There also exists an important relationship between the homographies
and the fundamental matrices . Let be a point in image . Then
is the corresponding point for the plane in image . Therefore, is located on the corresponding epipolar line; and,

Let be a line in image and let be the plane obtained by back-projecting into space. If
is the image of a point of this plane projected in image , then the corresponding point in image must be located on the corresponding epipolar line (i.e.
). Since this point is also located on the line it can be uniquely determined as the intersection of both (if these lines are not coinciding):
. Therefore, the homography
is given by
. Note that, since the image of the plane is a line in image , this homography is not of full rank. An obvious choice to avoid coincidence of with the epipolar lines, is
since this line does certainly not contain the epipole (i.e.
). Consequently,

(C29) |

Note that this is an important result, since it means that a projective camera setup can be obtained from the fundamental matrix which can be computed from 7 or more matches between two views. Note also that this equation has 4 degrees of freedom (i.e. the 3 coefficients of and the arbitrary relative scale between and ). Therefore, this equation can only be used to instantiate a new frame (i.e. an arbitrary projective representation of the scene) and not to obtain the projection matrices for all the views of a sequence (i.e. compute ). How this can be done is explained in Section 5.2.