SLIDE 1 COS 429: COMPUTER VISON
MULTI-VIEW GEOMETRY (1 lecture)
- Epipolar Geometry
- The Essential and Fundamental Matrices
- The 8-Point Algorithm
- Trifocal tensor
- Reading: Chapter 10
Many of the slides in this lecture are courtesy to Prof. J. Ponce
SLIDE 2
Reconstruction / Triangulation
SLIDE 3
(Binocular) Fusion
SLIDE 4 Epipolar Geometry
- Epipolar Plane
- Epipoles
- Epipolar Lines
- Baseline
SLIDE 5 Epipolar Constraint
- Potential matches for p have to lie on the corresponding
epipolar line l’.
- Potential matches for p’ have to lie on the corresponding
epipolar line l.
SLIDE 6
Epipolar Constraint: Calibrated Case Essential Matrix
(Longuet-Higgins, 1981)
SLIDE 7 Properties of the Essential Matrix
- E p’ is the epipolar line associated with p’.
- E p is the epipolar line associated with p.
- E e’=0 and E e=0.
- E is singular.
- E has two equal non-zero singular values
(Huang and Faugeras, 1989).
T T
SLIDE 8
Epipolar Constraint: Small Motions To First-Order: Pure translation: Focus of Expansion
SLIDE 9
Epipolar Constraint: Uncalibrated Case Fundamental Matrix
(Faugeras and Luong, 1992)
SLIDE 10 Properties of the Fundamental Matrix
- F p’ is the epipolar line associated with p’.
- F p is the epipolar line associated with p.
- F e’=0 and F e=0.
- F is singular.
T T
SLIDE 11 The Eight-Point Algorithm (Longuet-Higgins, 1981) |F | =1. Minimize: under the constraint
2
SLIDE 12
Non-Linear Least-Squares Approach (Luong et al., 1993) Minimize with respect to the coefficients of F , using an appropriate rank-2 parameterization.
SLIDE 13 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: q = T p , q’ = T’ p’.
- Use the eight-point algorithm to compute F from the
points q and q’ .
- Enforce the rank-2 constraint.
- Output T F T’.
T
i i i i i i
SLIDE 14
Data courtesy of R. Mohr and B. Boufama.
SLIDE 15
Without normalization With normalization Mean errors: 10.0pixel 9.1pixel Mean errors: 1.0pixel 0.9pixel
SLIDE 16
Trinocular Epipolar Constraints
SLIDE 17
Trinocular Epipolar Constraints These constraints are not independent!
SLIDE 18
Trinocular Epipolar Constraints: Transfer Given p and p , p can be computed as the solution of linear equations.
1 2 3
SLIDE 19
Trifocal Constraints
SLIDE 20
Trifocal Constraints All 3x3 minors must be zero! Calibrated Case Trifocal Tensor
SLIDE 21
Trifocal Constraints Uncalibrated Case Trifocal Tensor
SLIDE 22 Properties of the Trifocal Tensor Estimating the Trifocal Tensor
- Ignore the non-linear constraints and use linear least-squares
a posteriori.
- Impose the constraints a posteriori.
- For any matching epipolar lines, l G l = 0.
- The matrices G are singular.
- They satisfy 8 independent constraints in the
uncalibrated case (Faugeras and Mourrain, 1995).
2 1 3 T i 1 i
SLIDE 23
For any matching epipolar lines, l G l = 0.
2 1 3 T i
The backprojections of the two lines do not define a line!
SLIDE 24
Multiple Views (Faugeras and Mourrain, 1995)
SLIDE 25
Two Views Epipolar Constraint
SLIDE 26
Three Views Trifocal Constraint
SLIDE 27
Four Views Quadrifocal Constraint (Triggs, 1995)
SLIDE 28
Geometrically, the four rays must intersect in P..
SLIDE 29
Quadrifocal Tensor and Lines
SLIDE 30
Scale-Restraint Condition from Photogrammetry
