Two-view geometry

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

3. p p L2 L2 m1 m1 m1 C1 C1 C1 M M L1 L1 l1 l1 e1 e1 lT1 l2 e2 e2 Canonical representation: l2 m2 m2 m2 l2 l2 Fundamental matrix (3x3 rank 2 matrix) C2 C2 C2 Epipolar geometry Underlying structure in set of matches for rigid scenes • Computable from corresponding points • Simplifies matching • Allows to detect wrong matches • Related to calibration

4. The projective reconstruction theorem If a set of point correspondences in two views determine thefundamental matrix uniquely, then the scene and cameras may be reconstructed from these correspondences alone, and any two such reconstructions from these correspondences are projectively equivalent allows reconstruction from pair of uncalibrated images!

5. Objective • Given two uncalibrated images compute (PM,P‘M,{XMi}) • (i.e. within similarity of original scene and cameras) • Algorithm • Compute projective reconstruction (P,P‘,{Xi}) • Compute F from xi↔x‘i • Compute P,P‘ from F • Triangulate Xi from xi↔x‘i • Rectify reconstruction from projective to metric • Direct method: compute H from control points • Stratified method: • Affine reconstruction: compute p∞ • Metric reconstruction: compute IAC w

7. Epipolar geometry: basic equation separate known from unknown (data) (unknowns) (linear)

8. the singularity constraint SVD from linearly computed F matrix (rank 3) Compute closest rank-2 approximation

10. the minimum case – 7 point correspondences one parameter family of solutions but F1+lF2 not automatically rank 2

11. 3 F7pts F F2 F1 the minimum case – impose rank 2 (obtain 1 or 3 solutions) (cubic equation) Compute possible l as eigenvalues of (only real solutions are potential solutions)

12. ~10000 ~100 ~10000 ~100 ~10000 ~10000 ~100 ~100 1 Orders of magnitude difference Between column of data matrix  least-squares yields poor results ! the NOT normalized 8-point algorithm

13. (0,500) (700,500) (-1,1) (1,1) (0,0) (0,0) (700,0) (-1,-1) (1,-1) the normalized 8-point algorithm Transform image to ~[-1,1]x[-1,1] Least squares yields good results(Hartley, PAMI´97)

14. algebraic minimization possible to iteratively minimize algebraic distance subject to det F=0 (see book if interested)

15. Geometric distance Gold standard Sampson error Symmetric epipolar distance

16. Gold standard Maximum Likelihood Estimation (= least-squares for Gaussian noise) Initialize: normalized 8-point, (P,P‘) from F, reconstruct Xi Parameterize: (overparametrized) Minimize cost using Levenberg-Marquardt (preferably sparse LM, see book)

17. Gold standard Alternative, minimal parametrization (with a=1) (note (x,y,1) and (x‘,y‘,1) are epipoles) • problems: • a=0  pick largest of a,b,c,d to fix • epipole at infinity  pick largest of x,y,w and of x’,y’,w’ 4x3x3=36 parametrizations! reparametrize at every iteration, to be sure

18. Zhang&Loop’s approach CVIU’01

19. First-order geometric error (Sampson error) (one eq./point JJT scalar) (problem if some x is located at epipole) advantage: no subsidiary variables required

20. Symmetric epipolar error

21. Some experiments:

22. Some experiments:

23. Some experiments:

24. Some experiments: Residual error: (for all points!)

25. Recommendations: • Do not use unnormalized algorithms • Quick and easy to implement: 8-point normalized • Better: enforce rank-2 constraint during minimization • Best: Maximum Likelihood Estimation (minimal parameterization, sparse implementation)

26. Special case: Enforce constraints for optimal results: Pure translation (2dof), Planar motion (6dof), Calibrated case (5dof)

27. The envelope of epipolar lines What happens to an epipolar line if there is noise? Monte Carlo n=50 n=25 n=15 n=10

28. Other entities? Lines give no constraint for two view geometry (but will for three and more views) Curves and surfaces yield some constraints related to tangency

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

30. 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)

31. 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)

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

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

34. 3 3 2 2 4 4 1 5 1 5 Feature example Gives satisfying results for small image motions

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

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

37. (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

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

39. 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)

40. 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)

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

42. Next class: image pair rectificationreconstructing points and lines