We have developed a first order model of uncertainty prediction which takes account of all the errors involved in estimating the homography and using it for measurements. The validity has been demonstrated in a number of applications. The approach can be applied to any world measurement, lengths and parallelism have been demonstrated, and area and angles can also be shown.
It is worth noting that although the theory has been developed here for
plane to plane homographies, the same approach can be used for line to
line and 3-space to 3-space homographies. In particular the matrix
perturbation based derivation of covariance can be used where these
relations are estimated from an equation of the form
. This includes estimating epipolar geometry.
We are currently extending these ideas in several ways: 1. Investigation of the errors if the non-linear geometric estimation method described in section 3 is used; 2. If two views are used, then the validity of the planar assumption can also be verified. A ``plane plus parallax'' [ 8 ] approach can be used; 3. The camera has so far been modelled as a perfect perspective mapping. We are investigating the extent to which is necessary to correct for radial distortion; 4. Providing a consistency check on image-world correspondences by using a RANSAC [ 6 ] like approach to move from minimal subsets to the full set.
We would like to thank Andrew Fitzgibbon for the development and maintenace of much of the software used.
Antonio Criminisi