point of the existing control point network (Davies et al., 1996)
was used, in addition. This turned out to be essential for the
accuracy of the adjusted data. Object point coordinates agree
with those of this previous network within 2%. We obtained a
final accuracy of 0.5 pixels for image point coordinates (a priori
accuracy: 1 pixel), 1 km for the spacecraft position data, and
0.1-1 mrad for the camera pointing data.
3.1.2 Determination of conjugate image points: We
determined conjugate image points by an adaptive least squares
correlation algorithm (Gruen, 1985) in which the pattern of grey
values between patches in a reference and a search image are
compared. This algorithm was suggested to be less sensitive
against differing pixel resolution than cross-correlation
(Fórstner, 1995).
An optimum patch size must be chosen. The transformation
between the image patches assumes that the surface viewed by
the patches is approximately planar. This suggests that the
patch size be small, as otherwise, smoothing effects in the
terrain model, matching failures, or in the worst case,
topography blunders occur. On the other hand, the prevalence
of image noise and a required minimum of texture rather
suggest to choose large patch sizes.
Matching of images from different spectral filters failed;
therefore, only single-filter images were used to derive DTMs.
3.1.3 DTM generation: The coordinates of conjugate image
points (in terms of line, sample) in the two stereo images were
converted to ground coordinates (in terms of x,y,z) using
adjusted navigation data and applying the co-linearity equations
and least squares fitting. For comparison with the
photoclinometry models, the resulting three-dimensional cloud
of object points was then transformed into the frame of the
image which the photoclinometry model was based on, i.e., X,y,
and z were converted to line, sample, and height, where height
is measured with respect to a plane parallel to the image plane
that contains the center of Ida. Finally, a digital terrain model
was interpolated in image space using the "inverse distance"
approximation. This sometimes resulted in topography gaps if
the number of points required for the interpolation was too
small.
3.2 Two-dimensional photoclinometry
3.2.1 Approach: Topographic modeling of subareas of several
Galileo images of asteroids by photoclinometry was previously
carried out in order to facilitate crater-depth studies (Carr et al.,
1994; Sullivan et al., 1996). The method used is the two-
dimensional photoclinometry algorithm of Kirk (1987): The
surface shape is parameterized with finite elements in image
space; that is, the projection of each image pixel onto the
surface is an "element" and the displacements, measured toward
the camera, of the corners of the pixels are the topographic
unknowns being solved for. Standard finite-element techniques
are used to set up nonlinear equations relating the unknowns
(displacements) to the knowns (pixel brightnesses) via the
gradients of displacement and the photometric function. A
number of numerical techniques are then used to solve these
equations, such as iterative linearization of the nonlinear
equations (i.e., the Newton-Raphson method), iterative solution
of the linearized equations by the method of relaxation, as well
as multi-gridding to speed convergence of long-wavelength
portions of the topography.
Practical experience indicates that considerable judgement is
needed to determine when to change the number of relaxation
steps before relinearizing, the over/under-relaxation parameter,
and the conditions for changing grid resolution. All models
shown here were therefore generated with direct supervision of
iteration.
a ES
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
As the photoclinometric equations are underdetermined,
additional boundary conditions must be introduced. Here, we
use a measure of the "roughness" of the surface model, (dz/dx)?
+ (dz/dy)?, integrated over the surface, which is minimized at
the same time that the image is modeled. The quantities in this
squared gradient are in image coordinates; in particular, z is the
displacement towards the camera.
Determination of when the solution has converged adequately
could be problematic. In the multi-resolution algorithm, an
estimate of the trunctation error, i.e., the unavoidable error
introduced by coarsening the resolution, is generated at each
resolution except the finest one. The truncation error can,
however, be extrapolated to the finest resolution based on the
others. Iteration was continued until the residuals were less
than the truncation error at all resolutions.
3.2.2 Input Data: As the starting point of the iterations, a
global model of the shape of Ida (Thomas et al., 1996) with 2
degree resolution was used. It was also used to estimate the
surface scattering properties, i.e., the photometric function that
relates surface slopes to the image brightness, to be used in the
analysis. The global shape model was shaded with photometric
functions combining the Lommel-Seeliger (lunar) and Lambert
functions linearly in various proportions (McEwen, 1991), and
the proportion that best fit each observed image was adopted to
define the model photometric function for that image. This was
consistent with Hapke's physically based scattering model
(Hapke, 1993) with a low single-scattering albedo across a
range of phase angles (McEwen, 1991). These results lend
confidence that the surface scattering properties have been
modeled adequately.
4. RESULTS
4.1 Photogrammetry models
We derived digital terrain models from two stereo pairs of
images, respectively, in three distinct regions (termed I, II, and
III in the following) in which photoclinometric terrain models
were available (Fig.1). The first step of image correlation was
carried out successfully throughout most of the study area.
However, parts of region III show little texture and large
distortions in the topography (see the scarp in the lower part of
region III) which caused the matching to fail, and therefore
resulted in gaps in the terrain model (Fig.4a).
The effect of patch sizes of 10, 14, 18, and 22 pixels on
resulting topography was thoroughly analyzed: While large
patch sizes of 18 and 22 pixels resulted in smoothing effects
(i.e. small-scale features vanished and medium-sized craters
became flater), the topography became noisy at a patch size of
10 pixels. We therefore selected a patch size of 14 pixels for
the matching. t
4.2 Comparison with photoclinometry models
The first inspection of the terrain models (Fig. 2a, 3a, and 4a)
shows that photogrammetry and photoclinometry reflect the
surface features seen in the images rather differently. The
photoclinometry models are smoothly shaped but clearly show
craters at large and small scale. The photogrammetry models
are rough on small scale and resolve only large and medium
sized craters. Moreover, the large scale topography seems to
differ in both models, especially in regions I and II.
We attempted a more quantitative comparison between the two
models and computed height profiles along specific image lines
(Figs. 2b, 3b, and 4b). Apparently, height differences with
respect to the regional trends of up to 600 meters occur (cf. Fig.
2a). Topography of some large-scale features such as the large
crater in Fig.3b shows striking differences. This crater is more
246
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