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Computer Vision and Image Understanding
Volume 71, Issue 1, July 1998, Pages 1-18
 
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doi:10.1006/cviu.1997.0621    How to Cite or Link Using DOI (Opens New Window)
Copyright © 1998 Academic Press. All rights reserved.

Regular Article

On the Determination of Epipoles Using Cross-Ratios

Q. -T. Luonga and O. D. Faugerasb

a SRI International, 333 Ravenswood Avenue, Menlo Park, California, 94025 b I.N.R.IA. 2004, route de Lucioles, 06902, Sophia-Antipolis, France

Received 30 January 1995; 
accepted 19 February 1997. ;
Available online 10 April 2002.

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Abstract

We study the problem of computing the position of the epipoles in a pair of uncalibrated images. The approach, which is based on the invariance of the cross-ratio by theepipolar transformation, exploits algebraic constraints obtained from point correspondences and provides a solution in which only the epipoles are involved. This is in opposition to the methods based on the computation of the fundamental matrix. These notions are first presented as well as the newepipolar ordering constraint. Three families of methods are successively considered: the first uses statistics on closed-form solutions provided by the so-calledSturm method, the second uses intersect plane cubics through deterministic procedures, and the third is based on nonlinear minimizations of a difference of cross-ratios. We discuss the shortcomings of each and show, using numerous experimental comparisons, that there is a trade-off between elegance and robustness to noise. The cross-ratio based methods do not turn out to be a generally viable alternative to the method based on the fundamental matrix.

Abbreviations: projective geometryAbbreviations: uncalibrated imagesAbbreviations: motionAbbreviations: algebraic methods


 
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