tours do not sufﬁciently match with the pre scribed
directions deﬁned by the roof plane gradients and
their intersections. This can be an effect of the re-
gion growing me thod applied previously. Especially,
it occu rs when small roof facets are not detected
by RANSAC, so that corresponding regions must be
ﬁlled with adjacent facets. T he maximum to le rance ε
was chosen to be sma ll eno ugh to avoid inconsisten-
cies, but the behavior does not change if the tolerance
is moderately increased from three to six pixels with-
out increasing the number of vertices M.
5 CONCLUSIONS
We have in troduced MIPs that modify p olylines un-
der directional constraints. The applicability of the
programs has been demonstrated in the context of 3D
modeling of building roofs. In this scenario, we had to
deal with a large number of contours. Therefo re, short
running times of the individual MIPs were important.
Each contour was alrea dy simpliﬁed so that it could
be described with a f ew sampled polyline vertices. In
most cases, the MIPs did not reduce the nu mber of
vertices. The maximum reduction was 21 vertices.
This resulted in MIP running times of a few millisec-
onds. When applying the MIPs to polylines with more
vertices, longer running times can be expected.
Polyline simpliﬁcation based on normals is not
limited to 3D building reconstruction. An everyday
example is p ublic transport maps th at show general-
ized paths instead of exact ones.
ACKNOWLEDGEMENTS
The authors are grateful to Dagmar Schumacher for
proof-reading and to Udo Ha nnok and Philipp Blu-
menkamp from the Krefeld land registry ofﬁce for
providing us with oblique aerial images.
REFERENCES
˚
Arøe, A. L. (2022). Detection of Edge Points of Building
Roofs from ALS Point Clouds. Norwegian University
of Science and Technology (PhD thesis), Trondheim.
Aronov, B., Asano, T., Katoh, N., Mehlhorn, K., and
Tokuyama, T. (2005). Polyline ﬁtting of planar points
under min-sum criteria. In Fleischer, R. and Trippen,
G., editors, Proc. ISAAC 2004: Algorithms and Com-
putation, volume 3341 of LNCS, pages 77–88, Berlin,
Heidelberg. Springer.
Bode, L., Weinmann, M., and Klein, R. (2022). B oundED:
Neural boundary and edge detection in 3D point
clouds via local neighborhood statistics. arXiv,
arXiv.2210.13305:1–20.
Douglas, D. and Peucker, T. (1973). Algorithms for the re-
duction of the number of points required to represent
a digitized line or its caricature. The Canadian Car-
tographer, 10(2):112–122.
Funke, S., Mendel, T., Miller, A., Storandt, S ., and Wiebe,
M. (2017). Map simpliﬁcation with topology con-
straints: Exactly and in practice. In Fekete, S.
and Ramachandran, V., editors, Proc. 19th Workshop
on Algorithm Engineering and Experiments 2017
(ALENEX17), pages 185–196, Red Hook, NY. Curran
Associates.
Goebbels, S. and Pohle-Fr¨ohlich, R. (2017). Quality en-
hancement techniques for building models derived
from sparse point clouds. In Proc. 12th International
Joint Conference on Computer Vision, Imaging and
Computer Graphics Theory and Applications – Vol-
ume 1: GRAPP, (VISIGRAPP 2017), pages 93–104.
INSTICC, SciTePress.
Gr¨oger, G., Kolbe, T. H ., Nagel, C., and H¨afele, K. H.
(2012). OpenGIS City Geography Markup Language
(CityGML) Encoding Standard. Version 2.0.0. Open
Geospatial Consortium.
Imai, H. and Iri, M. (1986). An optimal algorithm for ap-
proximating a piecewise linear function. Journal of
Information Processing, 9(3):159–162.
Lang, T. (1969). Rules for r obot draughtsmen. The Geo-
graphical Magazine, 42(1):50–51.
Li, L., Songa, N., Sun, F., Liu, X. , Wang, R., Yaoa, J., and
Cao, S. (2022). P oint2roof: End-to-end 3D building
roof modeling from airborne LiDAR point clouds. IS-
PRS Journal of Photogrammetry and Remote Sensing,
193:17–28.
Nauata, N. and Furukawa, Y. (2020). Vectorizing world
buildings: Pl anar graph reconstruction by primitive
detection and relationship inference. In Vedaldi,
A., Bischof, H., Brox, T., and Frahm, J., editors,
Proc. Computer Vision–ECCV 2020: 16th Euro-
pean Conference, Part VIII, number 12353 in LNCS,
Cham. Springer.
Pinheiro, A. M. G. and Ghanbari, M. (2010). Piecewise
approximation of contours through scale-space selec-
tion of dominant points. IEEE Transactions on Image
Processing, 19(6):1442–1450.
Ramer, U. (1972). An iterative procedure f or the polygonal
approximation of plane curves. Computer Graphics
and Image Processing, 1(3):244–256.
Reumann, K. and Witkam, A. (1973). Optimizing curve
segmentation in computer graphics. In Gunther, A.,
Levrat, B., and Lipps, H., editors, Proc. International
Computing Symposium, Davos, pages 467–472, New
York, NY. Elsevier.
Visvalingam, M. and Whyatt, J. D. (1992). Line generalisa-
tion by repeated elimination of the smallest area. Car-
tographic Information Systems Research Group, Uni-
versity of Hull.
Zhao, Z. and Saalfeld, A. (1997). Linear-time sleeve-ﬁtting
polyline simpliﬁcation algorithms. In Proc. AutoCarto
13, Seattle, WA, pages 214–223, Maryland. American
Congress on Surveying and Mapping & American So-
ciety for Photogrammetry and Remote Sensing.
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