Seeliger law as a model for the surface reflectance 
properties, along with a distant point light source with 
known irradiance E, and known illumination direction. 
Each DTM mesh was divided into 10*10 object 
surface elements with constant albedo. Each of these 
Shaded relief images was then projected into the 
different images with known exterior orientation 
parameters by using a ray tracing algorithm. 
  
Figure 5: The same surface, shaded with the 
Lambertian (left) and the Lommel-Seeliger (right) 
photometric function 
Figure 5 shows the same surface as it is imaged from 
the same camera position, but with the two different 
surface reflectance functions. To approximate the 
imaging geometry of the HRSC camera near the 
closest approach to the Martian surface, the 
orientation parameters of the images were defined as 
follows (see figure 6): 
- three images on a straight line with a base length of 
163 km between neighbouring images; 
- flight altitude: 475 km; 
- stereo angle: 19° 
- object surface element size (ground resolution): 
19*19 m?. 
   
4575 km 
uet >. 
X427 surface Y 
Figure 6: Setup for the generation of the images 
4 
These images, along with known exterior orientation 
parameters and different initial values for the 
unknowns, were introduced to our algorithm in order 
to reconstruct the DTM heights and the surface 
albedo. 
   
    
648 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
4.2. Conducted experiments 
Four groups of experiments were conducted using the 
three images for each simulation. All experiments 
were stopped when the changes to the unknown 
heights and the albedo from one iteration to the next 
fell below a predefined threshold of 1 m for the 
heights and 1 grey value for the albedo, respectively. 
The following experiments were conducted: 
1) Surface reconstruction with known, error-free 
albedo and a horizontal plane as initial DTM. This 
experiment was carried out with the Lambert and 
the Lommel-Seeliger model. 
2) Surface reconstruction with known, error-free 
heights and a known, error-free albedo which is 
5% smaller than the correct value (only performed 
with the Lommel-Seeliger model); 
3) Reconstruction of a Lambert surface with the 
Lommel-Seeliger model. 
4) Surface reconstruction with the Lommel-Seeliger 
model, introducing a horizontal plane as initial 
height information, along with a wrong and 
unknown albedo. This experiment has been 
conducted with an albedo value which is 5%, 10%, 
15%, 20% and 50% smaller than the correct one 
which was used to generate the input images. 
4.3. Results 
All simulations show that the relative height 
differences between neighbouring surface elements can 
be computed after a few iterations; the absolute height 
offset of the whole surface is reconstructed more 
slowly. Changing the orientation of a surface element 
immediately changes its grey value (see equations (10) 
and (11)); a change of the absolute height offset 
causes the whole image of the surface to be shifted in 
image space, without changing the grey value 
differences between neighbouring surface elements to 
a large extent. As for the four groups of experiments, 
the following results were achieved: 
- Introducing a horizontal plane as initial height 
information, along with known and error-free albedo 
allows for a correct reconstruction of the Lommel- 
Seeliger surface; remaining height differences AZ 
between the correct DTM and the result of the 
surface reconstruction are in the order of AZ/h = 
10* and are caused by quantisation errors during the 
generation of the images. In the Lambert case, 
however, the surface is reconstructed incorrectly; 
while the surface inclinations in the vertical plane 
containing the light source direction s are correct, 
the inclinations perpendicular to this plane are 
wrong, causing a 'profiling' of the surface. Figure 7 
shows the reconstructed surface (left), along with 
the differences between correct and reconstructed 
heights (right); s denotes the illumination direction. 
Since the Lambert photometric function is 
   
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