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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-