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Triangulation and Multi-View Geometry Class 9

Triangulation and Multi-View Geometry Class 9. Read notes Section 3.3, 4.3-4.4, 5.1 (if interested, read Triggs’s paper on MVG using tensor notation, see http://www.unc.edu/courses/2004fall/comp/290/089/papers/Triggs-ijcv95.pdf ). (generate hypothesis). (verify hypothesis).

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Triangulation and Multi-View Geometry Class 9

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  1. Triangulation and Multi-View GeometryClass 9 Read notes Section 3.3, 4.3-4.4, 5.1 (if interested, read Triggs’s paper on MVG using tensor notation, see http://www.unc.edu/courses/2004fall/comp/290/089/papers/Triggs-ijcv95.pdf)

  2. (generate hypothesis) (verify hypothesis) Automatic computation of F 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

  3. Abort verification early OIOIIIIOIIIOIOIIIIOOIOIIIIOIOIOIIIIIIII OOOOOIOOIOOOOOIOOOOOOOIOOOOOIOIOOOOOOOO Given n samples and an expected proportion of inliers p,how likely is it that I have observed less than T inliers? abort if P<0.02 (initial sample most probably contained outliers) (inspired from Chum and Matas BMVC2002) (use normal approximation to binomial) To avoid problems this requires to also verify at random! (but we already have a random sampler anyway)

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

  5. Degenerate cases: • Degenerate cases • Planar scene • Pure rotation • No unique solution • Remaining DOF filled by noise • Use simpler model (e.g. homography) • Solution 1: Model selection (Torr et al., ICCV´98, Kanatani, Akaike) • Compare H and F according to expected residual error (compensate for model complexity) • Solution 2: RANSAC • Compare H and F according to inlier count (see next slide)

  6. RANSAC for (quasi-)degenerate cases 80% in plane 2% out plane 18% outlier • Full model (8pts, 1D solution) • (accept inliers to solution F) • Planar model (6pts, 3D solution) • Accept if large number of remaining inliers • (accept inliers to solution F1,F2&F3) • Plane+parallax model (plane+2pts) • closest rank-6 of Anx9 for all plane inliers • Sample for out of plane points among outliers

  7. 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)

  8. RANSAC for ambiguous matching • Include multiple candidate matches in set of potential matches • Select according to matching probability (~ matching score) • Helps for repeated structures or scenes with similar features as it avoids an early commitment, but also useful in general (Tordoff and Murray ECCV02)

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

  10. L2 X x2 C2 Triangulation (finally!) x1 C1 L1 Triangulation • calibration • correspondences

  11. L2 x1 C1 X L1 x2 C2 Triangulation • Backprojection • Triangulation Iterative least-squares • Maximum Likelihood Triangulation

  12. m1 l1 l1 l2 m1 m2 m2´ m1´ m2 l2 Optimal 3D point in epipolar plane • Given an epipolar plane, find best 3D point for (m1,m2) Select closest points (m1´,m2´) on epipolar lines Obtain 3D point through exact triangulation Guarantees minimal reprojection error(given this epipolar plane)

  13. m1 l2(a) l1(a) m2 Non-iterative optimal solution • 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 Note: only works for two views 3DOF (Hartley and Sturm, CVIU´97) (polynomial of degree 6) 1DOF

  14. Backprojection • Represent point as intersection of row and column • Condition for solution? Useful presentation for deriving and understanding multiple view geometry (notice 3D planes are linear in 2D point coordinates)

  15. Multi-view geometry (intersection constraint) (multi-linearity of determinants) (= epipolar constraint!) (counting argument: 11x2-15=7)

  16. Multi-view geometry (multi-linearity of determinants) (3x3x3=27 coefficients) (= trifocal constraint!) (counting argument: 11x3-15=18)

  17. Multi-view geometry (multi-linearity of determinants) (3x3x3x3=81 coefficients) (= quadrifocal constraint!) (counting argument: 11x4-15=29)

  18. Next class: rectification and stereo image I´(x´,y´) image I(x,y) Disparity map D(x,y) (x´,y´)=(x+D(x,y),y)

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