Geometry of Multiple Views 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, S Chapter 7

Epipolar geometry Epipolar plane defined by P, O, O’, p and p’ Baseline OO’ p’ lies on l’ where the epipolar plane intersects with image plane π’ l’ is epipolar line associated with p and intersects baseline OO’ on e’ e’ is the projection of O observed from O’ Epipolar lines l, l’ Epipoles e, e’

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

Epipolar Constraint: Calibrated Case M’=(R t) Essential Matrix (Longuet-Higgins, 1981) 3 ×3 skew-symmetric matrix: rank=2

Properties of essential matrix E is defined by 5 parameters (3 for rotation and 2 for translation) E p’ is the epipolar line associated with p’ E T p is the epipolar line associated with p Can write as l .p = 0 The point p lies on the epipolar line associated with the vector E p’

Properties of essential matrix (cont’d) E e’=0 and ETe=0 (E e’=-RT[tx]e=0 ) E is singular E has two equal non-zero singular values (Huang and Faugeras, 1989)

Epipolar Constraint: Small Motions To First-Order: Pure translation: Focus of Expansion e The motion field at every point in the image points to focus of expansion

Epipolar Constraint: Uncalibrated Case Fundamental Matrix (Faugeras and Luong, 1992) are normalized image coordinate

Properties of fundamental matrix F has rank 2 and is defined by 7 parameters F p’ is the epipolar line associated with p’ in the 1st image F T p is the epipolar line associated with p in the 2nd image F e’=0 and F T e=0 F is singular

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: a, b, c, d

Weak calibration In theory: E can be estimated with 5 point correspondences F can be estimated with 7 point correspondences Some methods estimate E and F matrices from a minimal number of parameters Estimating epipolar geometry from a redundant set of point correspondences with unknown intrinsic parameters

The Eight-Point Algorithm (Longuet-Higgins, 1981) Homogenous system, set F33=1 |F | =1. Minimize: under the constraint 2

Least-squares minimization |F | =1 Minimize: under the constraint Error function:

Non-Linear Least-Squares Approach (Luong et al., 1993) 8 point algorithm with least-squares minimization ignores the rank 2 property First use least squares to find epipoles e and e’ that minimizes |FT e|2 and |Fe’|2 Minimize with respect to the coefficients of F , using an appropriate rank-2 parameterization (4 parameters instead of 8)

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 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’ i i i i i i T

Weak calibration experiment

Trinocular Epipolar Constraints Optical centers O1O2O3 defines a trifocal plane Generally, P does not lie on trifocal plane formed Trifocal plane intersects retinas along t1, t2, t3 Each line defines two epipoles, e.g., t2 defines e12, e32, wrt O1 and O3 These constraints are not independent!

Trinocular Epipolar Constraints: Transfer Given p1 and p2 , p3 can be computed as the solution of linear equations. Geometrically, p1 is found as the intersection of epipolar lines associated with p2 and p3

Trifocal Constraints P 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) Recall

Trifocal Constraints Calibrated Case All 3×3 minors must be zero! P Calibrated Case All 3×3 minors must be zero! line-line-line correspondence Trifocal Tensor

Trifocal Constraints Calibrated 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 intersect point-line-line correspondence

Trifocal Constraints P Uncalibrated Case

Trifocal Constraints Uncalibrated Case Trifocal Tensor

Trifocal Constraints: 3 Points Pick any two lines l and l through p and p . 2 3 2 3 Do it again. T( p , p , p )=0 1 2 3

Properties of the Trifocal Tensor For any matching epipolar lines, l G l = 0 The matrices G are singular Each triple of points  4 independent equations Each triple of lines  2 independent equations 4p+2l ≥ 26  need 7 triples of points or 13 triples of lines The coefficients of tensor satisfy 8 independent constraints in the uncalibrated case (Faugeras and Mourrain, 1995)  Reduce the number of independent parameters from 26 to 18 2 1 3 T i i 1 Estimating the Trifocal Tensor Ignore the non-linear constraints and use linear least-squares a posteriori Impose the constraints a posteriori

Multiple Views (Faugeras and Mourrain, 1995) All 4 × 4 minors have zero determinants

Two Views 6 minors Epipolar Constraint

Three Views 48 minors Trifocal Constraint

Quadrifocal Constraint (Triggs, 1995) Four Views 16 minors Quadrifocal Constraint (Triggs, 1995)

Geometrically, the four rays must intersect in P..

Quadrifocal Tensor and 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

Scale-Restraint Condition from Photogrammetry Trinocular constraints in the presence of calibration or measurement errors