Initializing structure

Once two projection matrices have been fully determined the matches can be reconstructed through triangulation. Due to noise the lines of sight will not intersect perfectly. In the uncalibrated case the minimizations should be carried out in the images and not in projective 3D space. Therefore, the distance between the reprojected 3D point and the image points should be minimized:

$$D({\tt m}_1,{\bf P}_1 {\tt M})^2 + D({\tt m}_2,{\bf P}_2 {\tt M})^2$$ (E4)

It was noted by Hartley and Sturm [53] that the only important choice is to select in which epipolar plane the point is reconstructed. Once this choice is made it is trivial to select the optimal point from the plane. A bundle of epipolar planes has only one parameter. In this case the dimension of the problem is reduced from 3-dimensions to 1-dimension. Minimizing the following equation is thus equivalent to minimizing equation (5.4).

$$D({\tt m}_1,{\tt l}_1(\alpha))^2+D({\tt m}_2,{\tt l}_2(\alpha))^2$$ (E5)

with ${\tt l}_1(\alpha)$ and ${\tt l}_2(\alpha)$ the epipolar lines obtained in function of the parameter $\alpha$ describing the bundle of epipolar planes. It turns out (see [53]) that this equation is a polynomial of degree 6 in $\alpha$. The global minimum of equation (5.5) can thus easily be computed. In both images the point on the epipolar line ${\tt l}_1(\alpha)$ and ${\tt l}_2(\alpha)$ closest to the points ${\tt m}_1$ resp. ${\tt m}_2$ is selected. Since these points are in epipolar correspondence their lines of sight meet in a 3D point.