Maike Buchin

An algorithmic framework for segmenting trajectories based on spatio-temporal criteria

In this paper we address the problem of segmenting a trajectory such that each segment is in some sense homogeneous. We formally define different spatio-temporal criteria under which a trajectory can be homogeneous, including location, heading, speed, velocity, curvature, sinuosity, and curviness. We present a framework that allows us to segment any trajectory into a minimum number of segments under any of these criteria, or any combination of these criteria. In this framework, the segmentation problem can generally be solved in O(n log n) time, where n is the number of edges of the trajectory to be segmented.

Constrained free space diagrams: a tool for trajectory analysis

Time plays an important role in the analysis of moving object data. For many applications it is not sufficient to only compare objects at exactly the same times, or to consider only the geometry of their trajectories. We show how to leverage between these two approaches by extending a tool from curve analysis, namely the free space diagram. Our approach also allows us to take further attributes of the objects like speed or direction into account. We demonstrate the usefulness of the new tool by applying it to the problem of detecting single file movement. A single file is a set of moving entities, which are following each other, one behind the other. Our algorithm is the first one developed for detecting such movement patterns. For this application, we analyse demonstrate the performance of our tool both theoretically experimentally.

Finding long and similar parts of trajectories

A natural time-dependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly self-intersecting lines with time stamps. For the case when a minimum duration of the subtrajectories is specified and the subtrajectories must start at corresponding times, we give a linear-time algorithm. The algorithm is based on a result of independent interest: We present a linear-time algorithm to find, for a piece-wise monotone function, an interval of at least a given length that has minimum average value. In the case that the subtrajectories may start at non-corresponding times, it appears difficult to give exact algorithms, even if the duration of the subtrajectories is fixed. For this case, we give (1+@e)-approximation algorithms, for both fixed duration and when only a minimum duration is specified.

Detecting Commuting Patterns by Clustering Subtrajectories

In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Frechet distance and the discrete Frechet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the ‘longest’ subtrajectory cluster is as hard as MaxClique to compute and approximate.

Computing the Fréchet distance between simple polygons in polynomial time

We present the first polynomial-time algorithm for computing the Fréchet for a non-trivial class of surfaces: simple polygons. For this, we show that it suffices to consider homeomorphisms that map an arbitrary triangulation of one polygon to the other polygon such that diagonals of the triangulation are mapped to shortest paths in the other polygon.

Detecting single file movement

We study the problem of detecting a single file behavior in a set of trajectories. A group of entities is moving in single file if they are following each other, one behind the other. This movement pattern occurs often, among animals, humans, and vehicles. It is challenging to detect because it does not have a fixed layout. In this paper we first model the notion of following behind, on which we base our definition of single file. We present efficient algorithms for detecting following behind and single file behaviors. We test and evaluate these algorithms on real and generated test data.

Analysis of movement data

The study of movement is progressing rapidly as a subdiscipline in Geographic Information Science (GIScience). At the fulcrum of this new research area in GIScience are movement observations. Movement observations may be understood as spatiotemporal signals, which carry information on the movement of dynamic entities and the underlying mechanisms that drive their movement. These observations are key to the study and understanding of movement. Technological advancements in global positioning systems (GPS) and related satellite tracking technologies have resulted in significant increases in the availability of highly accurate data on moving phenomena, dramatically outpacing the development of appropriate methods with which to analyze them. In addition to increased spatial accuracy and temporal resolution of the locational information, improvements are being made to accelerometers and ‘biologgers’ that enable the collection of ancillary behavioral and physiological information. This special issue emerged from a pre-conference event associated with the GIScience 2014 conference held in Vienna: a workshop organized by the authors on ‘Analysis of Movement Data’ (AMD 2014). The workshop and this special issue explore recent trends in the study of movement and novel methods for analyzing and contextualizing movement data. A broad range of topics is covered concerning movement analysis, representation, and modeling. The studies presented use movement data from different domains, such as transportation (vehicles, marine traffic), cyclists and athlete tracking data, storm events, and movement ecology (birds, mammals, etc.). This editorial intends to frame and position the papers included in this special issue and to provide recommendations for future directions in the analysis of movement data. In order to frame the work presented here, we use the overarching research framework for the study of movement proposed by Dodge (2015). This framework, shown in an adapted version in Figure 1, posits that the study of movement consists of a continuum of research ranging from understanding movement to construct knowledge of the behavior of dynamic objects, to using this knowledge for modeling and prediction of movement. Visualization facilitates this process through data exploration, hypothesis generation, and communication of the outcomes (Andrienko et al. 2013, Wood et al. 2011, Zhang et al. 2013, Xavier and Dodge 2014). The framework relies on an iterative validation process, where analytics and models are parameterized, calibrated, and improved using real movement observations. Understanding movement, shown on the right side of Figure 1, entails development of methods for quantification of movement and its parameters (Dodge et al. 2008, Long and Nelson 2013, Laube 2014, Demšar et al. 2015); analysis of its context INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SCIENCE, 2016 VOL. 30, NO. 5, 825–834 http://dx.doi.org/10.1080/13658816.2015.1132424

Can We Compute the Similarity between Surfaces

A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all.Using a discrete approximation, we show that it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a decreasing sequence of rationals converging to the Fréchet distance. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below a specified value, is recursively enumerable.Furthermore, we show that a relaxed version of the Fréchet distance, the weak Fréchet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Fréchet distance in a geometric data structure called the Free Space Diagram.

Computing the Fréchet distance between simple polygons

We present the first polynomial-time algorithm for computing the Frechet distance for a non-trivial class of surfaces: simple polygons, i.e., the area enclosed by closed simple polygonal curves, which may lie in different planes. For this, we show that we can restrict the set of maps realizing the Frechet distance, and develop an algorithm for computing the Frechet distance using the algorithm for curves, techniques for computing shortest paths in a simple polygon, and dynamic programming.

Context-Aware Similarity of Trajectories

The movement of animals, people, and vehicles is embedded in a geographic context. This context influences the movement. Most analysis algorithms for trajectories have so far ignored context, which severely limits their applicability. In this paper we present a model for geographic context that allows us to integrate context into the analysis of movement data. Based on this model we develop simple but efficient context-aware similarity measures. We validate our approach by applying these measures to hurricane trajectories.