next up previous contents
Next: Geometric interpretation of constraints Up: Self-calibration Previous: Self-calibration   Contents


A counting argument

To restrict the projective ambiguity (15 degrees of freedom) to a metric one (3 degrees of freedom for rotation, 3 for translation and 1 for scale), at least 8 constraints are needed. This thus determines the minimum length of a sequence from which self-calibration can be obtained, depending on the type of constraints which are available for each view. Knowing an intrinsic camera parameter for $n$ views gives $n$ constraints, fixing one yields only $n-1$ constraints.

\begin{displaymath}n \times (\char93 known) + (n-1) \times (\char93 fixed) \ge 8 \end{displaymath}

Of course this counting argument is only valid if all the constraints are independent. In this context critical motion sequences are of special importance (see Section 6.2.5).

Therefore the absence of skew (1 constraint per view) should in general be enough to allow self-calibration on a sequence of 8 or more images (this was shown in [114,60,99]). If in addition the aspect ratio is known (e.g. $f_x=f_y$) then 4 views should be sufficient. When the principal point is known as well a pair of images is sufficient.


next up previous contents
Next: Geometric interpretation of constraints Up: Self-calibration Previous: Self-calibration   Contents
Marc Pollefeys 2002-11-22