The aim of this paper is to explore a linear geometric algorithm for recovering the three dimensional motion of a moving camera from image velocities. Generic similarities and differences between the discrete approach and the differential approach are clearly revealed through a parallel development of an analogous motion estimation theory previously explored in [24, 26]. We present a precise characterization of the space of differential essential matrices, which gives rise to a novel eigenvalue-decomposition-based 3D velocity estimation algorithm from the optical flow measurements. This algorithm gives a unique solution to the motion estimation problem and serves as a differential counterpart of the well-known SVD-based 3D displacement estimation algorithm for the discrete case. Since the proposed algorithm only involves linear algebra techniques, it may be used to provide a fast initial guess for more sophisticated nonlinear algorithms [13]. Extensive simulation results are presented for evaluating the performance of our algorithm in terms of bias and sensitivity of the estimates with respect to di erent noise levels in image velocity measurements.

In this paper, a geometric theory of camera self-calibration is developed. The problem of camera self-calibration is shown to be equivalent to the problem of recovering an unknown (Riemannian) metric of an appropriate space. This observation leads to a new account of the necessary and sufficient condition for a unique calibration. Based on this understanding, we obtain a new and complete critical motion analysis without introducing a projective space. A complete list of geometric invariants associated to an uncalibrated camera is given. Due to a new characterization of fundamental matrices, the Kruppa equations are re-derived and directly associated to the basic (co)invariants of the uncalibrated camera. We study general questions about the solvability of the Kruppa equations and show that, in some special cases, the Kruppa equations can be renormalized so as to allow for linear self-calibration algorithms. A further study of these cases not only reveals generic difficulties in convent...

Based upon an axiomatic formulation of vision system in a general Riemannian manifold, this paper provides a unied study of multiple view geometry for three dimensional spaces of constant curvature, including: Euclidean space, spherical space and hyperbolic space. It is shown that multiple view geometry for Euclidean space can be interpreted as a limit case when curvature of a nonEuclidean space approaches to zero. In particular, we show that epipolar constraint in the general case is exactly the same as that known for the Euclidean space. Generalized conditions on algebraic and geometric dependency among multilinear constraints are presented. A triangulation method for non-Euclidean spaces is also introduced using trigonometry laws from Absolute Geometry. The developed non-Euclidean multiple view geometry can be useful for geographical or astronomical purposes, when curvature of the space can no longer be ignored due to large scale. It may also be applied to situations when optical m...

Multiview geometry has been traditionally developed in the framework of projective geometry, which is technically rather algebraic. In this paper, we show an alternative approach which uses notation and concepts from differential geometry. We review all projective (multilinear) constraints and Euclidean invariants associated with the problem of structure and motion recovery from n views. As a consequence of the study of projective constraints we show geometric dependency of the trilinear and quadrilinear constraints on the bilinear ones and associated conditions on motions which guarantee the dependency. The study of Euclidean invariants leads us to a new derivation and interpretation of Kruppa's equations as a inner product coinvariant of Euclidean transformations in a space with unknown metric. The differential geometric approach allows us to establish the results in an elegant and concise way and reveal intrinsic geometric meaning of the problems. New results and new algorithms fall...

In this paper, we investigate the problem of using computer vision as a sensor to control the landing of an unmanned aerial vehicle (UAV). The vision problem we address is a differential version of the structure from motion problem for a planar scene, where one uses image velocities to recover the motion of the camera and the orientation of a (landing) plane. We propose a new estimation scheme for solving this problem. The autonomous landing task we address is for a vertical take-off and landing (VTOL) aircraft equipped with an on-board camera. Since there is an inherent scale ambiguity when using vision as a sensor for state estimation, we design a vision based landing controller which is robust to this unknown scale. We present simulation results for both the proposed vision algorithm and the landing controller. Keywords: unmanned aerial vehicle, autonomous landing, structure from motion 1 Introduction Our objective is to use computer vision as a sensor to control the landing of a ...

In this paper we study the problem of Euclidean structure and motion recovery from m-frames in the case of calibrated cameras. We formulate the problem in the "joint image space" and first review the existing multi-linear constraints between m-images of n-points using exterior algebraic notation. It is well known that the projective constraints capture the information about the motion between individual frames and are used to recover it up to a scale. We show how the structural scale information which is lost during the projection process can be recovered using additional Euclidean constraints and propose a linear algorithm for obtaining compatible scales of the joint image matrix entries. The presented theory and algorithms are developed for both the discrete and differential case. We outline how the approach can be extended for the hybrid case where for particular image locations both optical flow information and point correspondences in the consecutive frames are available. 1 Intr...

In this paper, vision theory for Euclidean, spherical and hyperbolic spaces is studied in a uniform framework using differential geometry in spaces of constant curvature. It is shown that the epipolar geometry for Euclidean space can be naturally generalized to the spaces of constant curvature. In particular, it is shown that, in the general case, the bilinear epipolar constraint is exactly the same as in the Euclidean case; also, there are only bilinear, trilinear and quadrilinear constraints associated with multiple images of a point. Differential (continuous) case is also studied. For the structure from motion problem, 3D structure can only be determined up to a universal scale, the same as the Euclidean case. Approaches are proposed to reconstruct 3D structure with respect to a normalized curvature.