eecs 274 computer vision
DESCRIPTION
EECS 274 Computer Vision. Geometry of Multiple Views. Geometry of Multiple Views. Epipolar geometry Essential matrix Fundamental matrix Trifocal tensor Quadrifocal tensor Reading: FP Chapter 10. Epipolar geometry. Epipolar plane OPO ’. Baseline OO ’. - PowerPoint PPT PresentationTRANSCRIPT
EECS 274 Computer Vision
Geometry of Multiple Views
Geometry of Multiple Views• Epipolar geometry
– Essential matrix– Fundamental matrix
• Trifocal tensor• Quadrifocal tensor• Reading: FP Chapter 10
• Epipolar plane OPO’
• Epipoles e, e’
• Epipolar lines l, l’
• Baseline OO’
Epipolar geometry
l’ is epipolar line associated with p and intersects baseline OO’ on e’
e’ is the projection of O observed from O’
• 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.
Epipolar constraint
Epipolar Constraint: Calibrated Case
Essential Matrix(Longuet-Higgins, 1981)3 ×3 skew-symmetric
matrix: rank=2
• E is defined by 5 parameters (3 for rotation and 2 for translation
• E T p’ is the epipolar line associated with p’
• E p is the epipolar line associated with p
• E e’=0 and E T e=0
• E is singular
• E has two equal non-zero singular values (Huang and Faugeras, 1989)
Properties of essential matrix
Epipolar Constraint: Small MotionsTo First-Order:
Pure translation:Focus of Expansion
Epipolar Constraint: Uncalibrated Case
Fundamental Matrix(Faugeras and Luong, 1992)are normalized image coordinate pp ˆ,ˆ
• F has rank 2 and is defined by 7 parameters
• F p’ is the epipolar line associated with p’
• F T p is the epipolar line associated with p
• F e’=0 and F T e=0
• F is singular
Properties of fundamental matrix
Rank-2 constraint• F admits 7 independent parameter• Possible choice of parameterization
using e=(α,β)T and e’=(α’,β’)T and epipolar transformation
• Can be written with 4 parameters
''''''''
badcacbddccdbaab
F
The Eight-Point Algorithm (Longuet-Higgins, 1981)
|F | =1.
Minimize:
under the constraint2
Least-squares minimization
• Error function: |F | =1.
Minimize:under the constraint
),'(')',(')',( ppdppdppppe TT FFF
Non-Linear Least-Squares Approach (Luong et al., 1993)
Minimize
with respect to the coefficients of F , using an appropriate rank-2 parameterization
The Normalized Eight-Point Algorithm (Hartley, 1995)
• Estimation of transformation parameters suffer form poor numerical condition problem
• Center the image data at the origin, and scale it so themean 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 thepoints q and q’
• Enforce the rank-2 constraint
• Output T F T’
T
i i i i
i i
Trinocular Epipolar Constraints
These constraints are not independent!
Trinocular Epipolar Constraints: Transfer
Given p and p , p can be computedas the solution of linear equations.1 2 3
Trifocal Constraints
The set of points that project onto an image line l is the plane L that contains the line and pinhole
Point P in L is projected onto p on line l (l=(a,b,c)T)
)(
1
tRPp z
KMM
Recall
Trifocal Constraints
All 3x3 minorsmust be zero!
Calibrated Case
Trifocal Tensorline-line-line correspondence
Trifocal ConstraintsCalibrated Case
Given 3 point correspondences, p1, p2, p3 of the same point P, and two lines l2, l3, (passing through p2, and p3), O1p1 must intersect the line l, where the planes L2 and L3
point-line-line correspondence
Trifocal ConstraintsUncalibrated Case
Trifocal ConstraintsUncalibrated Case
Trifocal Tensor
Trifocal Constraints: 3 Points
Pick any two lines l and l through p and p .Do it again.
2 3 2 3T( p , p , p )=01 2 3
Properties of the Trifocal Tensor
Estimating the Trifocal Tensor
• Ignore the non-linear constraints and use linear least-squaresa 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 theuncalibrated case (Faugeras and Mourrain, 1995).
2 1 3T i
1i
Multiple Views (Faugeras and Mourrain, 1995)
All 4 × 4 minors have zero determinants
Two Views
Epipolar Constraint
Three Views
Trifocal Constraint
Four Views
Quadrifocal Constraint(Triggs, 1995)
Geometrically, the four rays must intersect in P..
Quadrifocal Tensorand Lines
Given 4 point correspondences, p1, p2, p3, p4 of the same point P, and 3 lines l2, l3, l4 (passing through p2, and p3, p4), O1p1 must intersect the line l, where the planes L2 , L3, and L4