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Stilla, Uwe

Figure 5. Parallel line segments (green) form decomposition
lines (red), rendering short segments in between (orange)
Under the general assumption that ridge and eaves lines should
strictly run horizontally, many roof shapes require the ground
shape of cells to be trapezoids or rhomboids. Otherwise not all
roof faces will be planar and must be split into triangles to form
valid solids. Figure 6 shows an extreme example of a cell with a
Berliner roof shape where none of the four sides of the ground
shape are parallel. The middle face of the roof must be split into
two triangles, which is generally not acceptable and should be
avoided if possible. Due to the averaging process, the set of
resulting decomposition lines are not guaranteed to be parallel.
We therefore adjust the decomposition lines slightly so that
parallelism and rectangularity are enforced for pairs of
decomposition lines with small directional deviations. The same
Berliner roof shape of Figure 6 with a trapezoid ground shape
results in a valid solid after adjustment.
Roof Shape Determination
Now that a cell decomposition of the footprint is available, the
parameterized roof shapes of all cells need to be found. We do
this by examining the normal vectors of all points inside the
same cell. As point normal vectors are usually not given in
surface models, they first have to be generated. If the surface
model is structured as a grid, we compute the normal vector of
each point from the eight triangles fanned around it and average
their normal vectors. However, if the raw data is available in
form of an unstructured point cloud, we estimate a point’s local
plane of regression from its five nearest neighbours and take the
resulting surface normal vector.
For the construction of the building’s roof, we classify the roof
shapes that we use in our approach into three types: basic,
connecting and manual shapes. Whereas the shapes of the first
two classes can be determined in an automatic process, the last
Figure 6. Extreme example of a Berliner roof primitive with a
non-parallel before and a trapezoid roof shape after adjustment.
Once the decomposition lines have been generated, a rectangle
approximately two times the minimal bounding rectangle is
taken and split by these lines, forming nonintersecting cells in
the process. Then the cells are compared with the original
footprint, and the ones with a low overlap value are discarded.
Large cells assure that this classification fails only in few cases.
Figure 7 shows an example cell decomposition of a given
footprint. Cells with a low overlap with the original footprint
were discarded in the process. The four “horizontal” lines are
pair wise parallel, whereas the five “vertical” lines are all
Figure 8. Flat, shed, gabled, hipped and Berliner roof shape.
As not all houses have only one section, there is a need to
connect the roofs of the sections with specific junction shapes.
Figure 9 shows a small selection of connecting roof shapes.
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Figure 9. Examples of connecting roof shapes.