The Controversy Surrounding the Application of Projective Geometry to Stereo Vision

Abstract

Although the success of the projective geometry applications in stereo vision, a number of criticisms have been raised in the literature about its use. Most of them concern the performance of the eight-point algorithm that is used to find the fundamental matrix F. And few directly target the application of the projective geometry in computer vision. This paper first, reports on some of these criticisms followed by some flawed derivations of the fundamental matrix equation. Then, in a simple and unquestionable analysis, it demonstrates that the equation of the fundamental matrix mTrFml = 0 does not hold for all image points ml and mr. The matrix F is independent of the scene structure, it depends only on the rotation and translation of the second camera with respect to the first one; (F ≜ [t]xR). The vectors ml and mr are not orthogonal for every point M. And because the dot product (mTrF) · ml) is equal to zero if and only if the two vectors mTrF and ml are orthogonal, the equation is invalid.