International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
Structures, which describe Z- or L-shapes, cannot be handled by
mathematical morphology. For them, a diffusion part is
necessary, which is termed "curvature-space" in (Mayer 1998).
[n contrast to mathematical morphology only specific segments
are shifted. The choice which segment has to be moved in what
direction depends on the local curvature. Curvature is equated
with the occurrence of short segments and (Mayer 1998)
distinguishes different treatments for U-, L- and Z-structures
(cf. Fig.6).
2.3 Scale-Spaces for 3D Building Data
Our first idea for a 3D generalization also consisted of a
sequential processing of mathematical morphology and
curvature space. Implemented in Visual C++, using the ACIS
class library (www.spatial.com), mathematical morphology was
easy to realize by an incremental movement of all facets
inwards or outwards with respect to the direction of the normal
(cf. Fig.5).
Figure 5: From left to right: original building, result after
erosion (split), and result after dilation (merge) in 3D.
Again split, merge, and the elimination of protrusions and
notches are possible. Box-structures and step- / stair-structures,
which correspond to the L- and Z-structure of the 2D case (cf.
Fig.6), have to be handled with curvature space.
U-structure L-structure
2D t
Curvature Space
Z-structure
Hr
Morphology
3D
Protrusion Box-structures Step-structure
Figure 6: U-, L-, and Z-structures in 2D versus protrusions,
box-, and step- / stair-structures in 3D
In (Forberg and Mayer 2002) a complex procedure for
curvature space in 3D was introduced. It is based on the
analysis of the convexity and concavity of vertices and their
relations within facets. Complicated rules were devised, but it
was still not guaranteed, that the result is satisfying. Due to the
more complex geometry of 3D objects we were not able to
consider all cases.
Therefore, instead of using a sequential combination of
mathematical morphology and curvature space, a new approach
has been developed. lt integrates the treatment of external
events, protrusions, as well as box- and step- / stair-structures in
one single procedure, following a rather simple principle:
Parallel facets are determined and the distance between them is
computed. If the distance is under a certain threshold, the facets
are shifted towards each other until they merge into one facet,
regardless of the direction of the facet's normal with respect to
the inside or outside of the object (cf. Fig.7).
à + e |
^ yp
|
e
|
Figure 7. New approach for the generalization of 3D building
models. Parallel facets under a certain distance are shifted
towards each other.
This rather simple procedure is very general and therefore
suitable to cover also complex combinations of orthogonal
structures. Besides its simple implementation, it has got the
advantage, opposite, e.g., to the approach of (Thiemann 2002),
that small structures do not simply vanish, but a shape
adjustment takes place (cf. Fig. 8).
Figure 8: Object parts are not only eliminated, but they are
adjusted, so that the characteristic shape is kept.
To intensify this shape-preservation, the distance of the
movement for each facet can be weighted depending on the
area-relation between the two parallel facets. By now, no
weighting algorithm was found, that is suitable for all kinds of
orthogonal building structures. Because of that, for the results
presented in the remainder of this paper a simple half distance
movement was employed. Examples for external events
obtained with the approach are given in Figure 9 and 10. Figure
11 shows the elimination of protrusions and a notch, whereas in
Figure 12 protrusions and an inward pointing box-structure
disappear.
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