Fundamentals of Computer Vision
Lecture
• Epipolar geometry.
• Essential matrix.
• Fundamental matrix.
• 8-point algorithm.
Overview of today’s lecture
Slide credits
Most of these slides were adapted from:
• Kris Kitani (15-463, Fall 2016, Fall 2017), Ioannis Gkioulekas (16-385, Spring 2019), Robert Colin (454, Fall 2019s).
Some slides were inspired or taken from: • Fredo Durand (MIT).
Cross product
Can also be written as a matrix multiplication
Skew symmetric
Camera-camera transform just like world-camera transform
If these three vectors are coplanar then
Given a point in one image,
multiplying by the essential matrix will tell us the epipolar line in the second view.
Epipolar constraint
Potential matches for lie on the epipolar line
Assumption:
points aligned to camera coordinate axis (calibrated camera)
putting it together
rigid motion coplanarity
Essential Matrix
[Longuet-Higgins 1981]
Properties of the E matrix
Longuet-Higgins equation
Epipolar lines
Epipoles
(points in normalized coordinates)
How do you generalize to uncalibrated cameras?
The Fundamental matrix
The
Fundamental matrix
is a
generalization
of the
Essential matrix,
where the assumption of
calibrated cameras
is no longer valid
Given a point in one image,
multiplying by the essential matrix will tell us the epipolar line in the second view.
Assumption:
Epipolar constraint
points aligned to camera coordinate axis (calibrated camera)
The Essential matrix operates on image points expressed in
normalized coordinates
(points have been aligned (normalized) to camera coordinates)
Camera coordinate
camera Pixel coordinate point
image point
(row, col)
To use image (pixel) coordinates we must consider the
INTRINSIC camera
image plane
Recall?
K =
parameters
𝛼$ 𝑠 𝑝$ 0 𝛼( 𝑝(
001
camera coordinate system image coordinate system
The Essential matrix operates on image points expressed in
normalized coordinates
(points have been aligned (normalized) to camera coordinates)
K* + ,
To use image (pixel) coordinates we must consider the
INTRINSIC camera
camera point
image point
Camera coordinate
Pixel coordinate (row, col)
parameters
𝛼$ 𝑠 𝑝$ 0 𝛼( 𝑝(
001
image plane
Recall?
K =
camera coordinate system image coordinate system
The Essential matrix operates on image points expressed in
normalized coordinates
(points have been aligned (normalized) to camera coordinates)
K* + ,
camera image point point
Writing out the epipolar constraint in terms of image coordinates
Same equation works in image/pixel coordinates!
it maps pixels to epipolar lines