When users zoom in or out on a digital map, the map should change correspondingly to present geographical information at proper levels. A way to help map users better keep track of their interested objects is to change the map smoothly instead of discretely switching between several levels of detail. This paper focuses on the problem of providing smooth merging of area objects. We propose to merge multiple areas simultaneously to share their animation durations. In this way, each merging operation can be prolonged, and it is visually smoother. We present a greedy algorithm to decide which areas should be merged at each step. The merging process is pre-computed and is recorded into a space-scale cube (SSC). When a user accesses our web map, the SSC is sent to the client side so that the map can be generated by slicing the SSC in the graphics processing unit (GPU). We also explain how to snap the zooming to valid states so that the zooming will not stop halfway of the merging operations. Our case study shows that it is visually smoother to merge simultaneously than to sequentially merge each pair of areas.

To provide users with maps of different scales and to allow them to zoom in and out without losing context, automatic methods for map generalization are needed. We approach this problem for land-cover maps. Given two land-cover maps at two different scales, we want to find a sequence of small incremental changes that gradually transforms one map into the other. We assume that the two input maps consist of polygons, each of which belongs to a given land-cover type. Every polygon on the smaller-scale map is the union of a set of adjacent polygons on the larger-scale map. In each step of the computed sequence, the smallest area is merged with one of its neighbors. We do not select that neighbor according to a prescribed rule but compute the whole sequence of pairwise merges at once, based on global optimization. We have proved that this problem is NP-hard. We formalize this optimization problem as that of finding a shortest path in a (very large) graph. We present the Ag algorithm and integer linear programming to solve this optimization problem. To avoid long computing times, we allow the two methods to return non-optimal results. In addition, we present a greedy algorithm as a benchmark. We tested the three methods with a dataset of the official German topographic database ATKIS. Our main result is that Ag finds optimal aggregation sequences for more instancesthan the other two methods within a given time frame.

Computing trajectory similarity is a fundamental operation in movement analytics, required in search, clustering, and classification of trajectories, for example. Yet the range of different but interrelated trajectory similarity measures can be bewildering for researchers and practitioners alike. This paper describes a systematic comparison and methodical exploration of trajectory similarity measures. Specifically, this paper compares five of the most important and commonly used similarity measures: dynamic time warping (DTW), edit distance (EDR), longest common subsequence (LCSS), discrete Fréchet distance (DFD), and Fréchet distance (FD). The paper begins with a thorough conceptual and theoretical comparison. This comparison highlights the similarities and differences between measures in connection with six different characteristics, including their handling of a relative versus absolute time and space, tolerance to outliers, and computational efficiency. The paper further reports on an empirical evaluation of similarity in trajectories with contrasting properties: data about constrained bus movements in a transportation network, and the unconstrained movements of wading birds in a coastal environment. A set of four experiments: a. creates a measurement baseline by comparing similarity measures to a single trajectory subjected to various transformations; b. explores the behavior of similarity measures on network-constrained bus trajectories, grouped based on spatial and on temporal similarity; c. assesses similarity with respect to known behavioral annotations (flight and foraging of oystercatchers); and d. compares bird and bus activity to examine whether they are distinguishable based solely on their movement patterns. The results show that in all instances both the absolute value and the ordering of similarity may be sensitive to the choice of measure. In general, all measures were more able to distinguish spatial differences in trajectories than temporal differences. The paper concludes with a high-level summary of advice and recommendations for selecting and using trajectory similarity measures in practice, with conclusions spanning our three complementary perspectives: conceptual, theoretical, and empirical.

Traditionally, the content for vario-scale maps has been created using a ‘one fits all’ approach equal for all scales. Initially only the delete/merge operation was used to create the vario-scale data using the importance and the compatibility functions defined at class level (and evaluated at instance level) to create the tGAP structure with planar partition as basis. In order to improve the generalization quality other operators and techniques have been added during the past years; e. g. simplify, collapse (change area to line representation), split, attractiveness regions and the introduction of the concept of linear network topology. However, the decision which operation to apply has been hard coded in our software, making it not very flexible. Further, we want to include awareness of the current scale when deciding what generalization operation to apply. For this purpose we propose the scale dependent framework (SDF), which at its core contains the encoding of the generalization knowledge in the SDF conceptual model. This SDF model covers the representation of scale dependent class importance, scale dependent class compatibility values, scale dependent attractiveness regions and last but not least specification of generalization operations that are scale and class dependent. By changing the settings in the SDF configuration and re-running the vario-scale generalization process, we can easily experiment in order to find best settings (for specific map user needs). In this paper we design the SDF conceptual model and explicitly motivate and define the scope of its expressiveness. We further present the improved scale dependent tGAP creation software and present initial results in the form of better created vario-scale map content.