ing object “best” is up to the user who can incorporate
background information about the underlying object.
In case of the underlying building represented by D
1
(see Figures 7 and 8), for instance, we could not de-
cide whether the object is a single building complex
or if there are two separate buildings, possibly linked
by a corridor, if we solely rely on the information pro-
vided by the point set.
6 CONCLUSION AND FUTURE
WORK
In this article, we presented two methods based on
the Concave Hull Algorithm to create straight, angu-
lar and non-convex outlines of 2D point sets and pos-
sibly contained holes. The outlines of point sets rep-
resenting man-made structure, like buildings, created
by means of our methods are “better” than concave
hulls or boundaries obtained from α-shapes to the ef-
fect that our methods capture straight edges more ac-
curately and feature only few, distinct angles.
The employment of straightening concave hulls
by means of dominant directions to generate outlines
is preferable if the orientation of the underlying shape
is unknown or cannot be provided, or if the shape is
known to have edges running in only two distinct di-
rections, e.g., in case of rectangular buildings. Ad-
ditional directions could be obtained by introducing
an angle of tolerance ϕ
max
. If ](e
i
,d
ϕ
) > ϕ
max
∀ϕ ∈
{
β,γ
}
, the corresponding label vector entry is set to
zero (see Section 3.1). These edges would not be af-
fected by the straightening procedure, which is help-
ful if the building contains a narrowing or constric-
tions of angles other than the determined dominant
ones.
If information about the orientation of the shape
is available, our approach based on orthogonal pro-
jection can produce more faithful boundary polygons,
especially if the underlying shape features edges run-
ning in more than two distinct directions. More-
over, this approach forgoes the computation of con-
cave hulls or any other intermediate polygon.
The most challenging task is the reliable detection
of holes. We encountered this problem by means of
a heuristic that is subject to sampling problems and
may have several theoretical limitations. Although we
did not encounter major problems due to a rather spe-
ciﬁc, practical application, we would like to improve
on these potential issues. For example, holes might
be located close to the outer boundary of the associ-
ated point set, and they might thus share boundaries
with the exterior outline. This could cause our current
algorithms for bordering holes to fail, because the out-
line might “leave” the hole and intersect the exterior
boundary.
Moreover, we only considered “isolated” point
sets so far. In the application of building reconstruc-
tion, for example, this requires prior segmentation of
the source data. We would like to forgo this step
and determine spacings or gaps between distinct clus-
ters of points in larger point sets to segment these
while creating the corresponding outlines. This may
be accomplished by a combination of modifying and
constraining the neighborhoods employed by our ap-
proaches or the Concave Hull Algorithm, and adjust-
ments of our heuristic for hole detection to this prob-
lem, but remains future work to do.
REFERENCES
Asaeedi, S., Didehvar, F., and Mohades, A. (2013).
Alpha Convex Hull, a Generalization of Convex
Hull. The Computing Research Repository (CoRR),
abs/1309.7829.
Bendels, G. H., Schnabel, R., and Klein, R. (2006). De-
tecting Holes in Point Set Surfaces. Proceedings of
The 14th International Conference in Central Europe
on Computer Graphics, Visualization and Computer
Vision, 14.
de Berg, M., Cheong, O., van Krevald, M., and Overmars,
M. (2008). Computation Geometry. Springer, 3rd edi-
tion.
Douglas, D. and Peucker, T. (1973). Algorithms for the Re-
duction of the Number of Points Required to Repre-
sent a Digitized Line or Its Caricature. The Canadian
Cartographer, 10(2):112 –– 122.
Duckham, M., Kulik, L., Worboys, M., and Galton, A.
(2008). Efﬁcient Generation of Simple Polygons for
Characterizing the Shape of a Set of Points in the
Plane. Pattern Recognition, 41(10):3224–3236.
Edelsbrunner, H., Kirkpatrick, D., and Seidel, R. (1983).
On the Shape of a Set of Points in the Plane. IEEE
Transactions on Information Theory, 29(4):551 – 559.
Galton, A. and Duckham, M. (2006). What is the Region
Occupied by a Set of Points? In Proceedings of the 4th
International Conference on Geographic Information
Science, GIScience’06, pages 81–98, Berlin, Heidel-
berg. Springer-Verlag.
Jarvis, R. A. (1973). On the Identiﬁcation of the Convex
Hull of a Finite Set of Points in the Plane. Information
Processing Letters, 2(1):18 – 21.
Moreira, A. and Santos, M. Y. (2007). Concave hull: A
k-nearest Neighbours Approach for the Computation
of the Region Occupied by a Set of Points. Proceed-
ings of the 2nd International Conference on Computer
Graphics Theory and Applications (GRAPP), pages
61 – 68.
Ramer, U. (1972). An Iterative Procedure for the Polygonal
Approximation of Plane Curves. Computer Graphics
and Image Processing, 1(3):244 – 256.
GRAPP 2016 - International Conference on Computer Graphics Theory and Applications
70