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 computati onal models which can be used for estimating 3D motion from an image sequen ce. 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 mathemati cal constraints, there exist two principles relating a sequence of images t aken by a moving camera. One is the "epipolar constraint," applied to motio n fields, and the other the "positive depth" constraint, applied to normal flow fields. 3D motion estimation amounts to optimizing these constraints o ver the image. A statistical modeling of these constraints leads to functio ns which are studied with regard to their topographic structure, specifical ly as regards the errors in the 3D motion parameters at the places represen ting 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 ro tational errors on the image plane are perpendicular to each other. Further more, the estimated projection of the translation on the image lies on a li ne through the origin and the projection of the real translation. The situa tion is different for a camera with a full (360 degree) field of view (achi eved 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 th e translational or the rotational error becomes zero, while in the case of a restricted held of view both errors are non-zero. Although some ambiguiti es still remain in the full field of view case, the implication is that vis ual navigation tasks, such as visual servoing, involving 3D motion estimati on are easier to solve by employing panoramic vision. Also, the analysis ma kes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rota tion, algorithms that do the opposite, and algorithms that estimate all mot ion parameters simultaneously, thus providing a sound framework for the obs ervability of 3D motion. Finally, the introduced framework points to new av enues for studying the stability of image-based servoing schemes.