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-
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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.