~ image size (calibrated) Planar rectification (standard approach) Distortion minimization (uncalibrated) Bring two views to standard stereo setup (moves epipole to ) (not possible when in/close to image)

  23. 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

  24. polar rectification: example

  25. polar rectification: example

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

  27. Example: Béguinage of Leuven

  28. Exploiting motion and scene constraints • Epipolar constraint (through rectification) • Ordering constraint • Uniqueness constraint • Disparity limit • Disparity continuity constraint

  29. Ordering constraint surface slice surface as a path 6 5 occlusion left 4 3 2 1 4,5 6 1 2,3 5 6 2,3 4 occlusion right 1 3 6 1 2 4,5

  30. Uniqueness constraint • In an image pair each pixel has at most one corresponding pixel • In general one corresponding pixel • In case of occlusion there is none

  31. use reconsructed features to determine bounding box Disparity constraint surface slice surface as a path bounding box disparity band constant disparity surfaces

  32. Disparity continuity constraint • Assume piecewise continuous surface • piecewise continuous disparity • In general disparity changes continuously • discontinuities at occluding boundaries

  33. 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)

  34. Hierarchical stereo matching Allows faster computation Deals with large disparity ranges Downsampling (Gaussian pyramid) Disparity propagation (Falkenhagen´97;Van Meerbergen,Vergauwen,Pollefeys,VanGool IJCV‘02)

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

  36. Example: reconstruct image from neighboring images

  37. Multi-view depth fusion (Koch, Pollefeys and Van Gool. ECCV‘98) • Compute depth for every pixel of reference image • Triangulation • Use multiple views • Up- and down sequence • Use Kalman filter Allows to compute robust texture

  38. Assignment 2 (due by Wednesday 19/03/03) Compute F automatically from image pair (matches, 7-point, RANSAC, 8-point, iterative, more matches, epipolar lines, etc.)

  39. Next class: reconstructing points and lines

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