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. EXPERIMENTS 
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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Fig. 2. Experiment with smooth terrain. The noisy high resolution DTM and three lower scale levels are shown.