The Fundamental Matrix Song

If you like this, you'll probably also be interested in the RANSAC song.

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

You may also be interested in 2.71828183: The e Song.
Video with Japanese subtitles
Download a high-quality version of the video (about 25MB) (requires Xvid codec)
Download video with English subtitles
Download the MP3 (about 4MB)
Download the Opera House sequence

Thanks to:

Dr Peter Kovesi, whose code turned Matlab into a video effects tool,
Peter Pakulski, the AviSynth guru,
ilesoft, whose Sydney Harbour Bridge model is scarily accurate, and
Payashim, for the Japanese translation.

Related sites

Email: fmatrix at danielwedge dot com
Feel free to play this in lectures etc, you have my permission (though I'd be interested to hear from you if you do!)

Daniel Wedge
Song: 19th October, 2008.
Video: 25th March, 2009.

danielwedge.com home