ding lines in object space are necessary to solve 
the exterior orientation. The solution is based on 
fitting a bundle of planes to the corresponding 
straight lines in object space. 
Circular line: If the circular feature is given in 
object space, its 6 circle parameters are known. 
Consequently, 2 circular features are needed, 
including 4 redundant observations. The solution 
is based on fitting two cones to the corresponding 
circles in object space. 
In case of an incomplete description of the features in 
image or object space a number of further configur- 
ations for the solution of space resection is possible. 
For instance, if in case of circular features only the 
respective planes of the circles in object space are 
given, then 3 circular features are needed. Some 
examples for partial control information are given in 
Tab. 1, where for each of the 2D features the 
configurations are grouped according to the number 
of observations for the corresponding features in 3D 
object space. Furthermore, minimum configurations 
based on a combined use of points and lines can be 
derived by similar considerations. 
2.2 Relative orientation (2D-2D) 
Relative orientation involves the determination of the 
relative position and attitude of at least two images. 
The number of unknowns to be determined is 5 for 
the relative orientation of two images (generally 6 - i-7 
unknowns for i images) together with the parameters 
describing the corresponding features in model space. 
In the following generally 2 images and, if a solution 
is not possible, 3 images will be considered. 
Points in image space: 
- Point to point correspondence: 
A point is represented by 2 coordinates (observa- 
tions) per image and 3 model coordinates (un- 
knowns). 5 corresponding points in two images are 
needed in general to solve the relative orientation 
problem. The solution is based on the condition 
that the visual rays from the projection centres to 
the image points intersect in an identical point in 
3D (model) space. 
Lines in image space: 
- Line to line correspondence: 
A straight line is defined by 2 parameters (obser- 
vations) per image line and 4 parameters (un- 
knowns) in model space. Therefore, the relative 
orientation of two images cannot be solved with 
corresponding straight line features. The reason is 
that the corresponding projecting planes always 
intersect in a straight line (except for parallel or 
identical planes); however, no redundant informa- 
tion, which is needed for the determination of the 
orientation parameters, results from this intersec- 
115 
tion. Consequently, at least 3 images and 6 
corresponding straight line features in each of the 
images are necessary to simultaneously determine 
the relative orientation parameters of the images 
and the parameters describing the straight lines in 
3D (model) space, including one redundant 
observation. The solution is based on the condition 
that the corresponding three projecting planes 
intersect in an identical line in model space. 
A circular line feature is defined by 5 parameters 
(observations) per image ellipse and 6 parameters 
(unknowns) in model space. Consequently, 2 
corresponding circular features projected in the 2 
images are needed for a solution including 3 
redundant observations. The solution is based on 
the condition, that the projecting cones consisting 
of the projection centres and the ellipses in image 
space intersect in an identical circular feature. 
2.3 Absolute orientation (3D-3D) 
Absolute orientation, i.e. the transformation from a 
3D model coordinate system to a 3D reference system, 
is defined as a spatial similarity transformation by 7 
unknown parameters (3 translations, 3 rotations and 
a scale factor). 
- Point to point correspondence: 
3 corresponding points in the two systems are 
necessary to determine the 7 transformation 
parameters including 2 redundant observations. 
- Point to line correspondence: 
4 points in one system and the corresponding lines 
in the other system are needed to solve the 
transformation including 1 redundant observation. 
- Point to surface correspondence: 
7 points in one system and the corresponding 
surfaces in the other system are needed. 
- Line to line correspondence: 
For straight line features 2 straight lines in each of 
the two systems are needed to solve the trans- 
formation including 1 redundant observation. 
- Line to surface correspondence: 
For straight line features 4 straight lines and the 
corresponding planes are needed to solve the 
transformation including 1 redundant observation. 
- Surface to surface correspondence: 
For planar surface features 4 planes in each of the 
two systems are needed to solve the problem 
including 5 redundant observations. 4 planes are 
needed, because in case of 3 corresponding planes 
all configurations are degenerate.