it is
and
rain
off-
it to
10se
^ an
d in
hing
nage
nate
se in
ition
"lion
and
2). It
plied
ith a
nage
that,
cific
hing.
es of
NG
PHO
erial
-ate a
d is
obust
The
dings
vhich
egree
eters,
ation
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
e The grid width of the resulting elevation grid is a fundamental
parameter that is in general selected according to the
application-specific requirements for the DTM, within certain
limits given by the scale of the aerial images used for
matching. Selecting a larger grid size yields a smoothing
effect that helps to eliminate off-terrain points, but also
smoothes terrain structures that one might want to preserve.
The terrain type (flat, undulating, mountainous) has to be
selected in accordance with the actual terrain type to make
matching successful.
The degree of smoothing (high, medium, low) is the
parameter that is best suited for controlling whether to obtain
a DTM or a DSM. In this work, we selected the degree of
smoothing to be “high”, to obtain an initial elevation grid as
close as possible to the terrain.
The density of the original point cloud (dense, medium,
sparse) also influences the degree of smoothing: a sparse
point cloud results in a model closer to the terrain than a
dense point cloud.
MATCH-T can consider geomorphologic elements and
additional points in the interpolation process, typically
measured interactively by a human operator. The standard
deviation of surface points and break lines has an influence
on the weights of these additional observations in the
interpolation process.
MATCH-T delivers DSMs of good quality. If a DTM is
required, the algorithms for smoothing work well if the grid
width is not too small compared to the extents of groups of off-
terrain points in the original point cloud. For instance, groups of
trees and single buildings can be eliminated if the grid width is
in the range of about 5-10 m. However, if the grid width is
chosen smaller, e.g. 1.5 m, these objects remain in the matching
results, even if a high degree of smoothing is selected. Figure 1
shows a DSM generated by image matching with a resolution of
1.25 m. The remaining buildings are clearly visible.
Fionre 1 Shaded view of an elevation grid acquired by image
matching (Eggenburg east; cf. section 5).
As in general the grid width has to be chosen in dependence of
the proposed application of a DTM, there is only a small band
width for adapting this parameter. That is why we propose to
improve the image matching results by hierarchical robust linear
prediction in a post-processing step. Our good experience with
that technique gives us reason to believe that it should be
possible to eliminate buildings and groups of trees in high-
density DSMs delivered by image matching techniques.
3. HIERARCHICAL ROBUST LINEAR PREDICTION
We use the program SCOP--* (Briese et al., 2002) for the
interpolation of a hybrid raster DTM on the basis of irregular
415
point and vector data by linear prediction. This method is based
on the assumption that the heights of terrain points, after
removing a trend, are correlated, the correlation being a
function of the horizontal distance between the points (Kraus,
2000). Linear prediction will be fragile if gross errors occur, so
that a more robust approach has to be found. In this section, we
want to describe how this can be accomplished.
3.1 Robust Interpolation
Robust interpolation (Kraus and Pfeifer, 1998) was developed
for DTM generation from ALS-data in wooded areas. In this
process the elimination of gross errors and the interpolation of
the terrain are carried out simultaneously. This process consists
of three steps:
I. Interpolation of a surface model by linear prediction
considering individual weights for each point. At the
beginning all weights are assumed to be equal.
2. Calculation of the filter values, i.e., the vertical distances
from the interpolated surface to the measured points
Recomputation of the weights of the individual points in
dependence of the filter values, using a weight function
adapted to the stochastic properties of the filter values of the
off-terrain points.
Uu
The steps are repeated in an iterative process until all gross
errors are eliminated. The elimination of gross errors (off-
terrain points) is controlled by the weight function. This weight
function is controlled by 3 parameters (figure 2): Halfweight h
(the size of a filter value obtaining a weight of 0.5), slant s (co-
tangent of the slope at /=h), and the cut-off point 1.
| A p=p(f)
Eus h=0. 3m agi
Ï
Y =
E. Om mode, /
to foe
Figure 2. Weight function (Briese et al., 2002).
The values for 4, s, and t can be set independently for the
positive and the negative branches of the weight function, i.e.
for points above and below the surface interpolated in the
previous iteration. As a consequence, the weight function can
be asymmetric. This allows to favour points on or below the
intermediate surface (considered to be terrain points) and to
decrease the weights of points above the intermediate surface
that are more likely to be off-terrain points. The function is also
shifted by a value g. This also should compensate for the fact
that the intermediate surface is more likely to be above the
terrain than below it. By choosing the weight function to be
asymmetric and excentric, we model the actual distribution of
the errors of the off-terrain points with respect to the terrain.
Figure 2 shows a weight function for the elimination of off-
terrain points; note that in this case. points having a filter value
f « g are not affected by robust estimation (Briese et al., 2002).
3.2 Hierarchical Robust Interpolation
Robust interpolation relies on a ‘good mixture’ of terrain and
off-terrain points, but the algorithm is not able to eliminate
clusters of off-terrain points as they occur, e.g., in densely
developed urban areas. To meet this problem, robust
