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