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Degenerate cases

Figure 1: example of critical motion : vertical rotation axis

Horizontal rotation axis. We have and so, and (see eq:Sx) : eq:sys3 gives and and can then be computed with eq:sys5 and eq:sys6.

if we impose r=0, eq:sys2 gives . However, eq:sys1 can't give either or . if is considered, and can be calculated directly and solved to give r, and .

Vertical rotation axis. This is the case when , which is similar to the previous one. We have .

if r=0, we have and then all parameters except k and can be evaluated. if , leads to a total resolution of calibration

Rotation axis orthogonal to the image plane. We have ( and ).

It is the worst critical case in so far as , , k and r can be calculated, but remains always undetermined, whatever the constraint may be.

As a conclusion, we saw that the problem of affine-to-Euclidean calibration could be easily solved in particular cases (single motion, all parallel axes rotationsgif).

We also showed that using the constraint allowed us to avoid critical cases : there remains then just one real critical motion (rotation axes orthogonal to the image plane).

Bob Fisher
Mon Dec 7 13:48:06 GMT 1998