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