Initial frame

Two images of the sequence are used to determine a reference frame. The world frame is aligned with the first camera. The second camera is chosen so that the epipolar geometry corresponds to the retrieved ${\bf F}_{12}$:

$$\begin{array}{rcrcccl} {\bf P}_1 &=& [ & {\bf I}_{3 \times 3... ...}_{12} {\tt a}^\top &\vert& \sigma {\tt e}_{12} & ] \end{array}$$ (E3)

Equation 5.3 is not completely determined by the epipolar geometry (i.e. ${\bf F}_{12}$ and ${\tt e}_{12}$), but has 4 more degrees of freedom (i.e. ${\tt a} \mbox{ and } \sigma$). ${\tt a}$ determines the position of the reference plane (i.e. the plane at infinity in an affine or metric frame) and $\sigma$ determines the global scale of the reconstruction. The parameter $\sigma$ can simply be put to one or alternatively the baseline between the two initial views can be scaled to one. In [7] it was proposed to set the coefficient of ${\tt a}$ to ensure a quasi-Euclidean frame, to avoid too large projective distortions. This was needed because not all parts of the algorithms where strictly projective. For the structure and motion approach proposed in this paper ${\tt a}$ can be arbitrarily set, e.g. ${\tt a}=[0\,0\,0]^\top$.