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 particular problem of interest in geometric computer vision is inferring the essential matrix relating two images from point correspondences, especially from a minimal set of correspondences @cite_14 . Minimal solutions are useful for application in a random sample consensus (RANSAC) @cite_4 loop to robustly estimate the motion parameters and separate inliers from outliers. @cite_26 derived an efficient minimal solution from five point correspondences and et al @cite_6 later improved the accuracy of this method. In this work, we derive a solution for the essential matrix from at least three correspondences which applies when the camera undergoes spherical motion.
@cite_14 @cite_4 @cite_6 @cite_26
1598123022 1936156518 2138835141 1991544872
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. We propose a novel solution for computing the relative pose between two generalized cameras that includes reconciling the internal scale of the generalized cameras. This approach can be used to compute a similarity transformation between two coordinate systems, making it useful for loop closure in visual odometry and registering multiple structure from motion reconstructions together. In contrast to alternative similarity transformation methods, our approach uses 2D-2D image correspondences thus is not subject to the depth uncertainty that often arises with 3D points. We utilize a known vertical direction (which may be easily obtained from IMU data or vertical vanishing point detection) of the generalized cameras to solve the generalized relative pose and scale problem as an efficient Quadratic Eigenvalue Problem. To our knowledge, this is the first method for computing similarity transformations that does not require any 3D information. Our experiments on synthetic and real data demonstrate that this leads to improved performance compared to methods that use 3D-3D or 2D-3D correspondences, especially as the depth of the scene increases. Inferring scene geometry and camera motion from a stream of images is possible in principle, but is an ill-conditioned problem when the objects are distant with respect to their size. We have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step. An image stream can be represented by the 2FxP measurement matrix of the image coordinates of P points tracked through F frames. We show that under orthographic projection this matrix is of rank 3. Based on this observation, the factorization method uses the singular-value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively. Two of the three translation components are computed in a preprocessing stage. The method can also handle and obtain a full solution from a partially filled-in measurement matrix that may result from occlusions or tracking failures. The method gives accurate results, and does not introduce smoothing in either shape or motion. We demonstrate this with a series of experiments on laboratory and outdoor image streams, with and without occlusions. We propose a non-iterative solution to the PnP problem--the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences--whose computational complexity grows linearly with n. This is in contrast to state-of-the-art methods that are O(n 5) or even O(n 8), without being more accurate. Our method is applicable for all n?4 and handles properly both planar and non-planar configurations. Our central idea is to express the n 3D points as a weighted sum of four virtual control points. The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in O(n) time by expressing these coordinates as weighted sum of the eigenvectors of a 12×12 matrix and solving a small constant number of quadratic equations to pick the right weights. Furthermore, if maximal precision is required, the output of the closed-form solution can be used to initialize a Gauss-Newton scheme, which improves accuracy with negligible amount of additional time. The advantages of our method are demonstrated by thorough testing on both synthetic and real-data.