We describe a special case of structure from motion where the camera rotates on a sphere. The camera's optical axis lies perpendicular to the sphere's surface. In this case, the camera's pose is minimally represented by three rotation parameters. From analysis of the epipolar geometry we derive a novel and efficient solution for the essential matrix relating two images, requiring only three point correspondences in the minimal case. We apply this solver in a structure-from-motion pipeline that aggregates pairwise relations by rotation averaging followed by bundle adjustment with an inverse depth parameterization. Our methods enable scene modeling with an outward-facing camera and object scanning with an inward-facing camera.
A simple algorithm for computing the three-dimensional structure of a scene from a correlated pair of perspective projections is described here, when the spatial relationship between the two projections is unknown. This problem is relevant not only to photographic surveying1 but also to binocular vision2, where the non-visual information available to the observer about the orientation and focal length of each eye is much less accurate than the optical information supplied by the retinal images themselves. The problem also arises in monocular perception of motion3, where the two projections represent views which are separated in time as well as space. As Marr and Poggio4 have noted, the fusing of two images to produce a three-dimensional percept involves two distinct processes: the establishment of a 1:1 correspondence between image points in the two views—the ‘correspondence problem’—and the use of the associated disparities for determining the distances of visible elements in the scene. I shall assume that the correspondence problem has been solved; the problem of reconstructing the scene then reduces to that of finding the relative orientation of the two viewpoints. A new paradigm, Random Sample Consensus (RANSAC), for fitting a model to experimental data is introduced. RANSAC is capable of interpreting smoothing data containing a significant percentage of gross errors, and is thus ideally suited for applications in automated image analysis where interpretation is based on the data provided by error-prone feature detectors. A major portion of this paper describes the application of RANSAC to the Location Determination Problem (LDP): Given an image depicting a set of landmarks with known locations, determine that point in space from which the image was obtained. In response to a RANSAC requirement, new results are derived on the minimum number of landmarks needed to obtain a solution, and algorithms are presented for computing these minimum-landmark solutions in closed form. These results provide the basis for an automatic system that can solve the LDP under difficult viewing Abstract This paper presents a novel version of the five-point relative orientation algorithm given in Nister [Nister, D., 2004. An efficient solution to the five-point relative pose problem, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (6), 756–770]. The name of the algorithm arises from the fact that it can operate even on the minimal five-point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences, the algorithm returns up to 10 real solutions. The algorithm can also operate on many points. Like the previous version of the five-point algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper presents comparisons with other direct methods, including the previously developed five-point method, two different six-point methods, the seven-point method, and the eight-point method. It is shown that the five-point method is superior in most cases among the direct methods. The new version of the algorithm was developed from the perspective of algebraic geometry and is presented in the context of computing a Grobner basis. The constraints are formulated in terms of polynomial equations in the entries of the fundamental matrix. The polynomial equations generate an algebraic ideal for which a Grobner basis is computed. The Grobner basis is used to compute the action matrix for multiplication by a single variable monomial. The eigenvectors of the action matrix give the solutions for all the variables and thereby also relative orientation. Using a Grobner basis makes the solution clear and easy to explain. 