x2)
[Horn and Brooks, 86] where it is required for con-
vergence. However, in our experience it is not nec-
essary in the Frankot-Chellappa algorithm applied to
Mars or SPOT imagery.
e Improve each of the gradient estimates indepen-
dently by adjusting them towards the value minimis-
ing the difference between observed and computed
(from the gradients) intensity. The algorithm’s À pa-
rameter controls the rate of adjustment.
e Project the gradients to an integrable surface, min-
imising the sum of the squares of the adjustments
required. This is done in Fourier space.
The periodic wrap-round nature of the discrete Fourier
transform representation results in distortions at opposite
edges of the DEM. We overcome this by padding the im-
age to double size with a uniform grey-level equal to that
produced by a flat surface.
[Frankot and Chellappa, 88] used a central-differencing
differentiation operator. This results in a slightly corru-
gated appearence to the output DEM, as central differ-
encing cannot "see" corrugations. We obtain stripe-free
output using forward differencing.
3.2 Calibration
Given a radiometrically corrected Viking Orbiter image we
must still account for the effects of surface reflectance and
atmospheric scattering. Two numbers are required to ef-
fect a linear transformation between image grey levels and
the normalised range used internally. We currently specify
the "shadow current" and the "flat current". The shadow
current is the grey level with which a self-shadowed slope
will appear in the image; we generally estimate this to be
equal to the darkest pixel in the reseau-removed image
(N.B This is not equivelent to the camera's "dark current",
which would not compensate for atmospheric scattering).
The flat current is the grey level with which a level piece
of terrain will appear; we estimate this from the modal
grey-level in the scene. We believe this to be appropriate
as most of the scenes to which we have applied shape-
from-shading have contained large apparently flat areas
of almost uniform grey-level.
Errors in calibration generally result in long-range cur-
vature of the output DEM. We can compensate for this to
an extent by adjusting the calibration parameters in order
to eliminate long-range elevation trends. In future we ex-
pect to make use of information such as the USGS DEM.
Figure 7 shows the DEM obtained by applying shape-
from-shading to an extract from the orthoimage in fig-
ure 6. A Minnaert reflectance function with parameter
0.85 (compatible with [Davis and Soderblom, 84], given
the 53.2? phase angle for this image) was assumed. The
corresponding extract of the stereo-matched DEM is also
shown as a perspective view; the sloping terrain feature
may correspond to the eastern crater wall, but overall
there is little resemblance to a crater ulike the shape from
shading results. In future it is hoped to be able to re-
place selected areas of stereo-matched DEM containing
higher frequencies with the shape-from-shading derived
elevation models containing higher frequency intervals.
807
4 Conclusions
While matching techniques developed for SPOT do seem
to be applicable to Viking Orbiter, the resolution of the
imagery available over most of the Martian surface permits
only extraction of the largest terrain features. Shape from
shading offers a way of extracting features such as craters
in some regions.
In the near future, combined use of the existing USGS
DEM with the images available may permit gloibal cover-
age to be achieved even given the poor stereo coverage
of Viking Orbiter. Longer term, the Mars Observer Laser
Altimeter coupled with the MO camera and the HDSC on
Mars 94 will permit these techniques to be more fully ex-
ploited.
5 Acknowledgements
The EXODUS project is supported by SERC grants GR/F-
84-294 and GR/F-82-894 jointly with David Rothery, John
Murray and Gillian Thornhill at the Department of Earth
Sciences, Open University. We would like to thank Sher-
man Wu, Debbie Cook, Kay Edwards and Annie Howing-
ton at USGS Flagstaff for all their help and support and
Sue MacMahon at JPL.
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