ECE 549CS 543: COMPUTER VISON LECTURE 14

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ECE 549/CS 543 COMPUTER VISON LECTURE
14 MULTI-VIEW GEOMETRY II

• The Essential Matrix
• The Fundamental Matrix
• The 8-Point Algorithm
• The Trifocal Tensor

• Reading Chapter 10
• A list of potential projects is at
• http//www-cvr.ai.uiuc.edu/ponce/fall04/project
  s.pdf
• Homework Weak Calibration due Th. Oct. 21.
• http//www-cvr.ai.uiuc.edu/ponce/fall04/hw3/hw3
  .pdf

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Epipolar Constraint Calibrated Case
Essential Matrix (Longuet-Higgins, 1981)
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Properties of the Essential Matrix

• E p is the epipolar line associated with p.
• E p is the epipolar line associated with p.
• E e0 and E e0.
• E is singular.
• E has two equal non-zero singular values
• (Huang and Faugeras, 1989).

T
T
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Epipolar Constraint Uncalibrated Case
Fundamental Matrix (Faugeras and Luong, 1992)
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Properties of the Fundamental Matrix

• F p is the epipolar line associated with p.
• F p is the epipolar line associated with p.
• F e0 and F e0.
• F is singular.

T
T
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The Eight-Point Algorithm (Longuet-Higgins, 1981)
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Non-Linear Least-Squares Approach (Luong et al.,
1993)
Minimize
with respect to the coefficients of F , using an
appropriate rank-2 parameterization.
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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.

i
i
i
i
i
i
T
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Data courtesy of R. Mohr and B. Boufama.
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Mean errors 10.0pixel 9.1pixel
Without normalization
Mean errors 1.0pixel 0.9pixel
With normalization
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Trinocular Epipolar Constraints
These constraints are not independent!
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Trinocular Epipolar Constraints Transfer
Given p and p , p can be computed as the
solution of linear equations.
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2
3
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Trifocal Constraints
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Trifocal Constraints
Calibrated Case
All 3x3 minors must be zero!
Trifocal Tensor
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Trifocal Constraints
Uncalibrated Case
Trifocal Tensor
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Properties of the Trifocal Tensor
i
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Estimating the Trifocal Tensor

• Ignore the non-linear constraints and use linear
  least-squares
• a posteriori.
• Impose the constraints a posteriori.

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The backprojections of the two lines do not
define a line!