1. 3D Photography: Epipolar geometry
Kevin Köser, Marc Pollefeys
Sprint 2011

2. 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?

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

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

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

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

7. 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)

8. Example: converging cameras

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

10. Example: forward motion

11. 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

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

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

14. 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

15. 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)

16. Fundamental matrix for pure translation

17. Fundamental matrix for pure translation

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

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

20. The fundamental matrix F
relation to homographies
requires

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

22. Projective ambiguity of cameras given F
previous slide: at least projective ambiguity
this slide: not more!
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’
~ ~ ~
lemma:
(22-15=7, ok)

23. The projective reconstruction theorem
If a set of point correspondences in two views determine the fundamental 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!

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

25. Epipolar geometry?
courtesy Frank Dellaert

26. 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)

27. Computation of F (and E)
Linear (8-point)
Minimal (7-point)
Robust (RANSAC)
Non-linear refinement (MLE, …)
Practical approach
Calibrated 5-point
Calibrated + know vertical 3-point

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

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

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

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

35. 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%
(generate
hypothesis)
(verify hypothesis)
RANSAC
Step 4. Compute F based on all inliers
Step 5. Look for additional matches
Step 6. Refine F based on all correct matches
#inliers
90%
80%
70%
60%
50%
#samples 5 13 35
106
382

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

37. (Mostly) planar scene (see next slide)
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. 5-point relative motion
(Nister, CVPR03)
Linear equations for 5 points
Linear solution space
Non-linear constraints
scale does not matter, choose
10 cubic polynomials

40. 5-point relative motion
(Nister, CVPR03)
Perform Gauss-Jordan elimination on polynomials
represents polynomial of degree n in z -z -z -z

41. Minimal relative pose with know vertical
Fraundorfer, Tanskanen and Pollefeys, ECCV2010
Vertical direction can often be estimated
inertial sensor
vanishing point -g
5 linear unknowns  linear 5 point algorithm
3 unknowns  quartic 3 point algorithm

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