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Figure 4. Relative orientation using non homologous points
on straight line features.
The unknown parameters may be determined from above
equations by standard least squares adjustment, if sufficient
measurements are available. How many measurements
are needed to make the problem uniquely determined? Let
us assume we measure N points in both images for each of
the Mlines. This gives:
(1) Number of observation equations -
4N.M.
(2) Number of unknowns -
5 orientation unknowns:
(3 + 2) M unknowns per linear feature;
M(N-1) unknown distances S for points measured in
left image;
M N unknown distances S for points measured in right
image.
(3 Inequality condition -
(4 N M) equations = 5 + M (5 + 2N - 1) unknowns; or
M — >
2N-4
(8)
As two points measurements per line should suffice (N=2)
the required number of lines M follows to be infinite. Thus,
the relative orientation problem appears to be non-solvable
as stated above.
In a variation to the above formulation, we may assume,
that pairs of parallel lines are measured. We then ask, how
many of such pairs of parallel lines are required for relative
orientation. This variation of our argument follows from the
requirements of many constructions in projective geometry
to know the vanishing points of bundles of parallel lines
(Wylie, 1970). Also, in most industrial tools, parallel edges
are readily available and recognisable as such.
We now assume measurements of N points made in both
images for each of M pairs of parallel lines.
This gives:
(1) Number of observation equations -
8 NM.
(2) Number of unknowns -
5 orientation unknowns;
(6 + 2)M unknowns per pair of parallel lines;
2M(N-1) unknown distances S for points measured in
left image;
2M N unknown distances S for points measured in
right image.
689
(3) Inequality condition -
(8 zN M) equations = 5 + M(8 + 4N - 2) unknowns; or
5
My OS
E AN 6
(9)
For N equal to 2, it results M = 2,5; that means we need
three or more pairs of parallel lines in object space to
complete relative orientation.
Again, the above can be verified by a geometric argument.
Envisage a cube imaged from two stations.
Figure 5. Exemplification of Relative Orientation Process
The attitudes of the left image can be determined from the
shape of the triangle formed by the vanishing points, while
its relative position can be determined by the position of the
focal point with respect to the vanishing points (H. Wylie,
1970). It may however, be difficult to find three pairs of
parallel edges in an object. This condition can probably be
relaxed and further research is required on suitable minimal
measuring arrangements for relative orientation.