Self-calibration

The reconstruction obtained as described in the previous chapters is only determined up to an arbitrary projective transformation. This might be sufficient for some robotics or inspection applications, but certainly not for visualization. Therefore we need a method to upgrade the reconstruction to a metric one (i.e. determined up to an arbitrary Euclidean transformation and a scale factor).

In general three types of constraints can be applied to achieve this: scene constraints, camera motion constraints and constraints on the camera intrinsics. All of these have been tried separately or in conjunction. In the case of a hand-held camera and an unknown scene only the last type of constraints can be used.
Reducing the ambiguity on the reconstruction by imposing restrictions on the intrinsic camera parameters is termed *self-calibration* (in the area of computer vision). In recent years many researchers have been working on this subject. Mostly self-calibration algorithms are concerned with unknown but constant intrinsic camera parameters (see for example Faugeras et al. [32], Hartley [48], Pollefeys and Van Gool [116,118,104], Heyden and Åström [57] and Triggs [157]). Recently, the problem of self-calibration in the case of varying intrinsic camera parameters was also studied (see Pollefeys et al. [115,105,99] and Heyden and Åström [58,60]).

Many researchers proposed specific self-calibration algorithms for restricted motions (i.e. combining camera motion constraints and camera intrinsics constraints). In several cases it turns out that simpler algorithms can be obtained. However, the price to pay is that the ambiguity can often not be restricted to metric. Some interesting approaches were proposed by Moons et al. [89] for pure translation, Hartley [50] for pure rotations and by Armstrong et al. [2] (see also [28]) for planar motion.

Recently some methods were proposed to combine self-calibration with scene constraints. A specific combination was proposed in [117] to resolve a case with minimal information. Bondyfalat and Bougnoux [8] proposed a method of elimination to impose the scene constraints. Liebowitz and Zisserman [79] on the other hand formulate both the scene constraints and the self-calibration constraints as constraints on the absolute conic so that a combined approach is achieved.

Another important aspect of the self-calibration problem is the problem of critical motion sequences. In some cases the motion of the camera is not general enough to allow for self-calibration and an ambiguity remains on the reconstruction. A first complete analysis for constant camera parameters was given by Sturm [144]. Others have also worked on the subject (e.g. Pollefeys [99], Ma et al. [85] and Kahl [62]).

- Calibration

- Self-calibration
- A counting argument
- Geometric interpretation of constraints
- The image of the absolute conic
- Self-calibration methods
- Critical motion sequences

- A practical approach to self-calibration

- Conclusion