International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
to obtain an improved DTM which benefits from the existing 
ones. Typically, two DTMs, which represent a certain area on 
different scales or resolutions, are considered. For example, one 
DTM might be densely derived by image matching techniques 
from aerial or space images and a second one might be 
measured with GPS or with analytical photogrammetric 
instruments. The latter one is expected to be a sparse but more 
accurately measured DTM, which represents the terrain shape 
on a coarse scale. Other examples might be the integration of 
existing DTMs with new measured DTMs from radar or laser 
data. 
The next section briefly sketches the idea of multiscale wavelet 
processing. A concept for the integration of DTMs, which may 
differ in scale and accuracy, is presented in section 3. First 
experiments are presented to demonstrate the applicability of 
the approach (section 4) and give some insight into the accuracy 
behaviour of the integrated DTM over scale. 
2. WAVELET TRANSFORMATION AND 
MULTISCALE PROCESSING 
Wavelet theory is explained in detail for example in the 
textbook of Louis et al. (1997). In the following, we briefly 
outline the idea of multiscale analysis using wavelets for one 
dimensional signals. 
Splitting a signal into parts of varying detail is the key to the 
fast computation of the discrete wavelet transform (Mallat, 
1989). The decomposition of signal/= s+d, with s representing 
the smooth and d the detail part, can be carried out by discrete 
low-pass and high-pass filtering. By identifying s with the lower 
resolution signal / on scale level i and f A with the signal on the 
finer scale level i-1, the lower resolution signal will be obtained 
from f_x by low-pass filtering with a low-pass of response 
function /. The detail signal d is computed by high-pass filtering 
with impulse response h. I corresponds to the so-called scaling 
function, h to the wavelet function of the discrete wavelet 
transform. The convolutions can be elegantly formulated using 
multiplication with block-circulant matrices. L and H are the 
block-circulant matrices corresponding to the filter kernels l and 
h. The convolutions then read as follows: 
d i =Hf i _ l 
This process is referred to as the decomposition of the signal / 
and L and H are the decomposition operators. Recursive 
application of the decomposition formulas leads to further 
splitting of the smooth part over a selected number of M scale 
levels. By reversing this process the synthesis equation 
fi-l ~ L fi + H*d t 
of the wavelet transform is obtained. This reverse process is 
referred to as the reconstruction of the signal in which the finer 
representation is calculated from coarser levels by adding the 
details according to the synthesis equation. Decomposition 
includes a subsampling by a factor of two and reconstruction 
the corresponding oversampling. 
Based on this short description of the wavelet transform, we are 
now prepared to present our concept for DTM integration based 
on wavelets. Related work on multiresolution approximation in 
the area of physical geodesy is discussed in Li (1996). For a 
deeper understanding of the wavelet theory we refer to the 
textbook of Louis et al. (1997). 
3. A CONCEPT FOR DTM INTEGRATION BASED ON 
WAVELETS 
For the explanation of the concept, we restrict ourselves to two 
given DTMs. The generalisation to multiple DTMs will become 
obvious at the end of the discussion. Assume that one DTM 
with high resolution and a second one with low resolution are 
given. The first one is considered a fine scale representation of 
the surface of some terrain and the second one a coarse scale 
representation of the same terrain surface. We further assume 
that accuracy measures for both DTMs are given and 
represented by the corresponding covariance matrices. For 
simplicity, we presume that the grid width of both DTMs is 
related by a factor 2 M . 
3,1 Multiscale Representation of a DTM 
The first step of the integration process is to represent the high 
resolution DTM in a series of scales. The sequence of low-pass 
filtered and subsampled data results in a DTM pyramid 
generated by the wavelet transform. An example is plotted in 
Figure 1. Basically, it shows a familiar picture of a DTM 
pyramid, which is often generated in photogrammetry by image 
matching and finite element modelling using image pyramids 
(Ackermann and Hahn, 1991). But this, of course, does not 
imply any close relation between the generation processes of the 
wavelet representation and the finite element modelling. 
The formulas for the wavelet decomposition and reconstruction 
given in section 2 have to be adapted for the formulation of the 
DTM integration process. 
The wavelet decomposition for two-dimensional data can be 
conveniently formulated by using matrix notation based on the 
Kronecker product <£>. The wavelet DTM representation on level 
j calculated from the finer level DTM on level j-1 reads as 
follows (x is used to address the DTM, dx for the detail signal of 
the DTM) 
x“=(L®L)x“ 1 
di.“ = (L ® H)i“, = (H ® 
In addition to the decomposition of a DTM, its corresponding 
covariance matrix C must be also decomposed. For this second 
moment information, the law of error propagation applies 
giving