THE INTERMEDIARY MODEL
When it comes to surface fitting, the most important problem is
that the real surface is generally different from the model
surface. As mentioned previously, an index for the quality of the
adjustment is the variation of the distance between the points
that are used for the adjustment and the mathematically defined
surface. Assuming that the value of the variation (Go) is
0.001m° and that the points follow the normal distribution, this
means that 68% of the points are within a 3 cm distance from
the surface, 95% of the points are within 6cm and 99% of the
points are within 9cm. Furthermore, the error in the radius of
the model indicates the very same thing. Taking this into
account and assuming that a Mercator projection (Bugayevskiy
& Snyder, 1995) is going to be used, the derivatives of the
cartographic relationships with respect to the radius (R) of the
model, which indicate the way that the error in the radius affects
the x and y position of a point in the projection plane, are:
(5)
dx = À dR
dy 7 In(atan(z/4 + q/2)) dR
where dx, dy- the error in the position of a point on the
projection plane
dR - the error in the radius
A, @ = the longitude and latitude of a point
In the Figures (2) and (3), two graphs are presented to show the
magnitude of the error in the x, y position of a point on the
projection plane with respect to the values of the longitude and
latitude. In each graph three series of data are presented in order
for the reader to be able to estimate the magnitude of the error
in the position, with respect to errors of different magnitude in
the radius.
This fact obviously causes problems in mosaicing as the one-to-
one correspondence between the points of the surface of the
object and the points of the model surface cannot be ensured.
This very problem is indicated in Figure (4).
(PP)-
Figure 4. The problem caused due to the difference between
the model and the actual object
In Figure (4), PP is the projection plane, O is the object surface,
S is the surface of the sphere and Xg, Ys, Zs is the centre of the
sphere. Beginning from a random position of the projection
plane and applying the projection relationships, a point on the
surface of the sphere is defined. As shown in Figure (4), the
point that belongs on the sphere does not also belong on the
surface of the object and when applying the collinearity
equation for a stereo-pair, the images of two distinct points of
the object surface are obtained.
In order for mosaicing to be possible, such problems should not
appear. It is obvious that the problem could be solved if, insteaq
of a point that belongs on the sphere, a point that belongs on the
surface of the object could be obtained.
The solution to the problem is given by the creation of an
intermediary model, which is based on the DEM data. This
model helps to ensure the one-to-one correspondence between
the points of the two surfaces and it is entirely based on
directions.
After the new system is completely defined, the positions of the
points of the DEM are expressed in this very system in spherical
coordinates and the area of the object can be defined by the
minimum and maximum longitude and latitude. This
information is used for the creation of the intermediary model,
which basically is a matrix. The breadth of the values in
longitude and latitude is used for the determination of the size
of the matrix. In this matrix the rows correspond to integer
values of latitude whereas the columns correspond to integer
values of longitude. Furthermore, in order to avoid trimming the
edges of the object, the size of the matrix is increased. Each cell
of the matrix contains the mean distance between the surface of
the object and the centre of the model in the direction indicated
by its position in the matrix. The construction of the matrix is
done cell by cell. Beginning from the position of the cell (row,
column), the corresponding values of latitude and longitude are
calculated. The latitude and longitude are then used to detect all
the points of the DEM within a search area 3 degrees wide.
When the points of this area are detected, their mean distance
from the centre is calculated and this value fills the
corresponding cell. When no points of the DEM are detected in
the search area, the value of the cell is set to the radius of the
model. This way, a normalized model of the surface is obtained.
The whole idea might be very simple but it gave a very
satisfactory solution to the problem of mosaicing.
CARTOGRAPHIC DEVELOPMENT CREATION
The next stage is the creation of the developed images. Given a
specific cartographic projection and the DEM of an area of the
object, the corresponding area of the developed image on the
projection plane can be defined. For each position (x, y) the
colour has to be obtained; using the inverse cartographic
relationship, the corresponding latitude and longitude values are
calculated. Instead of using the radius of the model, the distance
between the surface of the object and the centre of the model is
used. This parameter is found by interpolation on the
intermediary model. This way, the full spherical coordinates of
the position are obtained and then expressed in the reference
system of the model in cartesian coordinates. Finally, the
position is expressed in the initial geodetic system, and using
the collinearity condition, the corresponding position on the
photographic plane is determined; using the parameters of the
interior orientation, the position on the digital image is found
and the colour is obtained by interpolation.
The equations that describe the relationship between a position
on the matrix of the intermediary model and the corresponding
direction in the 3D space are:
—466—
(4)
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n-l-fix(mi
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