International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
problem complex, the results shown in this paper clearly
demonstrate the potential of the employed method.
The extremely large number of elevations — equal to the nun?
ber of elevation cells — as unknown parameters is apt to pro-
duce substantial performance problems. The aim is to employ
an efficient numerical method guaranteeing processing time
approximately proportional to image size. Limitations to the
over-all process are mainly due to space-variant surface al
bedo, severe atmospheric disturbances, shadows and certain
illumination conditions. It may therefore be envisaged that
only a subset of Mars scenes will be amenable to data pro-
cessing by De-Re-Shading.
The next section is devoted to a brief introduction into our
general DRS approach. The fundamental SFS kernel will be
described in some detail both from the mathematical and nu
merical points of view. After an account on the present im-
plementation stage and future improvements emphasis will be
placed upon a discussion of the experimental results so far
obtained and conclusions to be drawn from them.
2. DE-RE-SHADING (DRS)
2.1 Conceptual Approach
The DRS process is based on the conditional comparison of
the irradiance of an ortho-image (£) with the modelled radi-
ance image (R) of the object scene; see the block diagram of
Fig. 1. We assume that the ortho-image was derived from an
original HRSC image (7) together with the orientation and
calibration state of the camera (C) during the time of expos-
ure and an intitial digital elevation model (Z) obtained by ste-
reo photogrammetry (shape from matching SFM). The condi-
tionality of the comparison stems from the constraint that the
global difference between initial DEM and corrected DEM
becomes minimal. Kernel is a Shape-from-Shading (SFS)
method mostly known from Computer Visualization.
Camera
Image Orientation Light
Irradiance Calibration DEM EQM BRDF Source
E
Ortho-Image Scene
Irradiance Radiance
("Albedop .
DRS (£c v
Figure 1. Block diagram of DRS data flow
Prerequisite for a definite solution is an initial DEM of suffr
cient quality. The stochastic model of the DEM may be given
by the global elevation variance (Var Z) as elevation quality
model EQM). Both initial DEM and EQM possess some po-
tential for the derivation of space variant albedo. Other input
parameters concern the BRDF (Bidirectional Reflectance
Distribution Function), i.c. an assumed reflectance law, and
the sun vector (s). The present experimental version utilizes
constant BRDF, hence diffuse (Lambert) reflection. The cor-
rected DEM may be used as new initial DEM thus defining
an iterative process, although real improvements may be ex-
pected only if the ortho-image will be computed anew as
well. Further, albedo is considered constant for each scene.
The dark solid lines indicate the present stage of the solution,
the dotted lines have not yet been implemented, and the light
solid lines belong to an a priori process external to SFS. Not
shown in Fig. 1 is the mosaicking of neighboring scenes.
2.2 Shape from Shading (SFS)
Depending on numerous, partly merely estimable geometrical
and physical factors such as surface reflectance, shadows,
light source distribution, image resolution, accuracy of initial
elevation map, etc., SFS applied to real-world imagery is a
non-trivial, generally ill-posed problem. SFS has been and
still is treated in an extensive mathematics and theory ork
ented literature covering mostly small and limited problems.
An excellent survey is given by (Zhang et al., 1999). An inr
portant aspect concerns the coupling of deterministic and
stochastic optimisation approaches (Crouzil et al., 2003) in
order to determine optimal values for the penalty factors of
the constraint terms.
From the methodical view point SFS is a problem of Vari-
ational Analysis with constraints. Historically such com
straints were predominantly incorporated to ensure conver
gence to a somewhat smoothed solution in order to omit dis-
continuities and make them mathematically manageable
(Horn, 1970). Although this is true in the continuous domain,
for discrete data there is a resolution limit given by the pixel
size. Real discontinuities can therefore not exist, hence
smoothing regularization constraints become obsolete.
In our present SFS-approach the basic (continuous) equation
is defined by the minimization (or optimisation) integral
v9
JG Ie —p R) +M(Z — g") jux dY — min (1)
9
taken over a scene (2. in the (X. Y) -plane. Equ.(1) is the un-
constrained form of the constrained minimization functional,
extended by the elevation constraint with a “penalty” factor
^. Z represents the surface of the desired DEM, Z'" the
initial DEM as approximation to Z(X.Y) which is to be de-
termined such that the functional J(Z) becomes minimal.
See Fig. 1 for the notation. R = R(p,g) as modelled reflec-
ted scene radiance is a function of the surface slopes
p — Z,(X,Y) and q — Z,(X,Y). The scene albedo factor
can be estimated by the expression
HEE £2)
and X is the ratio of the variances of E and Z9, viz.
G 2
Ref (3)
thus representing the global stochastic model part (Crouzil, et
al., 2003).
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