Observability of 3D Motion

This paper examines the inherent difficulties in observing 3D rigid motion from image sequences. It does so without considering a particular estimator. Instead, it presents a statistical analysis of all the possible computational models which can be used for estimating 3D motion from an image sequence. These computational models are classified according to the mathematical constraints that they employ and the characteristics of the imaging sensor (restricted field of view and full field of view). Regarding the mathematical constraints, there exist two principles relating a sequence of images taken by a moving camera. One is the “epipolar constraint,” applied to motion fields, and the other the “positive depth” constraint, applied to normal flow fields. 3D motion estimation amounts to optimizing these constraints over the image. A statistical modeling of these constraints leads to functions which are studied with regard to their topographic structure, specifically as regards the errors in the 3D motion parameters at the places representing the minima of the functions. For conventional video cameras possessing a restricted field of view, the analysis shows that for algorithms in both classes which estimate all motion parameters simultaneously, the obtained solution has an error such that the projections of the translational and rotational errors on the image plane are perpendicular to each other. Furthermore, the estimated projection of the translation on the image lies on a line through the origin and the projection of the real translation. The situation is different for a camera with a full (360 degree) field of view (achieved by a panoramic sensor or by a system of conventional cameras). In this case, at the locations of the minima of the above two functions, either the translational or the rotational error becomes zero, while in the case of a restricted field of view both errors are non-zero. Although some ambiguities still remain in the full field of view case, the implication is that visual navigation tasks, such as visual servoing, involving 3D motion estimation are easier to solve by employing panoramic vision. Also, the analysis makes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rotation, algorithms that do the opposite, and algorithms that estimate all motion parameters simultaneously, thus providing a sound framework for the observability of 3D motion. Finally, the introduced framework points to new avenues for studying the stability of image-based servoing schemes.