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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
The number of ambiguities using geometrically conditions alone,
was examined by Maas in (Maas, 1992) for different numbers of
images. The complexity of the matching strategy arise with the
number of images, but the high amount of ambiguities to be ex-
pected requires more images to be reduced. Therefore we use
n > 2 images and the constrain for an object point, that its im-
age points are seen in at least three images, to eliminate wrong
hypotheses.
The presented algorithm uses all images simultaneously, therefor
the test of hypotheses is realised in the object space. The geomet-
ric condition is that all projection rays of corresponding image
points intersect in one object point. Therefor we first find match-
ing hypotheses by using the epipolar constraints defining one im-
age as the starting image. The epipolarlines between the starting
image and all other images are calculated and every image point
close to the epipolarline is a hypotheses for a corresponding point
to the point in the starting image. The epipolarline can be shorten
by considering the height extension in object space. To get also
the points, which are not seen in the first image, but maybe in at
least three other images, this step should be calculated also for
other images as starting images. The number of starting images
depends on the constellation of the image system. Then we deter-
mine the object points belonging to these two point hypotheses.
The result is a 3D point cloud, where a group of at least m close
points define one object point. The number of points m depends
on the number of starting images. To test the hypotheses of corre-
spondences, a clustering of the point cloud is calculated using the
k-means algorithm. The resulting clusters containing at least m
points belong to one object point. The mean value of the points in
one cluster is a first approximation of the 3D point determination.
If a higher quality is required, all points can be finally determined
with by estimating a bundle adjustment. Therefor we use the im-
age point correspondences resulting from the points belonging to
one cluster. We summarize the algorithm into the following steps:
The main steps of the algorithm for 3D prediction of points
are:
1. Extraction of points x) in all n images, where j is the
number of the image and i the number of the point.
2. define one image as the starting image a.
3. for all points x? in image a determine hypotheses of
point correspondences using epipolar lines in all other
images.
4. define another image as second starting image b.
5. for all points x? in image 5 determine hypotheses of
point correspondences using epipolar lines in all other
images.
6. if necessary repeat point 4 and 5 for as much different
images as it is convenient.
7. clustering of the 3D point cloud P resulting from point
2 to 6 — approximated 3D object points X;,7 = 1..m.
8. final bundle adjustment of all matched points, using
X;,à = 1..m as approximated values — final object
points X;.
If the imaging system and the resulting images are not projective,
then there exist two different possibilities:
l. The specialized strict physical model of the mapping pro-
cess will be implemented in the algorithm, which is some-
times not possible, or the strict physical model can be very
complex and the computational time can rise in dependency
on the algorithm.
609
2. An approximation for the non projective mapping is used
for the matching process. For the a priori quality control of
the percentage reduction of computation complexity for the
replacement of the multi media geometry by a normalized
projective model see (Wolff and Fórstner, 2001).
4.2 Application for Non Projective Views
The application of the approximation by a virtual projective cam-
era presented in section 3 contains the following steps:
Implementation of the virtual camera for an effective 3D point
determination using non projective mappings
1. determine virtual projective mappings ^P for the obser-
vation space
2. matching the image points using ^P
3. final bundeladjustment using the strict model
S EXAMPLE AND QUALITY CONTROLS
5.1 Data: a surface of a fluvial sediment
Our work on using multi media geometry is motivated by in-
vestigations on the generation of fluvial sediments (Wolff and
Fórstner, 2000). The aim is to derive a physical model of the
underlying process of the dynamical sediment transport. The sur-
face of the water is smoothed by a perspex pane. We get the
standard case of multi media geometry: air, perspex and water
with plane interfaces. The observed sediment surface is shown
together with the extracted points of one image in Fig. 2 (for the
extraction of interest points see (Fórstner, 1994). The surface of
the sediment was formed by a jet of water hitting the sediment.
We used four Sony XC-77 CE cameras (748 x 564 pixel) for the
acquisition of the images.
Figure 2: Image of the sediment surface with extracted points.
5.2 Determine the reference data using the strict model
To get reference data for the quality analysis of the matched im-
age points and determined object points we carried out the pre-
sented algorithm using the strict multi media model. We use the
same software and values for its parameters to calculate reference
data and the approximated data.
5.3 Quality analysis
For the quality analysis of the 3D determination of points using
the approximation, we want to examine the following points:
