1. Camera calibration and epipolar geometry
Odilon Redon, Cyclops, 1914

2. Review: Alignment
What is the geometric relationship between pictures taken by cameras that share the same center?
How many points do we need to estimate a homography?
How do we estimate a homography?

3. Geometric vision Goal: Recovery of 3D structure
What cues in the image allow us to do this?

4. Merle Norman Cosmetics, Los Angeles
Visual cues
Shading
Merle Norman Cosmetics, Los Angeles
Slide credit: S. Seitz

5. The Visual Cliff, by William Vandivert, 1960
Visual cues
Shading
Texture
The Visual Cliff, by William Vandivert, 1960
Slide credit: S. Seitz

6. From The Art of Photography, Canon
Visual cues
Shading
Texture
Focus
From The Art of Photography, Canon
Slide credit: S. Seitz

7. Visual cues
Shading
Texture
Focus
Perspective
Slide credit: S. Seitz

8. Visual cues Shading Texture Focus Perspective Motion
Slide credit: S. Seitz

9. Our goal: Recovery of 3D structure
We will focus on perspective and motion
We need multi-view geometry because recovery of structure from one image is inherently ambiguous X? X? X? x

10. Our goal: Recovery of 3D structure
We will focus on perspective and motion
We need multi-view geometry because recovery of structure from one image is inherently ambiguous

11. Our goal: Recovery of 3D structure
We will focus on perspective and motion
We need multi-view geometry because recovery of structure from one image is inherently ambiguous

12. Recall: Pinhole camera model

13. Pinhole camera model

14. Camera coordinate system
Principal axis: line from the camera center perpendicular to the image plane
Normalized (camera) coordinate system: camera center is at the origin and the principal axis is the z-axis
Principal point (p): point where principal axis intersects the image plane (origin of normalized coordinate system)

15. Principal point offset
Camera coordinate system: origin is at the prinicipal point
Image coordinate system: origin is in the corner

16. Principal point offset

17. Principal point offset
calibration matrix

18. Pixel coordinates
Pixel size:
mx pixels per meter in horizontal direction, my pixels per meter in vertical direction
pixels/m m
pixels

19. Camera rotation and translation
In general, the camera coordinate frame will be related to the world coordinate frame by a rotation and a translation
coords. of point in camera frame
coords. of camera center in world frame
coords. of a point in world frame (nonhomogeneous)

20. Camera rotation and translation
In non-homogeneous coordinates:
Note: C is the null space of the camera projection matrix (PC=0)

21. Camera parameters Intrinsic parameters Principal point coordinates
Focal length
Pixel magnification factors
Skew (non-rectangular pixels)
Radial distortion

22. Camera parameters Intrinsic parameters Extrinsic parameters
Principal point coordinates
Focal length
Pixel magnification factors
Skew (non-rectangular pixels)
Radial distortion
Extrinsic parameters
Rotation and translation relative to world coordinate system

23. Camera calibration
Given n points with known 3D coordinates Xi and known image projections xi, estimate the camera parameters Xi xi

24. Camera calibration
Two linearly independent equations

25. Camera calibration
P has 11 degrees of freedom (12 parameters, but scale is arbitrary)
One 2D/3D correspondence gives us two linearly independent equations
Homogeneous least squares
6 correspondences needed for a minimal solution

26. Camera calibration
Note: for coplanar points that satisfy ΠTX=0, we will get degenerate solutions (Π,0,0), (0,Π,0), or (0,0,Π)

27. Camera calibration
Once we’ve recovered the numerical form of the camera matrix, we still have to figure out the intrinsic and extrinsic parameters
This is a matrix decomposition problem, not an estimation problem (see F&P sec. 3.2, 3.3)

28. Two-view geometry
Scene geometry (structure): Given corresponding points in two or more images, where is the pre-image of these points in 3D?
Correspondence (stereo matching): Given a point in just one image, how does it constrain the position of the corresponding point x’ in another image?
Camera geometry (motion): Given a set of corresponding points in two images, what are the cameras for the two views?

29. Triangulation
Given projections of a 3D point in two or more images (with known camera matrices), find the coordinates of the point O1 O2 x1 x2 X?

30. Triangulation
We want to intersect the two visual rays corresponding to x1 and x2, but because of noise and numerical errors, they don’t meet exactly R1 R2 O1 O2 x1 x2 X?

31. Triangulation: Geometric approach
Find shortest segment connecting the two viewing rays and let X be the midpoint of that segment X x2 x1 O1 O2

32. Triangulation: Linear approach
Cross product as matrix multiplication:

33. Triangulation: Linear approach
Two independent equations each in terms of three unknown entries of X

34. Triangulation: Nonlinear approach
Find X that minimizes X?
x’1 x2 x1
x’2 O1 O2

35. Epipolar geometry Baseline – line connecting the two camera centersX x x’
Baseline – line connecting the two camera centers
Epipolar Plane – plane containing baseline (1D family)
Epipoles
= intersections of baseline with image planes
= projections of the other camera center
= vanishing points of camera motion direction
Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs)

36. Example: Converging cameras

37. Example: Motion parallel to image plane

38. Example: Forward motion
Epipole has same coordinates in both images.
Points move along lines radiating from e: “Focus of expansion”

39. Epipolar constraint X x x’
If we observe a point x in one image, where can the corresponding point x’ be in the other image?

40. Epipolar constraint X X X x x’ x’ x’
Potential matches for x have to lie on the corresponding
epipolar line l’.
Potential matches for x’ have to lie on the corresponding
epipolar line l.

41. Epipolar constraint example
Source: K. Grauman

42. Epipolar constraint: Calibrated caseX x x’
Assume that the intrinsic and extrinsic parameters of the cameras are known
We can multiply the projection matrix of each camera (and the image points) by the inverse of the calibration matrix to get normalized image coordinates
We can also set the global coordinate system to the coordinate system of the first camera

43. Epipolar constraint: Calibrated caseX x x’ t R
Camera matrix: [I|0] X = (u, v, w, 1)T
x = (u, v, w)T
Camera matrix: [RT | –RTt]
Vector x’ in second coord. system has coordinates Rx’ in the first one
The vectors x, t, and Rx’ are coplanar

44. Epipolar constraint: Calibrated caseX x x’
Essential Matrix
(Longuet-Higgins, 1981)
The vectors x, t, and Rx’ are coplanar

45. Epipolar constraint: Calibrated caseX x x’
E x’ is the epipolar line associated with x’ (l = E x’)
ETx is the epipolar line associated with x (l’ = ETx)
E e’ = 0 and ETe = 0
E is singular (rank two)
E has five degrees of freedom (up to scale)

46. Epipolar constraint: Uncalibrated caseX x x’
The calibration matrices K and K’ of the two cameras are unknown
We can write the epipolar constraint in terms of unknown normalized coordinates:

47. Epipolar constraint: Uncalibrated caseX x x’
Fundamental Matrix
(Faugeras and Luong, 1992)

48. Epipolar constraint: Uncalibrated caseX x x’
F x’ is the epipolar line associated with x’ (l = F x’)
FTx is the epipolar line associated with x (l’ = FTx)
F e’ = 0 and FTe = 0
F is singular (rank two)
F has seven degrees of freedom

49. The eight-point algorithm
x = (u, v, 1)T, x’ = (u’, v’, 1)T
Minimize:
under the constraint |F|2 = 1

50. The eight-point algorithm
Meaning of error sum of Euclidean distances between points xi and epipolar lines F x’i (or points x’i and epipolar lines FTxi) multiplied by a scale factor
Nonlinear approach: minimize

51. Problem with eight-point algorithm

52. Problem with eight-point algorithm
Poor numerical conditioning
Can be fixed by rescaling the data

53. The normalized eight-point algorithm
(Hartley, 1995)
Center the image data at the origin, and scale it so the mean squared distance between the origin and the data points is 2 pixels
Use the eight-point algorithm to compute F from the normalized points
Enforce the rank-2 constraint (for example, take SVD of F and throw out the smallest singular value)
Transform fundamental matrix back to original units: if T and T’ are the normalizing transformations in the two images, than the fundamental matrix in original coordinates is TT F T’

54. Comparison of estimation algorithms
8-point
Normalized 8-point
Nonlinear least squares
Av. Dist. 1
2.33 pixels
0.92 pixel
0.86 pixel
Av. Dist. 2
2.18 pixels
0.85 pixel
0.80 pixel

55. Epipolar transfer Assume the epipolar geometry is known
Given projections of the same point in two images, how can we compute the projection of that point in a third image?
? x3 x1 x2

56. Epipolar transfer Assume the epipolar geometry is known
Given projections of the same point in two images, how can we compute the projection of that point in a third image? x1 x2 x3
l31
l32
l31 = FT13 x1
l32 = FT23 x2
When does epipolar transfer fail?

57. Next time: Stereo