Trajectory triangulation: 3D reconstruction of moving points from a monocular image sequence
01 Apr 2000-IEEE Transactions on Pattern Analysis and Machine Intelligence (IEEE Computer Society)-Vol. 22, Iss: 4, pp 348-357
TL;DR: The problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera is considered, i.e., to reconstruct moving objects from line-of-sight measurements only, and the solutions for points moving along a straight-line and along conic-section trajectories are investigated.
Abstract: We consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera, i.e., to reconstruct moving objects from line-of-sight measurements only. The task is feasible only when some constraints are placed on the shape of the trajectory of the moving point. We coin the family of such tasks as "trajectory triangulation." We investigate the solutions for points moving along a straight-line and along conic-section trajectories, We show that if the point is moving along a straight line, then the parameters of the line (and, hence, the 3D position of the point at each time instant) can be uniquely recovered, and by linear methods, from at least five views. For the case of conic-shaped trajectory, we show that generally nine views are sufficient for a unique reconstruction of the moving point and fewer views when the conic is of a known type (like a circle in 3D Euclidean space for which seven views are sufficient). The paradigm of trajectory triangulation, in general, pushes the envelope of processing dynamic scenes forward. Thus static scenes become a particular case of a more general task of reconstructing scenes rich with moving objects (where an object could be a single point).
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Cites background from "Trajectory triangulation: 3D recons..."
..., depending on the trajectory assumption [6])....
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...Avidan and Shashua [5, 6] coined the term trajectory triangulation as a technique to reconstruct 3D points of the moving object when the object trajectory is known or satisfies a parametric form....
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...Instead of assuming that the object is moving along a line, Shashua et al. [144] assumed that the object is moving over a conic section....
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...Prior knowledge about the camera motion is not needed, although some approaches [5, 6] assume that the camera pose is available....
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"Trajectory triangulation: 3D recons..." refers methods in this paper
...We would like to thank Yair Ramati for pointing our attention to the literature on orbit determination and reference [3]....
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TL;DR: The formalism of the Grassmann-Cayley algebra is proposed to use as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed).
Abstract: We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to use the formalism of the Grassmann-Cayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed). We have a fairly complete picture of the situation in the case of points; there are only three types of algebraic relations which are satisfied by the coordinates of the images of a 3-D point: bilinear relations arising when we consider pairs of images among the N and which are the well-known epipolar constraints, trilinear relations arising when we consider triples of images among the N, and quadrilinear relations arising when we consider four-tuples of images among the N. In the case of lines, we show how the traditional perspective projection equation can be suitably generalized and that in the case of three images there exist two independent trilinear relations between the coordinates of the images of a 3-D line. >
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160 citations
01 Jul 1992
TL;DR: It is shown that the antipenumbra is, in general, a disconnected set bounded by portions of quadric surfaces, and an implemented O(n2) time algorithm that computes this boundary is described.
Abstract: We define the antiumbra and the antipenumbra of a convex areal light source shining through a sequence of convex areal holes in three dimensions. The antiumbra is the volume beyond the plane of the final hole from which all points on the light source can be seen. The antipenumbra is the volume from which some, but not all, of the light source can be seen. We show that the antipenumbra is, in general, a disconnected set bounded by portions of quadric surfaces, and describe an implemented O(n2) time algorithm that computes this boundary. The antipenumbra computation is motivated by visibility computations and might prove useful in rendering shadowed objects. We also present an implemented extension of the algorithm that computes planar and quadratic surfaces of discontinuous illumination useful for polygon meshing in global illumination computations.
159 citations