scattering 
luced for 
The total 
wards the 
yers: 
e | (6) 
) leads to 
(7) 
  
  
  
(top) in 
ion of the 
surfaces, 
st to the 
t surfaces. 
ce models 
widely used in planetary photometry, e.g. [Lumme, 
Bowell 1981; Hapke 1981, 1984, 1986, 1993], are 
extensions of the Lommel-Seeliger law. Hence, this 
model can be used to describe the reflectance 
properties of planetary surfaces, assigning a constant, 
known single-particle scattering albedo w to the whole 
surface. 
Figure 3 shows the Lommel-Seeliger law in 
comparison to the Lambert law. The significant 
increase in brightness for large emittance angles e is 
due to the fact that with increasing e the area of the 
imaged surface layer also increases by 1/cos e, and thus 
a greater part of the surface layer contributes to the 
brightness observed in the sensor. 
The connection between the image grey values g(x,y’) 
and the reflected radiance L, is given by substituting 
equation (7) into the camera equation, [Horn 1986], 
g(x,y’) » k: E(x,y) 8 
nec 
zk- Te Y up reg) i 
image irradiance 
diameter of the optical lens 
focal length of the sensor 
angle between optical axis and image ray 
rescaling constant 
e 
wx 0t 
combining all terms, which are independent of i and e 
into the reflectance coefficient Ag: 
g(x,y) = Ag(w)  —2L ©) 
COSI +COSse 
3. MULTI IMAGE SHAPE FROM SHADING 
Multi-image Shape from Shading has been introduced 
by [Heipke 1992; Heipke, Piechullek 1994]. It uses at 
least two images simultaneously to determine the 
heights of a predefined geometric object model. The 
grey values of the images are directly related to the 
unknown heights of a DTM which is defined in object 
space. The main characteristics of the method are: 
- perspective transformation from object space to 
image space; 
-no need for corresponding points, since 
neighbouring surface elements are assumed to have 
the same albedo; 
- least-squares estimation of the unknowns; 
- high accuracy potential. 
In the quoted references, the Lambertian reflectance 
model was used to describe the reflectance properties 
for the imaged surface, leading to the following non- 
linear observation equation per pixel per image: 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
veAQcosi(Z) - g(¥'(2)y'(Z)) (10) 
V least squares residual of observation equation 
2; unknown heights of the geometric surface model 
In this paper two extensions are presented. First, we 
have incorporated the Lommel-Seeliger law as 
photometric function for the surface. Second, the 
reflectance coefficient A, (and thus the albedo of the 
surface) is also considered unknown, and is estimated 
together with the unknown surface heights. The 
corresponding non-linear observation equation reads: 
DU cosi (Z,) 
3 cosi (Z.) «cose(2,) (11) 
-g(X(Z),y'(Z)) 
After linearization of the observation equations (10) 
and (11), respectively, the unknowns Z; and A; are 
estimated in an iterative least-squares adjustment. 
4. EXPERIMENTS AND RESULTS 
Some experiments on surface reconstruction using the 
Lambert and the Lommel-Seeliger law with varying 
initial information for the unknown surface 
parameters are presented in this chapter in order to 
evaluate the potential of both approaches. All 
experiments have been conducted using synthetic 
images which approximate the imaging geometry of 
the HRSC camera, [Albertz et al. 1993], when imaging 
near the closest approach to the Martian surface. 
4.1. Input data 
To generate the synthetic images, a continuous, hilly 
terrain with an area of 4940*4940 m? and a maximum 
height difference of about 1550 m was approximated 
by a DTM of 26*26 meshes with a mesh size of 
190*190 m? each (see figure 4). 
  
Figure 4: Reference DTM for the generation of the 
images 
From this DTM two shaded relief images were 
generated using 1) the Lambert and 2) the Lommel-