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EECS 274 Computer Vision

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 ’.

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EECS 274 Computer Vision

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  1. EECS 274 Computer Vision Geometry of Multiple Views

  2. Geometry of Multiple Views • Epipolar geometry • Essential matrix • Fundamental matrix • Trifocal tensor • Quadrifocal tensor • Reading: FP Chapter 10

  3. Epipolar geometry • Epipolarplane OPO’ • Baseline OO’ l’ is epipolar line associated with p and intersects baseline OO’ on e’ • Epipolarlines l, l’ • Epipolese, e’ e’ is the projection of O observed from O’

  4. 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.

  5. Epipolar Constraint: Calibrated Case Essential Matrix (Longuet-Higgins, 1981) 3 ×3 skew-symmetric matrix: rank=2

  6. Properties of essential matrix • E is defined by 5 parameters (3 for rotation and 2 for translation • E Tp’ is the epipolar line associated with p’ • E p is the epipolar line associated with p • E e’=0 and E Te=0 • E is singular • E has two equal non-zero singular values • (Huang and Faugeras, 1989)

  7. Epipolar Constraint: Small Motions To First-Order: Pure translation: Focus of Expansion

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

  9. Properties of fundamental matrix • 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

  10. 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

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

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

  13. Non-Linear Least-Squares Approach (Luong et al., 1993) Minimize with respect to the coefficients of F , using an appropriate rank-2 parameterization

  14. 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 TFT’ i i i i i i T

  15. Trinocular Epipolar Constraints These constraints are not independent!

  16. Trinocular Epipolar Constraints: Transfer Given p and p , p can be computed as the solution of linear equations. 1 2 3

  17. 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) Recall

  18. Trifocal Constraints Calibrated Case All 3x3 minors must be zero! line-line-line correspondence Trifocal Tensor

  19. Trifocal Constraints Calibrated Case Given 3 point correspondences, p1, p2, p3of 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

  20. Trifocal Constraints Uncalibrated Case

  21. Trifocal Constraints Uncalibrated Case Trifocal Tensor

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

  23. T i • For any matching epipolar lines, lGl = 0. • The matrices G are singular. • They satisfy 8 independent constraints in the • uncalibrated case (Faugeras and Mourrain, 1995). 2 1 3 Properties of the Trifocal Tensor i 1 Estimating the Trifocal Tensor • Ignore the non-linear constraints and use linear least-squares • a posteriori. • Impose the constraints a posteriori.

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

  25. Two Views Epipolar Constraint

  26. Three Views Trifocal Constraint

  27. Four Views Quadrifocal Constraint (Triggs, 1995)

  28. Geometrically, the four rays must intersect in P..

  29. 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

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