Progressive Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.Comment: 20 pages, 20 figure

Visual analytics of delays and interaction in movement data

Maximilian Konzack, Tim Ophelders, Michel A. Westenberg and Kevin Buchin are supported by the Netherlands Organisation for Scientific Research (NWO) under grant no. 612.001.207 (Maximilian Konzack, Michel A. Westenberg and Kevin Buchin) and grant no. 639.023.208 (Tim Ophelders).The analysis of interaction between movement trajectories is of interest for various domains when movement of multiple objects is concerned. Interaction often includes a delayed response, making it difficult to detect interaction with current methods that compare movement at specific time intervals. We propose analyses and visualizations, on a local and global scale, of delayed movement responses, where an action is followed by a reaction over time, on trajectories recorded simultaneously. We developed a novel approach to compute the global delay in subquadratic time using a fast Fourier transform (FFT). Central to our local analysis of delays is the computation of a matching between the trajectories in a so-called delay space. It encodes the similarities between all pairs of points of the trajectories. In the visualization, the edges of the matching are bundled into patches, such that shape and color of a patch help to encode changes in an interaction pattern. To evaluate our approach experimentally, we have implemented it as a prototype visual analytics tool and have applied the tool on three bidimensional data sets. For this we used various measures to compute the delay space, including the directional distance, a new similarity measure, which captures more complex interactions by combining directional and spatial characteristics. We compare matchings of various methods computing similarity between trajectories. We also compare various procedures to compute the matching in the delay space, specifically the Fréchet distance, dynamic time warping (DTW), and edit distance (ED). Finally, we demonstrate how to validate the consistency of pairwise matchings by computing matchings between more than two trajectories.Publisher PDFPeer reviewe

Progressive simplification of polygonal curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an O(n3m)-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fréchet and area-based distances, and enables simplification for continuous scaling in O(n5) time. To speed up this algorithm in practice, a technique is presented for efficiently constructing many so-called shortcut graphs under the Hausdorff distance, as well as a representation of the shortcut graph that enables us to find shortest paths in anticipated O(nlog⁡n) time on spatial data, improving over O(n2) time using existing algorithms. Experimental evaluation of these techniques on geospatial data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage

Visual exploration of migration patterns in gull data

We present a visual analytics approach to explore and analyze movement data as collected by ecologists interested in understanding migration. Migration is an important and intriguing process in animal ecology, which may be better understood through the study of tracks for individuals in their environmental context. Our approach enables ecologists to explore the spatio-temporal characteristics of such tracks interactively. It identifies and aggregates stopovers depending on a scale at which the data is visualized. Statistics of stopover sites and links between them are shown on a zoomable geographic map which allows to interactively explore directed sequences of stopovers from an origin to a destination. In addition, the spatio-temporal properties of the trajectories are visualized by means of a density plot on a geographic map and a calendar view. To evaluate our visual analytics approach, we applied it on a data set of 75 migrating gulls that were tracked over a period of 3 years. The evaluation by an expert user confirms that our approach supports ecologists in their analysis workflow by helping to identifying interesting stopover locations, environmental conditions or (groups of) individuals with characteristic migratory behavior, and allows therefore to focus on visual data analysis

Visual exploration of migration patterns in gull data

\u3cp\u3eWe present a visual analytics approach to explore and analyze movement data as collected by ecologists interested in understanding migration. Migration is an important and intriguing process in animal ecology, which may be better understood through the study of tracks for individuals in their environmental context. Our approach enables ecologists to explore the spatio-temporal characteristics of such tracks interactively. It identifies and aggregates stopovers depending on a scale at which the data is visualized. Statistics of stopover sites and links between them are shown on a zoomable geographic map which allows to interactively explore directed sequences of stopovers from an origin to a destination. In addition, the spatio-temporal properties of the trajectories are visualized by means of a density plot on a geographic map and a calendar view. To evaluate our visual analytics approach, we applied it on a data set of 75 migrating gulls that were tracked over a period of 3 years. The evaluation by an expert user confirms that our approach supports ecologists in their analysis workflow by helping to identifying interesting stopover locations, environmental conditions or (groups of) individuals with characteristic migratory behavior, and allows therefore to focus on visual data analysis.\u3c/p\u3