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### Relation between the fundamental matrix and image homographies

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,

 (C27)

should be verified. Moreover, equation (3.27) holds for every image point . Since the fundamental matrix maps points to corresponding epipolar lines, and equation (3.27) is equivalent to . Comparing this equation with , and using that these equations must hold for all image points and lying on corresponding epipolar lines, it follows that:
 (C28)

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)

corresponds to the homography of a plane. By combining this result with equations (3.16) and (3.17) one can conclude that it is always possible to write the projection matrices for two views as
 (C30)

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.

Next: Three view geometry Up: Two view geometry Previous: Two view geometry   Contents
Marc Pollefeys 2002-11-22