The First Order Expansion of Motion Equations in the Uncalibrated Case

In the present paper we address the problem of computing structure and motion, given set point correspondences in a monocular image sequence, consideringsmall motionswhen the camera isnot calibrated. We first set the equations defining the calibration, rigid motion, and scene structure. We then review the motion equation, the structure from motion equation, and the depth evolution equation, including the particular case of planar structures, considering a discrete displacement between two frames. As a further step, we develop the first order expansion of these equations and analyze the observability of the related infinitesimal quantities. It is shown that we obtain a complete correspondence between these equations and the equation derived in the discrete case. However, in the case of infinitesimal displacements, the projection of the translation (focus of expansion or epipole) is clearly separated from the rotational component of the motion. This is an important advantage of the present approach. Using this last property, we propose a mechanism of image stabilization in which the rotational disparity is iteratively canceled. This allows a better estimation of the focus of expansion, and simplifies different aspects of the analysis of the equations: structure from motion equation, analysis of ambiguity, and geometrical interpretation of the motion equation. This mechanism is tested on different sets of real images. The discrete model is compared to the continuous model. Projective reconstructions of the scene are provided.

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