Test Results using Simulated Data
For theoretical analysis, different types of simulated signals f such as city profile functions, city surface functions, step functions and so on, are used. In other word, each f is known in all tests. Therefore, one can determine the true error of A_jf and analyze its convergence characteristics. The conclusions will be given briefly as follows.

The algorithm is most effective and efficient, if a from-coarse-to-fine strategy is used to determine the locations of significant wavelet coefficients. The accuracy of such an approximation becomes better if a finer resolution (i.e. a larger j-value) is adopted, where j is a resolution index. The influence radius of the Gibbs phenomenon becomes smaller in a rate of 2^-j as the j-value becomes larger. The ratio of the maximal approximation error near a pseudo break point to the signal height difference at that break point is a constant. On the other hand, the Daubechies wavelet of 3^rd order is the best basis comparing to the other representative bases, namely the Haar wavelet, the Meyer wavelet and the Fourier basis. It is very suitable to describe different kinds of signals, e.g. a smooth signal, a rugged one and a signal with local break points.

Test Results using Real 3D-City Surface Data
The theoretical analyses are also done using a real geometrical city model in a suburb of Taipei. The model is measured on analytical plotter Leica BC3 using a stereo pair of aerial photos with a mean photo scale 1/5000, photo format 23cm x 23cm, and focal length 30cm. The analog photos are also scanned by photo scanner DSW200 with a pixel size of 25mm x 25m m. The measured 3D points in the stereo model are then edited to produce a geometrical surface in the test area shown in Figure 1. The geometrical surface is taken as a known real 3D-city surface that is used to examine the accuracy and characteristics of the wavelets-based approximation algorithm.

Figure 1. Aerial image in test area (left) and geometrical surface of the test area (right)

Some test results are shown in Figures 2 ~ 4. Apparently, approximation function A_jf can converge to f as the resolution becomes finer, namely the index j becomes larger. In addition, one can superimpose the digital image data on the computed geometrical surface and get a virtual reality of the 3D-city surface in test area, although the image information on all walls is evidently insufficient because only aerial images are used in these tests. Moreover, a dynamical stereoscopic view can also be made, if the related computer hardware and software are available. It is really an easy-to-learn and user-friendly 3D 'map'.
(a) Approximation A₀f (left) and its error function (right)



(b) Approximation A₂f (left) and its error function (right)



(c) Approximation A₄f (left) and its error function (right)

Figure 2. Approximations A_jf and their error functions with j = 0, 2, 4



(i)



(ii)

Figure 3. Curve of the root mean square error (i) and maximal absolute error (ii) of the approximations A_jf with j = 0 ~ 4



Figure 4. Virtual reality of the 3D-city surface in test area