Self-calibration methods

In this section some self-calibration approaches are briefly discussed. Combining Equation (6.7) and (6.8) one obtains the following equation:

$${\bf K}_i{\bf K}_i^\top \sim {\bf P}_i \Omega^* {\bf P}_i^\top$$ (F11)

Several methods are based on this equation. For constant intrinsic parameters Triggs [199] proposed to minimize the deviation from Equation (6.11). A similar approach was proposed by Heyden and Åström [70]. Pollefeys and Van Gool [148] proposed a related approach based on the transfer equation (i.e. Equation (6.9)) rather than the projection equation. These different approaches are very similar as was shown in [148]. The more flexible self-calibration method which allows varying intrinsic camera parameters [135] is also based on Equation (6.11).

The first self-calibration method was proposed by Faugeras et al. [45] based on the Kruppa equations (Equation (6.10)). The approach was improved over the years [104,222]. An interesting feature of this self-calibration technique is that no consistent projective reconstruction must be available, only pairwise epipolar calibration. This can be very useful is some cases where it is hard to relate all the images into a single projective frame. The price paid for this advantage is that 3 of the 5 absolute conic transfer equations are used to eliminate the dependence on the position of the plane at infinity. This explains why this method performs poorly compared to others when a consistent projective reconstruction can be obtained (see [134]).

When the homography of the plane at infinity ${{\bf H}^\infty_{ij}}$ is known, then Equation (6.9) can be reduced to a set of linear equations in the coefficients of $\omega _i$ or $\omega^*_i$ (this was proposed by Hartley [62]). Several self-calibration approaches rely on this possibility. Some methods follow a stratified approach and obtain the homographies of the plane at infinity by first reaching an affine calibration, based an a pure translation (see Moons et al. [117]) or using the modulus constraint (see Pollefeys et al. [134]). Other methods are based on pure rotations (see Hartley [64] for constant intrinsic parameters and de Agapito et al. [29] for a zooming camera).