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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
Instead of establishing the Euler-Lagrange equations assock
ated with Equ.(1) as, e.g., in (Dorrer, et al, 1998), and solve
the discretized system of partial differential equations, we
have solved the discretized Equ.(1) directly by variation of
Z until the functional / is minimal. This is possible by ex-
pressing the slopes simply as the convolutions of constant
difference filters, A and B , with Z , viz.
p23-4*2,-q(Z)BsZ. (4)
Then the scene radiance is R = R(p(Z),4(Z)) and Equ.(1)
becomes
Js ff e ~p RZ -XQ-Z9ylaxaY, (s)
which may now be solved iteratively by the method of con-
jugate gradients (Shewchuck, 1994). Main advantages of
CGM are that linearization of nonlinear functions is not re-
quired, convergence of the method normally is superlinear
(Beckmann, et al., 2001), and Z will be determined indir-
ectly when J approaches the minimum. Some prerequisites
for CGM such as convexity of J follow from 7‘ being
rather close to Z . In essence, CGM approaches the minimum
by successively searching for line minimums along consecut-
ive "conjugated" gradients
E -pR(Z)(R,(Z)* A--R(Z)«B
via ff] pR(Z))(R (2) * )« B)
2 dXdY
+ À (7 ze,
(6)
The principle of CGM is exhaustively explained in (Shew-
chuck, et al, 1994).
2.3 Present Implementation
The present experimental version of the DRS software under
development is a preliminary stand-alone version written in C
and runs both under MS-windows and Linux. In addition an
independent APL2-version is used for testing and further
evaluation purposes. It is expected that the final C-version
will be integrated in a comprehensive software package de
veloped by the DLR Co-Investigator team.
The still rudimentary property of the DRS version may be
characterized by the following limitations:
* Lambert reflection (constant BRDF)
e Constant scene albedo factor
e No shadows
e No atmospheric correction
e Simplified stochastic model of the DEM
e — Non-automated pre- and post processing
e — Autmated SFS-kernel only
e Rectangular scene regions
e Local cartesian 3D-coordinate system.
Despite these limitation the refinement of an initial photo-
grammetric, i.e. SFM-derived, DEM is remarkable as will be
demonstrated for two suitable scenes in the following section.
It is of course envisaged to successively improve the present
DRS version by dropping the limitations. This is however a
tedious way requiring not only new theoretical ideas but also
extensive experimental work with many different HRSC
scenes.
1301
3. DISCUSSION OF EXPERIMENTS
Prior to the availability of HRSC data we have first used a
rectangular subsection of a Mars Global Surveyor MOC
wide-angle scene (M000094 situated on Mars in the southern
part of Icaria Planum at 95° W longitude and 41° § latitude)
for which DLR had prepared the required ortho-image
(Fig.2a) and associated MOLA-DEM. The latter is shown as
modelled radiance image in Fig.2b and with 100-m contours
in Fig.2c. As may be seen from Fig. 2b, due to the relatively
wide gaps between adjacent MOLA profiles the DEM may
only be considered as rough approximation to the real sur-
face.
The ortho-image has size 476x402 with a pixel size of 232 m,
thus covering an area of 110 km by 93 km. This yields a rel
atively small printing scale of 1:2°500°000.
Fig. 2d Fig. 2¢
Figure 2. MOC wide-angle scene 110 km by 93 km
at printing scale 1:2°500°000
The SFS-refined result after some 20 iterations is shown in
Fig. 2d and Fig. 2e. In general the contour lines (Fig. 2e) de
rived from the SFS-refined DEM appear much more regu-
