1 / 40

Epipolar geometry Class 5

Epipolar geometry Class 5. Geometric Computer Vision course schedule (tentative). Two-view geometry. Three questions:. Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image?.

herronj
Download Presentation

Epipolar geometry Class 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Epipolar geometryClass 5

  2. Geometric Computer Vision course schedule(tentative)

  3. Two-view geometry Three questions: • Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image? • (ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views? • (iii) Scene geometry (structure): Given corresponding image points xi ↔x’i and cameras P, P’, what is the position of (their pre-image) X in space?

  4. The epipolar geometry C,C’,x,x’ and X are coplanar

  5. The epipolar geometry What if only C,C’,x are known?

  6. The epipolar geometry All points on p project on l and l’

  7. The epipolar geometry Family of planes p and lines l and l’ Intersection in e and e’

  8. The epipolar geometry epipoles e,e’ = intersection of baseline with image plane = projection of projection center in other image = vanishing point of camera motion direction an epipolar plane = plane containing baseline (1-D family) an epipolar line = intersection of epipolar plane with image (always come in corresponding pairs)

  9. Example: converging cameras

  10. Example: motion parallel with image plane (simple for stereo  rectification)

  11. Example: forward motion e’ e

  12. The fundamental matrix F algebraic representation of epipolar geometry we will see that mapping is (singular) correlation (i.e. projective mapping from points to lines) represented by the fundamental matrix F

  13. The fundamental matrix F geometric derivation mapping from 2-D to 1-D family (rank 2)

  14. The fundamental matrix F algebraic derivation (note: doesn’t work for C=C’  F=0)

  15. The fundamental matrix F correspondence condition The fundamental matrix satisfies the condition that for any pair of corresponding points x↔x’ in the two images

  16. The fundamental matrix F F is the unique 3x3 rank 2 matrix that satisfies x’TFx=0 for all x↔x’ • Transpose: if F is fundamental matrix for (P,P’), then FT is fundamental matrix for (P’,P) • Epipolar lines: l’=Fx & l=FTx’ • Epipoles: on all epipolar lines, thus e’TFx=0, x e’TF=0, similarly Fe=0 • F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2) • F is a correlation, projective mapping from a point x to a line l’=Fx (not a proper correlation, i.e. not invertible)

  17. Fundamental matrix for pure translation

  18. Fundamental matrix for pure translation

  19. Fundamental matrix for pure translation General motion Pure translation for pure translation F only has 2 degrees of freedom

  20. The fundamental matrix F relation to homographies valid for all plane homographies

  21. The fundamental matrix F relation to homographies requires

  22. Projective transformation and invariance Derivation based purely on projective concepts F invariant to transformations of projective 3-space unique not unique canonical form

  23. ~ ~ Show that if F is same for (P,P’) and (P,P’), there exists a projective transformation H so that P=HP and P’=HP’ ~ ~ Projective ambiguity of cameras given F previous slide: at least projective ambiguity this slide: not more! lemma: (22-15=7, ok)

  24. 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!

  25. 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

  26. Epipolar geometry? courtesy Frank Dellaert

  27. Other entities besides points? Lines give no constraint for two view geometry (but will for three and more views) Curves and surfaces yield some constraints related to tangency (e.g. Sinha et al. CVPR’04)

  28. Computation of F • Linear (8-point) • Minimal (7-point) • Robust (RANSAC) • Non-linear refinement (MLE, …) • Practical approach

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

  30. ~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

  31. (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] normalized least squares yields good results(Hartley, PAMI´97)

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

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

  34. 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)

  35. (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% RANSAC Step 4. Compute F based on all inliers Step 5. Look for additional matches Step 6. Refine F based on all correct matches

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

  37. Issues: • (Mostly) planar scene (see next slide) • 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)

  38. Computing F for quasi-planar scenes QDEGSAC 337 matches on plane, 11 off plane #inliers %inclusion of out-of-plane inliers 17% success for RANSAC 100% for QDEGSAC data rank

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

More Related