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

The error sensitivity in the estimation of the 3-D motion and the normal of a planar surface from an instantaneous motion field is studied. The statistical theory or the Cramer-Rao lower bound is used for error covariance in the estimated motion and structure parameters. This 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 measurement noise, size of the field of view, and the motion-geometry configuration. The coupling between translation and rotation 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. The relationships of the uncertainty bounds for every unknown motion parameter to the angle between translation and the plane-normal, the size of the field of view, and the distance from the perceived plane and the translation magnitude are discussed