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Critical motion sequences

One noticed very soon that not all motion sequences are suited for self-calibration. Some obvious cases are the restricted motions described in the previous section (i.e. pure translation, pure rotation and planar motion). However there are more motion sequences which do not lead to unique solutions for the self-calibration problem. This means that at least two reconstructions are possible which satisfy all constraints on the camera parameters for all the images of the sequence and which are not related by a similarity transformation.

Several researchers realized this problem and mentioned some specific cases or did a partial analysis of the problem [157,176,121]. Sturm [144,145] provided a complete catalogue of critical motion sequences (CMS) for constant intrinsic parameters. Additionally, he identified specific degeneracies for some algorithms [142].

However it is very important to notice that the classes of CMS that exist depend on the constraints that are enforced during self-calibration. The extremes being all parameters known, in which case almost no degeneracies exist, and, no constraints at all, in which case all motion sequences are critical.

In table 6.1 and 6.2 the most important critical motion sequences for self-calibration using the constraint of constant -but unknown- intrinsics respectively intrinsics known up to a freely moving focal length are listed. More details can be found in [99]. For self-calibration to be successful it is important that the global motion over the sequence is general enough so that it is not contained in any of the critical motion sequence classes.

Table 6.1: Critical motion sequences for constant intrinsic parameters
critical motion type ambiguity  
pure translation affine transformation (5DOF)  
pure rotationF3 arbitrary position for plane at infinity (3DOF)  
orbital motion projective distortion along rotation axis (2DOF)  
planar motion scaling axis perpendicular to plane (1DOF)  



Table 6.2: Critical motion sequences for varying focal length
critical motion type ambiguity  
pure rotationF4 arbitrary position for plane at infinity (3DOF)  
forward motion projective distortion along optical axis (2DOF)  
translation and scaling optical axis (1DOF)  
rotation about optical axis    
hyperbolic and/or elliptic motion one extra solution  



next up previous contents
Next: A practical approach to Up: Self-calibration Previous: Self-calibration methods   Contents
Marc Pollefeys 2002-11-22