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Seven-point algorithm

In fact the two view structure (or the fundamental matrix) only has seven degrees of freedom. If one is prepared to solve non-linear equations, seven points must thus be sufficient to solve for it. In this case the rank-2 constraint must be enforced during the computations.

A similar approach as in the previous section can be followed to characterize the right null-space of the system of linear equations originating from the seven point correspondences. This space can be parameterized as follows ${\bf v}_1 + \lambda {\bf v}_2 \mbox{ or } {\bf F}_1 + \lambda {\bf F}_2$ with ${\bf v}_1$ and ${\bf v}_2$ being the two last columns of ${\bf V}$ (obtained through SVD) and ${\bf F}_1$ respectively ${\bf F}_2$ the corresponding matrices. The rank-2 constraint is then written as

\begin{displaymath}
\det \left( {\bf F}_1 + \lambda {\bf F}_2 \right) = a_3 \lambda^3 + a_2 \lambda^2 + a_1 \lambda + a_0 = 0
\end{displaymath} (D8)

which is a polynomial of degree 3 in $\lambda$. This can simply be solved analytically. There are always 1 or 3 real solutions. The special case ${\bf F}_1$ (which is not covered by this parameterization) is easily checked separately, i.e. it should have rank-2. If more than one solution is obtained then more points are needed to obtain the true fundamental matrix.


next up previous contents
Next: More points... Up: Two view geometry computation Previous: Eight-point algorithm   Contents
Marc Pollefeys 2002-11-22