What Is Computed by Structure from Motion Algorithms?

In the literature we nd two classes of algorithms which, on the basis of two views of a scene, recover the rigid transformation between the views and subsequently the structure of the scene. The rst class contains techniques which require knowledge of the correspondence or the motion eld between the images and are based on the epipolar constraint. The second class contains so-called direct algorithms which require knowledge about the value of the ow in one direction only and are based on the positive depth constraint. Algorithms in the rst class achieve the solution by minimizing a function representing deviation from the epipolar constraint while direct algorithms nd the 3D motion that, when used to estimate depth, produces a minimum number of negative depth values. This paper presents a stability analysis of both classes of algorithms. The formulation is such that it allows comparison of the robustness of algorithms in the two classes as well as within each class. Speciically, a general statistical model is employed to express the functions which measure the deviation from the epipolar constraint and the number of negative depth values, and these functions are studied with regard to their topographic structure, speciically as regards the errors in the 3D motion parameters at the places representing the minima of the functions. 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. In the case of algorithms based on the epipolar constraint the solution for translation is biased toward the center of the image while in the case of direct algorithms it is biased away from the center. The analysis also makes it possible to compare properties of algorithms that rst 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. The framework introduced here, besides serving as a tool that enables the robustness analysis of 3D motion estimation algorithms, provides a formal relationship between a large number of published motion estimation algorithms.