In the following sections first briefly the different types of
measurements are described: tracking feature points, shape
from shading and tracking edges. The main subject of this
paper is the combination of the different types of measure-
ments and is presented in section 3.
2 TYPES OF MEASUREMENTS
2.1 Tracking feature points
Because a time sequence of many images is available, fea-
tures can be followed through the sequence of images. This
is far easier than the more common practice of trying to find
corresponding features in images recorded by camera’s with
large differences in viewing angles. The features in successive
images in the sequence will move and differ only slightly. If ac-
curate feature models and a model for the acquisition process
is available, Kalman filtering techniques can be used to obtain
accurate position estimations of the feature points. The ac-
curacy increases with the number of images that contains the
feature. And since each feature is present in many images,
the resulting accuracy will be much higher than would be ob-
tainable by just using stereo image pairs with similar image
resolution. An additional advantage is that it is rather simple
to take into account a complex camera model including lens
distortions. The image doesn’t have to be warped first, but
the estimated 3D feature points are projected back into the
image plane using the complex camera projection model.
2.2 Shape from shading
However, as stated earlier, such feature points may be sparse.
Hence an interpolation technique must be applied to obtain
the DEM for every point in the scene. Simple interpolation
techniques will give a very poor result, because of the few
available feature points.
Shading in the images gives a rough indication for the local
orientation of the surface. This can be used to improve the in-
terpolation process considerably. The estimated orientations,
however, are far less accurate than the measurements ob-
tained by feature tracking. These orientation measurements
are available also in areas without clear feature points, how-
ever.
2.3 Projection of edges
The edges of objects, e.g. fields, roads etc. also provide a
means to obtain height measurements. However, the shape
of the edges will change from image to image, and also their
orientation and positions. A 3-d model of surface patches
is used to project the edges back on the images and find
an optimal fit on the edges detected in the images. Since
edge features will differ just slightly in subsequent images,
the matching process is relatively simple. Once the 3-d mod-
els of these segment boundaries are known, also the height
and curvature of the surface locally on the boundaries are
known. The accuracies of these measurements will generally
be somewhere in between the accuracies of the other two
measurements.
2.4 Other measurements and foreknowledge
It is also possible to include other measurements, e.g. using
existing stereo techniques. Furthermore often a rough model
of the landscape is available, which can be used as an initial
estimate and often there is a good idea of the smoothness of
the surface.
3 DEM construction
The construction of a DEM means to find a model such that
it best fits the measurements and can be used to predict the
elevation for arbitrary positions:
£(p) = M(6,p) (1)
and:
6=" 4" SAP MOD) (2)
i
Where 2(p) is the estimate for the elevation at a position
p. and M is the digital elevation model function with pa-
rameters 0. (This only shows the fitting for feature point
measurements, but similar expressions can be obtained for
orientation measurements using SEL oM etc.)
Since land surfaces generally are rather complex, many pa-
rameters are required for an accurate description. A problem
with these many parameter models is that often there is not
a unique fit for the measured data.
Therefore, a different approach is taken here. Each measure-
ment is 'propagated' through the DEM, i.e. from it estima-
tions of elevation and surface orientations for all other posi-
tions are derived using the existing DEM and foreknowledge
about the local properties of the surface. Next the existing
DEM is updated using these new estimations.
3.1 Propagation of measurements
For simplicity first a 2 dimensional DEM (only x and z direc-
tions) is considered. Suppose it is known that the maximum
gradient of the surface is 27,4, (e.g. 1096, i.e. 10cm per me-
ter.) An elevation measurement at position zo now bounds
the possible elevations on all other positions by:
z(zo)— | = — zo | *Zmas € 2(z) € 2(zo)-- | z — to | *Zmas
(3)
The combination of different measurements in this case is
simply the intersection between the ranges. Each new mea-
surement further bounds the elevations. This is shown graph-
ically in fig.3.1.
A step further is not to assume a maximum surface gradient,
but a certain distribution for the gradient. Suppose that the
distribution of the surface gradient is normal with expectancy
0:
For a measurement at position zo with an accuracy that is
also given by a normal distribution G(zo, o3), the propagation
can then be described as follows:
Eíz(r)) 2zo--(r—z0)E {#(x)} =# (5)
and
o? (z(z)) = 0° (zo) -(z—20)^ 0? (z (z)) — iie et
820
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
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