mble over
be approx-
legree four
dge turned
id minimi-
es are flat)
ermined by
egree three
re no three
ALAS 1 it
| weakly in
ngly in the
position of
s not have
nals. Thus
s along the
is, tumbles
tend to be
nplied. In-
jined func-
0,1]. (9)
f the func-
the spring
applied to
vo subdivi-
it o-values
the net is
well. The
1 the same
on
le with the
ase a face
could not
rmanently,
n must be
ne they are
Figure 6: Approximation with surface energy minimization
used, it is sufficient to divide the patch that cannot fullfil the
criterion.
6 CONCLUDING REMARKS
Due to the parametrization of the surface over a TIN, this
approach obtains a universality which allows one to model
surfaces of abitrary topology. Because of the use of the sur-
face normals at the vertices of the triangles, it is furthermore
very flexible in the adaption of the subsistent data:
e Breaklines can be taken into account by giving two nor-
mal vectors to those surface points that are situated
along the breaklines, one for each side. Only points
lying on the appropriate side of the breakline may in-
fluence the estimation process of the surface normal.
e Structure lines, e.g. mountain ridges or the bottom of
a valley, may be taken into account by an appropriate
triangulation and choice of the surface normals and the
tangent plane field.
e Contour lines, e.g. digitized from a map, are perpen-
dicular to the surface normals, and hilltops are points
with known surface normals. This can be exploited in
the estimation of the normals by introducing constraints
for the derivatives.
This is a preliminary report on our progress in a larger re-
search project. Further studies are necessary, such as data re-
duction by appropriate preprocessing, hierarchical modelling,
refined estimation techniques and others.
Acknowledgements. This research has been supported by
the Austrian Science Foundation through project P274-PHY.
We would also like to thank H. Kager for fruitful and stimu-
lating discussions.
643
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
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