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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 with and being the two last columns of (obtained through SVD) and respectively the corresponding matrices. The rank-2 constraint is then written as

 (D8)

which is a polynomial of degree 3 in . This can simply be solved analytically. There are always 1 or 3 real solutions. The special case (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: More points... Up: Two view geometry computation Previous: Eight-point algorithm   Contents
Marc Pollefeys 2002-11-22