基本矩阵之歌

   上午在《Multiple View Geometry in Computer Vision》一书的主页上check errata,偶然发现一神曲《The fundamental matrix song》,竟然将computer vison里面外极几何和基本矩阵的内容融汇贯通于旋律之中……
   orz…太赞了~
   老外真能折腾,这玩意儿也能谱曲填词,简直高深莫测啊…
   直接看看视频吧:
  

    歌词如下:
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

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