between reference DTM and reconstructed DTM.
While in simulation 2) the difference between
observed and calculated grey value is due to a
constant factor, in this simulation the difference is
caused by a wrong assumption concerning the
relation between the grey values and the angles i
and e. It should be noted that in this case a large
number of iterations had to be performed before a
convergence was reached.
- The last group of simulations served to assess,
whether unknown albedo can be estimated together
with unknown DTM heights. The DTM heights
could be successfully reconstructed in all
experiments. The unknown albedo nearly reached
the correct value after the first iteration. The
remaining height offset and surface tilt are then
removed, until the correct values for the heights and
the albedo are achieved. In comparison to
simulation 2), where a wrong albedo caused a tilt of
the whole surface towards the light source, the grey
value differences here result in an improvement of
the albedo in order to remove this tilt. This is due
to the fact that the albedo is a constant factor,
which influences all grey value observations in the
same way. Since this behaviour is independent of the
difference between correct and initial albedo value,
all experiments reconstructed the surface heights
and the albedo after the same (few) number of
iterations.
4.4. Conclusions
Our SFS algorithm has been tested successfully using
synthetic images. In comparison to the well-known
Lambert law, the Lommel-Seeliger law was introduced,
along with unknown surface albedo, as a model for the
surface reflectance properties. The following
conclusions can be drawn from the results presented
above:
- The Lambert law turns out to be unstable, resulting
in singularities for the surface reconstruction, if
multiple images with identical illumination direction
are introduced. This behaviour is independent of the
exterior orientation of the images, since the image
grey values are not influenced by the camera
position;
- The Lommel-Seeliger law allows for a correct
surface reconstruction, even if all images are
introduced with identical illumination direction. The
image grey values are a function of camera position,
and therefore additional independent information is
available for the surface reconstruction;
- SFS using the Lommel-Seeliger law is a method to
reconstruct unknown heights correctly, even if poor
initial values, e.g. a horizontal plane, are introduced,
- Unknown albedo can be estimated correctly from
the images, along with unknown heights, even if an
extremely wrong initial albedo value is introduced;
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
this is due to the fact that the initial value for the
albedo affects all grey value observations in the
same way.
- Wrong albedo introduced as error-free results in an
incorrect inclination of the whole surface relative to
the illumination direction. The relative height
differences within the surface, however, can be
reconstructed with high accuracy;
- If the imaged surface does not obey the modeled
reflectance behaviour, completely wrong heights are
obtained. In contrast to a surface reconstruction
with wrong albedo, the resulting height differences
are also a function of the surface topography.
Therefore, SFS can only produce correct results, if
the reflectance properties of the surface are modeled
appropriately.
5. OUTLOOK
Multi-image SFS using the Lommel-Seeliger
photometric function turned out to yield good results,
even if poor initial values for the unknown surface
parameters are available only. Additionally, this
approach can overcome singularities which occur when
Lambert surfaces have to be reconstructed from
multiple images with identical illumination direction.
However, only synthetic images have been used;
therefore, the presented approach has to be tested
using real imagery of poorly textured surfaces, such as
planets or asteroids, in order to evaluate the
correctness of the surface reflectance model, and to
assess the robustness and reliability of the methods
when image noise and non-uniform albedo are
present. Furthermore, the demands of the MARS96
mission have to be met by implementing the three-line
scanner imaging geometry of the HRSC and WAOSS
sensors.
6. REFERENCES
Albertz J., Scholten F., Ebner H., Heipke C., Neukum G.
(1993): Two Camera Experiments on the Mars 94/96
Mission. GIS 6 (4), 11-16
Hapke B. (1981): Bidirectional Reflectance
Spectroscopy; 1. Theory. Journal of Geophysical
Research 86 (B4), 3039-3054
Hapke B. (1984): Bidirectional Reflectance
Spectroscopy; 3. Correction for Macroscopic
Roughness. Icarus 59, 41-59
Hapke B. (1986): Bidirectional Reflectance
Spectroscopy; 4. Extinction and the Opposition Effect.
Icarus 67, 264-280
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