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More points...

It is clear that when more point matches are available the redundancy should be used to minimize the effect of the noise. The eight-point algorithm can easily be extended to be used with more points. In this case the matrix ${\bf A}$ of equation 4.7 will be much bigger, it will have one row per point match. The solution can be obtained in the same way, but in this case the last singular value will not be perfectly equal to zero. It has been pointed out [52] that in practice it is very important to normalize the equations. This is for example achieved by transforming the image to the interval $[-1,1]\times[-1,1]$ so that all elements of the matrix ${\bf A}$ are of the same order of magnitude.

Even then the error that is minimized is an algebraic error which has nor real ``physical'' meaning. It is always better to minimize a geometrically meaningful criterion. The error measure that immediately comes to mind is the distance between the points and the epipolar lines. Assuming that the noise on every feature point is independent zero-mean Gaussian with the same sigma for all points, the minimization of the following criterion yields a maximum likelihood solution:

\begin{displaymath}
{\cal C}(F)= \sum \left( D({\tt m}',{\bf F} {\tt m})^2 +
D({\tt m},{\bf F}^\top {\tt m}')^2 \right)
\end{displaymath} (D9)

with $D({\tt m},{\tt l})$ the orthogonal distance between the point ${\tt m}$ and the line ${\tt l}$. This criterion can be minimized through a Levenberg-Marquard algorithm [122]. The results obtained through linear least-squares can be used for initialization.


next up previous contents
Next: Robust algorithm Up: Two view geometry computation Previous: Seven-point algorithm   Contents
Marc Pollefeys 2002-11-22