efﬁcient feature space for the clusters. Further, they
propose a Mahalanobis classiﬁer to detect anomalies
in the data. A self-organizing map is used to estimate
the similarities between trajectories. The algorithm is
validated by using different datasets, i.e. identiﬁca-
tion of sign language and video surveillance footage.
Melo et al. (2006) propose a feature space using
low-degree polynomials for the detection and classiﬁ-
cation of road lanes. For the clustering of similar tra-
jectories a K-Means algorithm is used. The different
lanes are further classiﬁed into different categories.
The proposed algorithm is tested with real data.
Dahlbom and Niklasson (2007) use splines to rep-
resent the main trajectory of a cluster. Therefore, the
underlying data is clustered using the mean of the nor-
malized distances between each trajectory point and
its nearest cluster point. Afterwards, the clusters are
estimated by using splines in order to reduce the com-
plexity of the representation.
Especially in the maritime domain, anomaly de-
tection is an active ﬁeld of research in itself, e.g.,
de Vries and van Someren (2012) use piecewise lin-
ear segmentation methods to compress trajectories of
maritime vessels. These compressed trajectories are
then clustered and anomaly detection is performed by
using kernel methods. Furthermore, expert domain
knowledge is incorporated. The algorithms are vali-
dated with a dataset from the Netherlands’ coast near
Rotterdam.
An algorithm that estimates a mean path for nor-
mal routes is proposed by Rosen and Medvedev
(2012). The mean path is deﬁned as the trajectory
which minimizes the euclidean distance to every other
trajectory in the same cluster. Anomalies are then de-
tected by comparing a new trajectory with this mean
path and an anomaly score is calculated. The algo-
rithm is evaluated by using simulated data as well as
a real dataset.
Guillarme and Lerouvreur (2013) propose an un-
supervised algorithm for modeling routes by using
data from a satellite based Automatic Identiﬁcation
System (AIS). First, the recorded trajectories are par-
titioned by using a stops and moves of trajectories al-
gorithm. The move parts of the trajectories are further
divided by using a piecewise linear segmentation or a
sliding window approach. This results in segments of
similar movement. These segments are clustered us-
ing the OPTICS algorithm. Afterwards, hand-picked
clusters are used for modeling the vessels’ motion-
patterns. First results for this algorithm using real data
are illustrated in the paper.
Shao et al. (2014) use a fuzzy k-nearest neighbors
and fuzzy c-means approach to conduct trajectory
correlation and clustering. Therefore, fuzzy logic is
utilized to model uncertainties in the tracks. The pro-
posed algorithms are evaluated using different types
of sensing systems.
Fischer et al. (2014) present a method to model
speciﬁc situations based on dynamic Bayesian net-
works. The main idea is to utilize expert knowledge
to describe situations of interest. For the evaluation
a speciﬁc situation, namely an incoming suspicious
smuggling vessel, is modeled and the results for dif-
ferent parameters are shown. The described situation
is translated from a situational dependency network
to a dynamic Bayesian network. Therefore, several
parameters must be chosen. Fischer et al. present a
possible approach to automatically specify these pa-
rameters.
3 ALGORITHM
A trajectory recorded by a surveillance system cannot
easily be compared to another due to
• different lengths,
• different sample rates, and
• different numbers of points.
In order to compare different trajectories several ap-
proaches are possible. E.g., dynamic time warping
is used by Vakanski et al. (2012) to learn trajectories
demonstrated by human in order to program a robot.
Laxhammar and Falkman (2011) use the Hausdorff
metric to compare two trajectories. Here, a trajectory
will be represented in a way, that its complexity will
be reduced and direct comparison between itself and
other trajectories will be possible.
A trajectory has a speciﬁc length n and consists of
multiple points p
i
= (p
i,lon
, p
i,lat
)
T
with i = 1, . . . , n,
where p
i,lon
is the longitude and p
i,lat
is the latitude
position of the i-th point. Therefore, a trajectory t is
given by t = {p
i
| i = 1, . . . , n}. The idea is now to
reduce the number of points, in such a way, that the
resulting trajectory can be compared by using e.g. the
euclidean distance.
Therefore, the trajectory is estimated by using a
b-spline representation. A b-spline interpolation con-
nects several (cubic) functions to interpolate a given
set of points. To assess a new trajectory, it has to be
compared with recorded normal trajectories. Hence,
a normal model of the trajectory in the observed area
will be generated by using machine learning algo-
rithms.
Anomaly Detection using B-spline Control Points as Feature Space in Annotated Trajectory Data from the Maritime Domain
251