# Motion of points and lines in the uncalibrated case

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## Abstract

In the present paper we address the problem of computing structure and motion, given a set point and/or line correspondences, in a monocular image sequence, when the camera is not calibrated.

Considering point correspondences first, we analyse how to parameterize the retinal correspondences, in function of the chosen geometry: Euclidean, affine or projective geometry. The simplest of these parameterizations is called the *FQs*-representation and is a composite projective representation. The main result is that considering *N*+1 views in such a monocular image sequence, the retinal correspondences are parameterized by 11 *N*−4 parameters in the general projective case. Moreover, 3 other parameters are required to work in the affine case and 5 additional parameters in the Euclidean case. These 8 parameters are “calibration” parameters and must be calculated considering at least 8 external informations or constraints. The method being constructive, all these representations are made explicit.

Then, considering line correspondences, we show how the the same parameterizations can be used when we analyse the motion of lines, in the uncalibrated case. The case of three views is extensively studied and a geometrical interpretation is proposed, introducing the notion of trifocal geometry which generalizes the well known epipolar geometry. It is also discussed how to introduce line correspondences, in a framework based on point correspondences, using the same equations.

Finally, considering the *F Qs*-representation, one implementation is proposed as a “motion module”, taking retinal correspondences as input, and providing and estimation of the 11 *N*−4 retinal motion parameters. As discussed in this paper, this module can also estimate the 3D depth of the points up to an affine and projective transformation, defined by the 8 parameters identified in the first section. Experimental results are provided.

## Keywords

Motion Parameter Geometrical Interpretation Projective Geometry External Information Projective Transformation## Preview

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