Example 2 - General Motion

:

• A point corresponding to always lies on its epipolar line , but in general does not lie on .

• A general motion can be thought of as a pure translation of the optical centre, followed by a rotation of the image plane about the optical centre. The effects of the rotation can be ``undone'' by a projective transformation

Computing : I Number of Correspondences Given perfect image points (no noise) in general position. Each point correspondence generates one constraint on .

This can be rearranged as where is a measurement matrix, and is the fundamental matrix represented as a 9-vector.

• 8 or more points in general position: The dimension of the null-space of is one. , and consequently , is determined uniquely up to scale.

• 7 points in general position: The dimension of the null-space is two.

  Suppose the 9-vectors and , corresponding to the matrices and respectively, span the null-space. Then, up to scale, there is a one-parameter family of matrices: . Imposing

  

  which is a cubic in , and thus has either one or three real solutions.