ECE 549CS 543: COMPUTER VISON LECTURE 13

1
ECE 549/CS 543 COMPUTER VISON LECTURE
13 MULTI-VIEW GEOMETRY I

• Epipolar Geometry
• The Essential Matrix
• The Fundamental Matrix
• The 8-Point Algorithm

• Reading Chapter 10
• A list of potential projects is at
• http//www-cvr.ai.uiuc.edu/ponce/fall04/project
  s.pdf
• Homework Photometric stereo (due Tue. Oct. 12)
• http//www-cvr.ai.uiuc.edu/ponce/fall04/hw2/hw2
  .txt

2
I will be out of town next week (Oct. 12 and
Oct. 14). Fred Rothganger will replace me for
these two lectures. He will continue multi-view
geometry and give a research lecture on object
recognition.
3
Reconstruction / Triangulation
4
(Binocular) Fusion
5
Epipolar Geometry

• Epipolar Plane

• Baseline

• Epipoles

• Epipolar Lines

6
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.

7
Epipolar Constraint Calibrated Case
Essential Matrix (Longuet-Higgins, 1981)
8
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
9
Epipolar Constraint Small Motions
To First-Order
Pure translation Focus of Expansion
10
Epipolar Constraint Uncalibrated Case
Fundamental Matrix (Faugeras and Luong, 1992)
11
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
12
The Eight-Point Algorithm (Longuet-Higgins, 1981)
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)

• 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
15
Data courtesy of R. Mohr and B. Boufama.
16
Mean errors 10.0pixel 9.1pixel
Without normalization
Mean errors 1.0pixel 0.9pixel
With normalization