The coupling of rotation and translation in motion estimation of planar surfaces

This paper studies the error sensitivity in the estimation of the 3D-motion and the normal of a planar surface from an instantaneous motion field. We use the statistical theory of the Cramer-Rao lower bound for the error covariance in the estimated motion and structure parameters which enables the derivation of results valid for any unbiased estimator under the assumption of Gaussian noise in the motion field. The obtained lower-bound-matrix is studied analytically with respect to the measurement noise, size of the field of view and the motion-geometry configuration. The main result of this analysis is the coupling between translation and rotation which is exacerbated if the field of view and the slant of the plane become smaller and the deviation of the translation from the viewing direction becomes larger. By-products of this study are the relationships of the uncertainly bounds for every unknown motion parameter to the angle between translation and the plane-normal, the size of the field of view, the distance from the perceived plane and the translation magnitude.