Trajectories and Their Representations - Mobility Data - page 9

(a) (b) (c)
Figure 1.4 The three kinds of representations of movement: continuous, discrete, and
stepwise.
implementations, such as the movement track. In order to satisfy applications
that need a continuous view of trajectories, the discrete implementation may be
enriched with interpolation functions that allow dynamically reconstructing a
continuous representation of the discretized trajectory.
Definition 1.3. A continuous representation of a trajectory (or continuous tra-
jectory in short) is a trajectory representation that describes in a continuous way
the movement of the object for the time interval [ t Begin ,t End ] of the trajectory. It
usually consists of a finite sequence of spatio-temporal positions, and the inter-
polation functions that enable the computation of the spatio-temporal position
of the moving object for any instant in [ t Begin ,t End ].
Whenever the movement track is too sparse for inferring the original con-
tinuous movement of the object, or the applications do not need the continuous
movement, the finite sequence of spatio-temporal positions is used as a discrete
representation of a trajectory. Currently this is the case, for example, of the
applications that use the movement tracks generated by social networks (see
Chapter 16 ).
Definition 1.4. A discrete representation of a trajectory (or discrete trajectory
in short) is a trajectory representation that is made up of the finite list of spatio-
temporal positions for the time interval [ t Begin ,t End ] of the trajectory, but not
providing the continuity of the movement of the object.
Figure 1.4 a visualizes (as a line) a continuous representation of a trajectory.
Figure 1.4 b visualizes (as a set of points) a discrete representation. Figure 1.4 c
visualizes a stepwise (segmented) representation (see Section 1.3 ).
To complete the basic picture we briefly introduce two trajectory concepts,
holes and semantics gaps, which address the understanding of missing points
at the conceptual level. These concepts contribute to a more complete vision of
trajectories. The reader has to be aware that they only play an important role in
a limited number of application cases, which explains why researchers rarely
take these concepts into account.
The term missing point denotes the existence, within a movement track, of an
abnormal (longer than the sampling rate) temporal gap between two consecutive