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