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Fusion of sensor data, knowledge sources and algorithms for extraction and classification of topographic objects
Baltsavias, Emmanuel P.

International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999
3.2 Integration of DTMs
The integration concept consists of three steps:
Step 1: Calculate a wavelet decomposition with M levels for the
high resolution DTM and its associated covariance matrix. The
number of levels is chosen in such a way that the coarsest scale
level fits to the resolution of the second DTM.
Step 2: The low-pass filtered coarse scale representation of the
first DTM and the second DTM are merged together using
weighted least squares averaging. The weights result from the
inverted covariance matrices of both DTMs. This means that for
the low-pass filtered DTMl the corresponding low-pass filtered
covariance matrix Cl is used (Eqs. 1 and 2) and for DTM2 the
given second DTM with its covariance matrix C2.
Weighted least squares averaging is of course only one
possibility among others. Other widely applied techniques for
DTM interpolation like least squares prediction or Wiener
filtering may be used, if they promise better results.
Step 3: The merging result of both DTMs together with the
calculated covariance matrix can be considered as the
representation of the integrated DTM on scale level M. By
using the reconstruction formulas (Eqs. 3 and 4) the estimates
for the integrated DTM together with its accuracy can be
calculated on all scales and in particular on the finest scale.
Generalisation from two to several DTMs can be simply carried
out by repeating the three steps. As soon as two DTMs are
integrated the resulting height and accuracy data are taken as a
new DTM which can be integrated with a third one. A fourth
and fifth DTM and so on may follow. From the viewpoint of
computational efficiency it might be reasonable to integrate
coarser DTMs first and move on by integrating the higher
resolution DTMs. But more important for the computational
efficiency of the overall processing is the implementation of the
wavelet decomposition and reconstruction formulas. The
elaboration with the block-circulant matrices according to Eqs.
1-4 is very convenient to understand the structure of the
transform but for efficiency reasons a recursive implementation,
essentially based on discrete convolutions, is required. For more
details about the fast wavelet transform refer to Mallat (1989).
In the next section, we visualise results of the wavelet
decomposition and reconstruction, in particular those relating to
the first and third step of the integration process and discuss the
achievements by focussing on the accuracy.
4.1 Illustration of Integration Results
In the experimental investigation we first want to visualise the
overall DTM integration process, in particular for the terrain
surface, over a number of scale levels. For that purpose, we use
two DTM samples. The first one represents a rather smooth
area, the other one a region with undulating terrain. The densely
sampled rather noisy DTMs for both areas can be seen in
Figures 2 and 4 together with another three lower scale levels of
the DTM pyramid. On the coarse level, merging with a second
DTM is carried out by least squares averaging as outlined in
section 3. The second DTM given on the lower scale is not
shown, as it looks fairly similar to the integration result on that
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400 -
200 >.
Fig. 2. Experiment with smooth terrain. The noisy high resolution DTM and three lower scale levels are shown.