Computer Vision: Epipolar Geometry and Stereo in EECS 442, Study notes of Electrical and Electronics Engineering

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EECS 442 – Computer vision Epipolar Geometry

• Why is stereo useful? • Epipolar constraints • Essential and fundamental matrix • Estimating F • Examples Reading: [AZ] Chapters: 4, 9, 11 [FP] Chapters: 10

Pinhole perspective projection

Recovering structure from a single view

C

O

w

P

p Calibration rig Scene Camera K From calibration rigFrom points and lines at infinity+ orthogonal lines and planes → structure of the scene, K → location/pose of the rig, K Knowledge about scene ( point correspondences, geometry of lines & planes, etc…

Recovering structure from a single view

Intrinsic ambiguity of the mapping from 3D to image (2D) Courtesy slide S. Lazebnik

Two eyes help!

• Find X that minimizes

2 2 2 1 1 2

X P x d X P x d

O

1

O

2

x

1

x

2 X Triangulation

Stereo-view geometry

Scene geometry: Find coordinates of 3D point fromits projection into 2 or images. - Correspondence: Given a point in one image, how can I find the corresponding point x’ in another one? - Camera geometry: Given corresponding points intwo images, find camera matrices, position and pose.

Example: Converging image planes

O

1

O

2 X e 2 x 1 x 2 e 2 Example: Parallel image planes

• Baseline intersects the image plane at infinity • Epipoles are at infinity • Epipolar lines are parallel to x axis

Example: Forward translation

• The epipoles have same position in both images • Epipole called FOE (focus of expansion)

O

2 e 1 e 2 O 2

• Two views of the same object - Suppose I know the camera positions and camera matrices - Given a point on left image, how can I find the corresponding point on right image? Epipolar Constraint

⎤ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢⎣ = → u v^1 P M P [ ] 0 I K M =

O

O’

p p’ P R, T Epipolar Constraint [ ] T R K ' M = ⎤ ⎥ ⎥ ⎥⎦ ⎡ ⎢ ⎢ ⎢⎣ ′ ′ = ′ → u v^1 P M P

O

O’

p p’ P R, T

[

]

0 ) p R ( T p

T

×

K 1 and K 2 are known (calibrated cameras)

Epipolar Constraint

Perpendicular to epipolar plane

O

O’

p p’ P R, T

[

]

0 ) p R ( T p

T

×

[

]

p

R

T

p

T

×

E

= essential matrix(Longuet-Higgins, 1981)

Epipolar Constraint

• E x 2 is the epipolar line associated with x 2 (l 1 = E x 2

• E

T x 1 is the epipolar line associated with x 1 (l 2

= E

T x 1

• E is singular (rank two) • E e 2

and

E

T e 1

• E is 3x3 matrix; 5 DOF Epipolar Constraint O 1

O

2 x 2 X x 1 e 1 e 2 l 1 l 2