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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
between albedo and topographical variations as reason for grey 
value changes in image space. 
For the mathematical modelling of the bi-directional reflectance 
(BDR), we use the Lunar-Lambert model (McEwen, 1991) for 
describing the object surface. This radiometric model is a linear 
combination of two different models, the Lambert and the 
Lommel-Seeliger model. 
The Lambert law is one of the simplest and most frequently 
used reflectance models. Specific descriptions and equations of 
the Lambert law can be found in (Horn, 1986; Hapke, 1993; 
Zhang et al., 1999). The Lambert law describes a surface, which 
emits the incoming irradiance uniformly in all directions. The 
model is based on the assumption that the brightness of a 
surface depends only on the incidence angle i, the angle 
between the direction of illumination $ and the surface normal 
ñ (figure 1). This means, that a surface looks equally bright 
from every viewing direction. The Lambert model characterizes 
the reflectance from bright surfaces very well. 
The second used radiometric model is the Lommel-Seeliger 
law. In order to extend the assumption that light reflection 
occurs at the boundary surface between two media only, the 
Lommel-Seeliger law was derived by Seeliger (Horn et al., 
1989; Hapke, 1993; Rebhan, 1993). In this model, light 
scattering is assumed to take place at the individual particles 
within a layer of infinite thickness below the apparent surface; 
the irradiance observed at a sensor comes from light scattered 
by all particles in the medium lying within the field of view of 
the sensor. Therefore, the Lommel-Seeliger law does not only 
contain i; but also the emittance angle e between viewing 
direction v and # (figure 1). The significant increase in 
brightness for large e is due to the fact that with increasing e the 
area of the imaged surface also increases, and consequently a 
greater part of the surface layer contributes to the brightness 
observed in the sensor. In contrast to the Lambert law, the 
Lommel-Seeliger law describes dark surfaces better. 
Using first only the Lommel-Seeliger reflectance the model 
grey value G(x y) in image space can be formulated based on 
the well-known camera equation (e.g. Horn, 1986): 
  
c ar d 
G(x', v') = k e. a ar uS 
Y 1 cos C ) f 
nv ns 
cos (e) - ——- and cos (i) = — 
FA 
with 
where G(x’,y’) model grey value at image point P^ 
X} image coordinates of P' (proj. of P into image space) 
k rescaling constant for transformation of image 
irradiance into model grey value G(x’,y’) 
a exponent of light fall-off 
y angle between optical axis and the ray through P and 
p? 
d aperture of optical lens 
f focal length of optical lens 
Es scene irradiance 
XX,Y) albedo of the object surface at P(X,Y,Z) 
n normal vector of the object surface at P(X,Y,Z) 
Ki vector in illumination direction at P(X,Y,Z) 
y vector in viewing direction at P(X,Y,Z) 
8 
9 
The scene irradiance Eg, the parameters (a, d, f, k and y) and the 
albedo p are assumed to be constant values and are merged into 
a so called reflectance coefficient A: 
  
  
DN" VIL. 
Ap = k — cos (7) v E, p(^ x ) (2) 
4 of, 
Thus, the model grey value G depends on Ap, 5 , 9 and ñ : 
, D 2 cos (i) : 
G(x s ) = f (AR.5.%,7) = Ap A M T (3) 
cos (i) + cos (e) 
Eo P. 
j pall 
i ; 
» 
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à nr s 
EN s ys d, 
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> / “u $^ P' 
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X 
Figure 1. Configuration of multiple images, camera parameters 
and the relationship between $ , nn, V 
In the case of the Lunar-Lambert model the model grey value G 
looks as follows: 
s cos) c 
G(x', y") = A, |2A— 
Mealy “ cos(i)+cos(e) 
*(1-A^)cos(i) (4) 
The parameter A controls the weighting between the Lambert 
and the Lommel-Seeliger term. The light source in this 
approach is assumed to be a distant point with a known 
position. The influences of a possibly existing atmosphere are 
considered to be constant, and thus part of Ap. Moreover, the 
parameters of the interior and the exterior orientation are 
assumed to be known from a camera calibration and a 
previously carried out bundle adjustment. 
For the purpose of the object surface description, a geometrical 
and a radiometrical surface model are introduced. The 
mathematical description of the geometric model is given by 
means of a DTM with a simple grid structure, which is defined 
in the XY-plane of the object space. The roughness of the terrain 
is the decisive point for the choice of the mesh size of the grid. 
An independent height Z(X,, Yj) is assigned to each grid point 
(X,Yj) of the DTM. A height Z at an arbitrary point is 
interpolated from the neighbouring grid heights, e.g. by bilinear 
interpolation. At each point of the object surface, 5? and thus 
the angles i and e become a function of the neighbouring Z, ;. 
A radiometric surface model is introduced to establish the 
connection between the geometric surface model and the 
reflectance behaviour of the surface. Each DTM grid mesh is 
divided into several object surface elements of constant size. The 
size is chosen approximately equal to the pixel size multiplied by 
the average image scale factor. Each object surface element is 
assigned the same reflectance coefficient Ag.