I am learning how to estimate the Fundamental Matrix via the 8-point algorithm:
From N > 8 corresponding points that satisfy $xFx' = 0$, we need to solve the LS problem:
$kron(x,x')f = 0$
Where $kron(x,x')$ is a N by 9 matrix made of the elements of the corresponding points, and $f$ is the matrix $F$ as a column vector.
However, I came a cross a statement that if the set of N corresponding points is coplanar, or lying on the same 3D plane, then the LS problem has no solution.
Can someone please explain why the fundamental matrix can't be estimated from a coplanar set of points? Both an algebraic and geometric explanation will be welcome.
In the book, Three-dimensional computer vision by Olivier Faugeras (Page 271) is explained why you can't have coplanar points.
Basically it's because the matrix $kron(x,x^\prime)$ has to be of rank more or equal to 8 to compute the inverse of the eight first columns of $kron(x,x^\prime)$ and recover the essential matrix (as 8x1 vector) as $$ f=-F_{33}(U_8)^{-1}C_9 $$ Where $U_8$ represent the $n\times8$ matrix formed by the first eight columns of $kron(x,x^\prime)$ and $C_9$ is the ninth colimn vector of $kron(x,x^\prime)$.