Scene knowledge

The knowledge of (relative) distances or angles in the scene can be used to obtain information about the metric structure. One of the easiest means to calibrate the scene at a metric level is the knowledge of the relative position of 5 or more points in general position. Assume the points ${\tt M}'_l$ are the metric coordinates of the projectively reconstructed points ${\tt M}_l$, then the transformation ${\bf T}$ which upgrades the reconstruction from projective to metric can be obtained from the following equations

$${\tt M}'_l \sim {\bf T} {\tt M}_l \mbox{ or } \lambda_l {\tt M}'_l = {\bf T} {\tt M}_l$$ (F1)

which can be rewritten as linear equations by eliminating $\lambda_l$. Boufama et al. [14] investigated how some Euclidean constraints could be imposed on an uncalibrated reconstruction. The constraints they dealt with are known 3D points, points on a ground plane, vertical alignment and known distances between points. Bondyfalat and Bougnoux [13] recently proposed a method in which the constraints are first processed by a geometric reasoning system so that a minimal representation of the scene is obtained. These constraints can be incidence, parallelism and orthogonality. This minimal representation is then fed to a constrained bundle adjustment.

The traditional approach taken by photogrammetrists [20,56,179,57] consists of immediately imposing the position of known control points during reconstruction. These methods use bundle adjustment [21] which is a global minimization of the reprojection error. This can be expressed through the following criterion:

$${\cal C}_{bundle} = \sum^n_{i=1} \sum_{l \in I_i} \left( (x_... ..._i({\tt M}_l))^2 + (y_{li} - {\bf P}_i({\tt M}_l))^2 \right)$$ (F2)

where $I_i$ is the set of indices corresponding to the points seen in view $i$ and ${\bf P}_i({\tt M}_l)$ describes the projection of a point ${\tt M}_l$ with camera ${\bf P}_i$ taking all distortions into account. Note that ${\tt M}_l$ is known for control points and unknown for other points. It is clear that this approach results in a huge minimization problem and that, even if the special structure of the Jacobian is taken into account (in a similar way as was explained in Section 5.4.2, it is computationally very expensive.