Figure 4: Basic concept of free-form curves in bundle
block adjustment
Note that the shape of S is completely described by a
set of node points K. Parameter t describes the position
of the foot (S(K,t) ) of point P on S.
So, the unknowns of our least squares adjustment
problem are the co-ordinates of the node points (K), the
positions of the points along the curve (1) and the object
co-ordinates of the curve points (P), and the orientation
parameters (the rotational parameters and the
projection centres) of the images. If the photographs
have been taken with a calibrated camera, the
elements of the inner orientation are well known.
Considering parameters t unknown in the adjustment
process enables optimisation of the positions of the
nodes both laterally and along the curve (Forkert, 1993,
pp. 221-228). If the distance between to consecutive
nodes falls below a user specified threshold, one of the
nodes will be deactivated automatically.
The problem of “relative orientation“ can be solved if
object curves exist which are visible in three or more
images. For the following considerations let's
concentrate on one object curve, only. If the images
are not oriented, the bundles of rays running through
the image curves do not intersect uniquely. So, the
bundles have to be shifted and rotated until they
intersect correctly in one spatial curve. At the same
time, the shape of this tie curve will be determined.
Consequently, the relative orientation of three or more
images can be found by using an arrangement of tie
curves. Note, that in general it is not possible to find
homologous points in different images of the curve. So,
our method is based on homologous curves instead of
on homologous points.
Questions for the minimum configuration of tie curves
can be answered by replacing the curves by their
significant tangents. A U-formed curve, for instance,
consists of two significant tangents. Every tangent is
described by four parameters in object space and by
two parameters in image space, respectively. So, for
example, the eleven parameters describing the relative
orientation of three images can theoretically be
determined by at least three U-formed curves.
198
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
4 IMAGE ORIENTATION
4.1 Determination of approximate values
In this test project, the local reference co-ordinate
system was chosen arbitrarily by fixing seven
orientational elements. The scale of this local co-
ordinate system was estimated corresponding to the
real object dimensions in order to get realistic r.m.s.
errors in object space.
An appropriate configuration of provisional tie curves,
extracted from the images as described above, allows
to determine initial values for the image orientation. The
real image orientation will be carried out later on (see
section 4.2), using tie curves cutted out from the
provisional curves.
In order to get provisional values for the image
orientation parameters, the first step of calculation is
carried out conventionally with points: the end points of
the image curves are used as tie points. End points of
curves imaged only partially might cause significantly
large residuals and have to be eliminated by robust
estimation. The results of the adjustment can be seen
in table 1.
No. images No. tie points r.m.s. error in
image
12 84 5mm
Table 1
Thereafter, the initial node positions of a provisional tie
curve can be obtained through an approximation
algorithm: Approximate curve points in object space are
found by intersecting their image rays with the cone
surface formed by the rays of another image.
Afterwards, the algorithm distributes the nodes in
regular intervals along the polygon of curve points. In
this test project, the initial number of nodes was chosen
automatically dependent on the number of curve points
(one node per fifty curve points). In order to improve
the initial node arrangement, the approximation process
is completed by a curve adjustment with the
approximated object points assumed to be constant.
For the time being image orientation is quite inaccurate.
Nevertheless 29 out of 40 provisional tie curves could
be approximated as shown in figure 5.
Experience shows, that it is advisable to refine the tie
curve's shape before starting image orientation. The
unknowns of this "curve reconstruction" task are: the
co-ordinates of the node points, the positions of the
points along the curve and the object co-ordinates of
the curve points. The actual values of the orientational
parameters are assumed to be constant. The
automated iterative process of curve reconstruction
consists of two main steps alternately applied:
1) adjustment with a given number of nodes
2) insertio
containing
So, the fl
by step b
additional
residuals,
obviously
tie curve «
Fig
Up to nc
order to
the pro
paramete
adjustme
the obje
points ar
The posi
considere
computa:
iterations
