e model
each point of
' constant size
'y the average
; of the vector
albedo A, the
(3)
(4)
ns (3) and (4)
1 image j with
pace using the
es at this posi-
ns. The corre
at a common
(5)
Tativc compi-
uation (6) the
(6)
veen the image
DTM meshes
del can also be
e by the intro
rather than ab-
In order to ob-
i object spact,
nd in SFS a
ant albedo and
Christian Piechullek
3 THEORETICAL ANALYSIS OF MULTI IMAGE SHAPE FROM SHADING
In order for this method to work noticeable local grey value gradients dg/dZ ,
dg /0Zk,1=0g / dr dr / 0Zk,! + dg / dc dc / O0Zk, 1 (7)
in line with the employed reflectance model must be available in image space (r and c stand for the pixel coordinates
row and column). Otherwise, the matrix of normal equations within the least squares adjustment will become singular.
While this condition is the same for least squares image matching and MI-SFS, it is more critical in the case discussed
here due to the generally poor texture in areas suitable for SFS, i. e. areas with constant albedo. As can be seen from
equation (7) this condition has two important implications: (1) a shift in the DTM heights Z4, must result in a noticeable
shift in image space, i.e. or / dZ orac/ 0Z must be noticeable, and (2) such a shift must in turn result in noticeable local
grey value difference dg/dr and dg / dc between the neighbouring pixels.
- À number of restrictions follow from this condition when applying the proposed method to a planetary scenario:
(1) Planar surfaces cannot be reconstructed. In the Lambert law the BDR only depends on the incidence angle i which
is constant for the whole plane. Thus, all images of this plane have uniform brightness, dg / r and 3g / àc are both
zero. The situation is somewhat better for the Lommel-Seeliger law, because the BDR also depends on the emit-
tance angle e. However, for all realistic planetary applications due to the small instantaneous field of view of the
employed sensors neighbouring pixels will have nearly the same emittance angle, and thus nearly the same grey
value. Again, no local grey values are present.
(2) Undulated surfaces, on the other hand can be reconstructed from one or multiple images. However, in case only
one vertical image is available, i. e. the optical axis is more or less parallel to the Z-axis of the object coordinate
system, areas near the nadir point can create problems regardless of the employed reflectance model, because
àr / 0Z and dc /dZ might be both too small. This problem can be resolved by changing the direction of the axes of
the object coordinate system.
(3) A small incidence angle i will generate only small grey value gradients (see figure 2 and 3) and can create problems
for the Lambert case, especially if only one image is available and stereoscopic correspondence cannot be used. For
the Lommel-Seeliger case this issue occurs as well, but can be at least partly solved by choosing a large emittance
angle e. If possible a large incidence angle should be used.
(4) The BDR for Lommel-Seeliger type surfaces is relatively flat (see again figure 3), especially for values of i and e
being similar and below approximately 60 degrees. If both angles are identical the BDR even becomes independent
of i and e, and thus of the terrain inclination, and takes on the value of 0.5 A;s (see equation 2). Thus, when the
emittance angle can be influenced a value different from the incidence angle and as large as possible should be cho-
sen.
(5) Since the BDR depends linearly on the albedo (see equations 1 and 2) surfaces should be imaged with as much
irradiance as possible, i. e. with as large an exposure time as possible.
(6) Radiometric manipulations of the images will have different results with respect to the resulting surface shape. A
constant change in overall surface brightness (offset) will primarily change the estimated surface albedo, but the
normal vectors will also be effected. A positive offset will lead to smaller variations of the surface normal, a nega-
tive offset will lead to larger ones. A linear change in surface contrast (gain factor) on the other hand will have no
effect, because the resulting linear contrast enhancement will be completely compensated for by the estimated sur-
face albedo. Brightness and contrast enhancement of individual images, however, can have unpredictable effects.
(7) Due to the generally poor image texture, noise can be a particularly difficult problem for the Lambert and the
Lommel-Seeliger case. In order to reduce the noise level it is advisable to filter the images using a lowpass prior to
the surface reconstruction.
In extensive simulation studies (Piechullek et al. 1998; Piechullek 2000) all mentioned points were verified. While the
points (1) to (6) follow directly from the given equations, the influence of noise was investigated in more detail. Three
error-free synthetic images from different viewpoints were generated using an undulated terrain. Using these error-free
images stable results for the object surface were achieved with a range of different DTMs as initial values for the un-
known heights. Subsequently, the images were contaminated by Gaussian noise with a standard deviation of 6 grey
values. While the results obtained from these noisy images were clearly a function of the DTM used as initial values,
lowpass-filtering and thus noise reduction prior to surface reconstruction produced the desired results: convergence
towards the correct result was again achieved. The radius of convergence was found to be approximately 20 pixels for
both, the Lambert and the Lommel-Seeli ger case. A comparison between the results from the error-free and the noisy,
filtered images revealed that the effect of filtering onto the geometric accuracy of the reconstruction was negligible.
The simulations were also carried out for individual images. As was to be expected, in this case the accuracy of the
obtained results was worse by a factor of about 2 compared to the solution with three images due to the lower redun-
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 727
