Initial projection matrices

This method is adapted from the method proposed in [9]. In a metric frame, valid choices for the projection matrices of a camera with intrinsic parameters ${\bf K}$ in successive positions related by rotation matrix ${\bf R}$ and translation vector ${\bf t}$ are, see equation 3.7, ${\bf P}_1^M = {\bf K} [ {\bf I} \vert {\bf0}]$ and ${\bf P}_2^M = {\bf K} [ {\bf R}^\top \vert -{\bf R}^\top {\tt t}]$. To establish a quasi-metric frame, ${\bf P}_1$ and ${\bf P}_2$ are set ``close'' to the form of ${\bf P}_1^M$ and ${\bf P}_2^M$, using approximate values for the camera intrinsics ${\bf K}^*$ and the rotation ${\bf R}^*$.

The images should be normalized ${\tt m} \leftarrow {\bf K^*}^{-1} {\tt m}$. Equation (5.1) can be filled in. In this case ${\tt\pi}$ should be chosen so that ${\bf H} = [{\tt e}_{12}]_\times {\bf F}_{12} + {\tt e}_{12} {\tt\pi}^\top \approx {\bf R}^*$. This is achieved by orthogonal projection of ${\bf R}$ onto the subspace of possible ${\bf H}$.

Since two consecutive views are in general not too far apart, often ${\bf R^*=I}$ is a good approximation. For most of the intrinsics a good approximation is known beforehand (i.e. principal point in the center of the image, aspect ratio one and no skew). This is not always the case for the focal length. If no a priori guess for the focal length is available different are tried out and the one with most points reconstructed in front of the camera is selected. If more values have a similar amount of valid points, the smallest value is taken. Although the frame obtained as such can still be far from the real metric frame, it is in general sufficient to allow for the approach to succeed. In fact this approach is based on the concept of cheirality of Harley [62] or the oriented projective geometry introduced in computer vision by Laveau [96]. How to recover the quantitative metric properties of space will be explained in the next chapter.