The International Archives of the Photogrammetrw Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
1033 
tion (5) - involving bilinear interpolation for surfels within a 
particular DTM facet. See Gehrke & Haase (2006) for formal 
derivation and further details. 
R = R° + 
<3R° X-X 0 
SX z°-z 0 
H- 
SR° Y - Y 0 
SY Z°-Z 0 
dZ 
(5) 
The entire Facets Stereo Vision algorithm has to be carried out 
iteratively, starting with the resampling using enhanced DTM 
heights, until no significant height changes remain. It has been 
proven effective to start up with a comparatively coarse model 
(MOLA DTM resolution) and increase it step by step (Fig. 2). 
3.2 Regularization Aspects 
The basic algorithm becomes weak in smooth regions that com 
prise only little texture or small contrast differences - areas that 
frequently occur on the dust-covered Martian surface. Thus, the 
adjustment has to be stabilized by additional assumptions. This 
may be realized by introducing terrain-dependent restrictions to 
surface curvature (Wrobel et al., 1992; Gehrke 2007). 
While such regularization significantly reduces noise and over 
comes DTM outliers, in particular when locally weighted by 
surface properties, such conditions are only fictitious. This lack 
could be overcome by fully using all radiometric image infor 
mation, i.e., the observed radiance factors of each pixel, and to 
combine matching in object space with photoclinometry. 
Initial Heights 
(MO LA) 
HRSC 
image Data 
Г 
Orientation 
\ 
L. 
Data 
✓ 
о 
£ 
О 
«я 
Ф 
ОС 
о __ 
Ф ге 
«л <~ 
та Ф 
ф а 
Definition of 
Facet Sizes 
Differential 
Rectification 
i 
r Pseudo > 
^ Orthoimages J 
- r ^—' 
Atmospheric 
Modeling 
Atmospheric 
Optical Depth 
Radiometric 
Adaptation 
i 
Photo- 
* Matching in 
clinometry 
Object Space 
Integrated Modeling Approach 
* 
t 
> 
DTM 
J 
Reflectance 
Parameters 
4 
Orthoimages 
Hapke 
Modeling 
~~r~ 
Material 
Properties 
Fig. 2: Integrated approach for geometric and radiometric mo 
deling of the Martian surface. 
4. THE INTEGRATED APPROACH 
4.1 Combination of Facets Stereo Vision and Photoclino 
metry 
Radiometric image formation is described by a surface reflec 
tance model combined with an atmospheric model (see 4.3 for 
the latter). Assuming, the reflectance model and its parameters 
are known, local surface normal vectors can be derived - see 
equation (1). Since this equation is in the form of (5), it readily 
allows for the integration of photoclinometry into Facets Stereo 
Vision (5), once local surfel inclinations in (1) are expressed in 
terms of dZ, again, with bilinear interpolation within the DTM. 
Such an approach nicely combines the advantages of both mat 
ching, which provides reliable absolute heights for well textured 
image parts (global accuracy), and photoclinometry, which 
leads to relative heights (local accuracy) and is suited to bridge 
matching gaps. This goal is achieved by weighting the two al 
gorithms in the combined least squares adjustment depending 
on local image texture, practically carried out using radiance 
factor gradients in X and Y that are already part of (5). 
4.2 Radiometric Surface Modeling 
In most cases, the assumption of known reflectance parameters 
does not hold true; they have to be determined. Here lies the 
major advantage of the combined approach: If matching leads 
to (some) reliable heights that constrain inclination, reflectance 
parameters may be derived within the photoclinometry part of 
the adjustment. This ability requires certain relief energy as 
well as good image texture. 
However, the Hapke parameters, described in chapter 2, tend to 
be highly correlated and the derivation along with DTM and or 
thoimage will very likely fail. Therefore, the empirical two- 
parameter Lunar-Lambert-Model - a suitable approximation for 
Hapke, proven throughout various theoretical and practical in 
vestigations (McEwen, 1991; Kirk et al., 2004) - is chosen as 
the radiometric surface description in this context: 
R = A 
2L- 
COS0: 
COS 0: +COS0, 
+ (l-L)cos0¡ 
(6) 
The normal albedo A in (6) is linked to the particle single scat 
tering albedo w, the limb darkening parameter L to the surface 
roughness in the Hapke Model (1). Thus, based on the empirical 
reflectance resulting from the adjustment, physically mea 
ningful Hapke parameters can be retrieved afterwards (Fig. 2). 
4.3 Atmospheric Modeling 
For an appropriate surface modeling, attenuation of the atmo 
sphere (optical depth) and the influence of ambient light are to 
be regarded - cp. equation (5). While the latter can be integra 
ted into the adjustment as an additional parameter, the optical 
depth is derived in advance since the related multiplicative term 
in (5) would highly correlate with the albedo factor in (6). 
Optical depth can be derived by the Stereo Method (Hoekzema 
et al., 2006), which uses HRSC stereo bands: different angles 
result in different atmospheric path lengths and, according to 
equation (3), different influences of optical depth, which causes 
different image contrasts. Then - assuming identical ambient 
light and allowing only linear differences in surface reflectance 
between stereo angles - the optical depth could be derived from 
contrast differences, ideally from nadir and the outermost stereo 
band(s) as those feature the biggest path length difference.