The fundamental matrix
Used in stereo geometry
A matrix with nine entries
It's square with size 3 by 3
Has seven degrees of freedom
It has a rank deficiency
It's only of rank two
Call the matrix F and you'll see...
Two points that correspond
Column vectors called x and x-prime
x-prime transpose times F times x
Equals zero every time
The epipolar constraint
Involves epipolar lines
Postmultiplying F by x
Results in vector l-prime
It's the epipolar line
In the other view passing through x-prime
A three component vector
Of homogeneous design
The left and right nullspaces of F
Are the epipoles e-prime and e
All of the epipolar lines
Should pass through these
Here's a linear estimation example:
Take a set of 8 point samples
Construct a matrix, take the SVD
And the elements of F are in the last column of V
If you try to estimate
F with a coplanar set of points
Your sample set will be degenerate
And will not bring you joy
When doing the estimation
If you don't perform rank deprivation
Your epipolar lines
And the epipoles will not coincide
But if your scene has three views
The trifocal tensor is what you'd use
Constraints from the third view act like glue
That can't be determined from just two views
[Credit to: Daniel Wedge 2009]
