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The International Archives of the Photogrammetrv. Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
polation. Facets of an orthoimage - usually the model with the 
highest resolution - are named surfels (surface elements); their 
size should be related to the ground resolution of the original 
imagery. The radiometry of an orthoimage is influenced by sur 
face geometry and material reflectance properties in combina 
tion with illumination and viewing conditions (Fig. 1). 
The two terms in equation (3) are related to forward and back 
ward scattering, both probability distributions in terms of ellip 
ses whose shape is defined by b with 0 < b < 1; b = 0 (circle) 
refers to evenly distributed scattering, b —» 1 to pronounced 
lobes. Forward and backward scattering are weighted by c with 
0 < c < 1. Although these parameters are empirical, they are 
suited to describe particle shape, surface, and interior, i.e., the 
density of internal scatterers (cp. Fig. 5). 
For HRSC, radiance factors of all image pixels can be derived 
from the recorded intensities (Jaumann et al., 2007); they are 
treated as observables: Rhrsc- These values result from surface 
reflectances R, influenced by atmospheric attenuation and am 
bient light. The first effect is multiplicative, depending on the 
atmospheric optical depth x and the path length, i.e., the reflec 
tance angle 0r(Level) with respect to the surface normal (level sur 
face). Ambient light causes an additive contribution AR a . Then, 
the radiometric relation between surface reflectances and obser 
ved values may be modeled (e.g., Hoekzema et al., 2006): 
Fig. 1: HRSC imaging configuration and models for the Mar 
tian surface. 
For remote sensing applications, which are based on observed 
reflectance, material properties are usually described as para 
meters of a bidirectional reflectance model. A physically mea 
ningful description of planetary surfaces is achieved through the 
Hapke Model (Hapke, 1993) that can be written as: 
f 
^•HRSC = ^' ex P 
V 
\ 
T 
cos 0r(Level) , 
+ AR a 
(3) 
The geometric relation between an object point (X,Y,Z) and an 
image point (x,y) are the well-known collinearity equations: 
c r,i(X-X 0 ) + r,|(Y-Y„) + r 3 |(Z-Z 0 ) 
■b(X-X 0 ) + r 23 (Y-Y 0 ) + r 33 (Z-Z 0 ) 
yi2(X-X 0 ) + r 22 (Y-Y 0 )+r 32 (Z-Z 0 ) 
y C r, 3 (X-X 0 ) + r 23 (Y-Y 0 ) + r 33 (Z-Z 0 ) 
HRSC interior orientation consists of the focal length c; posi 
tion and attitude are given through (X,Y,Z) and the elements 
(r] ],...,r 33 ) of the rotation matrix, respectively. It has to be poin 
ted out that each individual image line features its own exterior 
orientation. Both exterior and interior orientations are known 
from bundle adjustment (SPIEGEL & Neukum, 2007). 
3. FACETS STEREO VISION 
R = 
COS0: 
4 cosO: +COS0, 
-{P(a)+H(0.)H(e r )-i}s(e) 
0) 
R is the radiance factor (RADF) as defined by Hapke (1993). It 
depends on illumination and observation geometries - local 
incidence angle 0;, viewing angle 0 r , and phase angle a - and 
material properties: particle single scattering albedo w (bright 
ness) in combination with the angular distribution P, multiple 
inter-particle scattering modeled by the geometry-dependent H- 
Functions, and macroscopic roughness, i.e., the average surface 
tilt 0. Equation (2) does not take into account the opposition 
effect (a strong reflectance surge for a —> 0), as the necessary 
observations are not available by HRSC unless a larger number 
of suitable orbits is used (cp. Jehl et al., 2008). That is, how 
ever, outside the scope of this investigation (see chapter 6). 
A widely used description of the angular distribution of particle 
scattering is the Double Henyey-Greenstein phase function: 
P(a) = ( 
1-b 2 
1-b 2 
- + (l-c)- 
(l-2bcosa + b 2 j 2 |l + 2bcosa + b 2 j 
(2) 
Facets Stereo Vision is a powerful approach for matching in 
object space. It has been developed since the late 1980s, mainly 
by Wrobel (1987) and Weisensee (1992). The application of 
the basic algorithm to HRSC on Mars Express images is discus 
sed in detail by Gehrke & Haase (2006) and Gehrke (2007). 
3.1 The Approach for HRSC Data Processing 
Geometric surface modeling from HRSC image data in this 
context is based on the indirect algorithm of Facets Stereo Vi 
sion according to Weisensee (1992), which has been adapted to 
Mars Express orbit and HRSC line scanner geometry. 
The algorithm starts with the definition of appropriate orthoim 
age surfels and DTM facet sizes. Then, so-called pseudo ortho 
images (pseudo observables) R are resampled for each HRSC 
band using equations (4); the necessary starting heights Z° are 
taken from the Mars Orbiter Laser Altimeter (MOLA) DTM 
(Smith et al., 2001). After local contrast and brightness adap 
tation, an average orthoimage R° is derived as the average of all 
HRSC bands. Depending on DTM quality, the individual pseu 
do orthoimages will show lateral displacements (therefore the 
term “pseudo”). These are reduced by deriving corrections dZ 
for all DTM posts by least squares adjustment based on equa-