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Discretized Approaches to Schematization
[article]

2016
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arXiv
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pre-print

To produce cartographic maps, simplification is typically used to reduce complexity of the map to a legible level. With schematic maps, however, this simplification is pushed far beyond the legibility threshold and is instead constrained by functional need and resemblance. Moreover, stylistic geometry is often used to convey the schematic nature of the map. In this paper we explore discretized approaches to computing a schematic shape S for a simple polygon P. We do so by overlaying a plane

arXiv:1606.06488v1
fatcat:h33mxq7mobdyxlaars6v33novm
## more »

... h G on P as the solution space for the schematic shape. Topological constraints imply that S should describe a simple polygon. We investigate two approaches, simple map matching and connected face selection, based on commonly used similarity metrics. With the former, S is a simple cycle C in G and we quantify resemblance via the Fréchet distance. We prove that it is NP-hard to compute a cycle that approximates the minimal Fréchet distance over all simple cycles in a plane graph G. This result holds even if G is a partial grid graph, if area preservation is required and if we assume a given sequence of turns is specified. With the latter, S is a connected face set in G, quantifying resemblance via the symmetric difference. Though the symmetric difference seems a less strict measure, we prove that it is NP-hard to compute the optimal face set. This result holds even if G is full grid graph or a triangular or hexagonal tiling, and if area preservation is required. Moreover, it is independent of whether we allow the set of faces to have holes or not.##
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Map Matching with Simplicity Constraints
[article]

2013
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arXiv
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pre-print

We study a map matching problem, the task of finding in an embedded graph a path that has low distance to a given curve in R^2. The Fr\'echet distance is a common measure for this problem. Efficient methods exist to compute the best path according to this measure. However, these methods cannot guarantee that the result is simple (i.e. it does not intersect itself) even if the given curve is simple. In this paper, we prove that it is in fact NP-complete to determine the existence a simple cycle

arXiv:1306.2827v1
fatcat:4ov3wqstkner3gwzvpug3qgynm
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... n a planar straight-line embedding of a graph that has at most a given Fr\'echet distance to a given simple closed curve. We also consider the implications of our proof on some variants of the problem.##
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Locally Correct Frechet Matchings
[article]

2012
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arXiv
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pre-print

*Meulemans*and B. Speckmann are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707. ...

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Obstructing Classification via Projection
[article]

2021
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arXiv
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pre-print

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space R^d and each point is labeled with k binary-valued properties. A priori

arXiv:2105.09047v1
fatcat:jbqdt2n2qvc5dmp5qgm2guubtq
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... e assume that it is "easy" to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space R^m (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to R^(d-1) suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear "inseparability" of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities.##
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Near-Delaunay Metrics
[article]

2021
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arXiv
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pre-print

We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which

arXiv:2106.11621v1
fatcat:b2cwie7ybfdwveg72enwqivmti
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... ulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.##
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Locally correct Fréchet matchings

2018
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Computational geometry
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*Meulemans*and B. Speckmann are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707. ...

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Short Plane Supports for Spatial Hypergraphs
[article]

2018
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arXiv
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pre-print

A graph G=(V,E) is a support of a hypergraph H=(V,S) if every hyperedge induces a connected subgraph in G. Supports are used for certain types of hypergraph visualizations. In this paper we consider visualizing spatial hypergraphs, where each vertex has a fixed location in the plane. This is the case, e.g., when modeling set systems of geospatial locations as hypergraphs. By applying established aesthetic quality criteria we are interested in finding supports that yield plane straight-line

arXiv:1808.09729v1
fatcat:i22h4hs4obh2jg6dqonpjb4p7q
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... ngs with minimum total edge length on the input point set V. We first show, from a theoretical point of view, that the problem is NP-hard already under rather mild conditions as well as a negative approximability results. Therefore, the main focus of the paper lies on practical heuristic algorithms as well as an exact, ILP-based approach for computing short plane supports. We report results from computational experiments that investigate the effect of requiring planarity and acyclicity on the resulting support length. Further, we evaluate the performance and trade-offs between solution quality and speed of several heuristics relative to each other and compared to optimal solutions.##
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Route Reconstruction from Traffic Flow via Representative Trajectories
[article]

2020
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arXiv
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pre-print

Understanding human mobility is an important aspect of traffic analysis and urban planning. Trajectories provide detailed views on specific routes, but typically do not capture all traffic. Loop detectors capture all traffic flow at specific locations instead, but provide no information on individual routes. Given a set of loop-detector measurements and a set of representative trajectories, our goal is to investigate how one can effectively combine these two partial data sources to create a

arXiv:2012.05019v1
fatcat:72ynplxjlbf75oln72aqli6q7e
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... complete picture of the underlying mobility. Specifically, we want to reconstruct a realistic set of routes from the loop-detector data, using the given trajectories as representatives of typical behavior. We model loop-detector data as a network flow that needs to be covered by the reconstructed routes and we capture realism of the routes via the Fr\'echet distance to the representatives. We prove that several forms of the resulting problem are NP-hard. Hence we explore heuristics that decompose the flow well while following the representatives to varying degrees. First we propose the Fr\'echet Routes (FR) heuristic which generates candidates routes with bounded Fr\'echet distance. Second we describe a variant of multi-commodity min-cost flow (MCMCF) which is loosely coupled to the trajectories. Lastly we consider global min-cost flow (GMCF) which is essentially agnostic to the representatives. We evaluate these approaches on synthetic and real-world trajectory data with a map-matched ground truth. We find that GMCF explains the flow best, but produces a large number of routes (significantly more than the ground truth); these routes are often nonsensical. MCMCF produces a large number of mostly realistic routes which explain the flow reasonably well. In contrast, FR produces significantly smaller sets of realistic routes that still explain the flow well, albeit with a higher running time.##
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A Framework for Algorithm Stability
[article]

2018
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arXiv
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pre-print

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We

arXiv:1704.08000v2
fatcat:3iuaqeswyfbdtk2czx5nr4g2dm
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... ocus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.##
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Partitioning Polygons via Graph Augmentation
[chapter]

2016
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Lecture Notes in Computer Science
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*Meulemans*is supported by Marie Sk lodowska-Curie Action MSCA-H2020-IF-2014 656741. ...

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Area-Preserving Subdivision Schematization
[chapter]

2010
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Lecture Notes in Computer Science
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We describe an area-preserving subdivision schematization algorithm: the area of each region in the input equals the area of the corresponding region in the output. Our schematization is axis-aligned, the final output is a rectilinear subdivision. We first describe how to convert a given subdivision into an area-equivalent rectilinear subdivision. Then we define two area-preserving contraction operations and prove that at least one of these operations can always be applied to any given simple

doi:10.1007/978-3-642-15300-6_12
fatcat:33umlt5fnfahxfibwp5sosbera
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... ctilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff-and Fréchet-distance) that show that better schematizations might result in worse shapes.##
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Exploring Curved Schematization

2014
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2014 IEEE Pacific Visualization Symposium
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Reimer and

doi:10.1109/pacificvis.2014.11
dblp:conf/apvis/GoethemMSW14
fatcat:ajktulcoizfe3pnxwlxl3qotgi
*Meulemans*[21] conjecture that parallelism drives straight-line schematization. Automated curved schematization has only recently emerged as a research topic. Van Goethem et al. ...##
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Locally Correct Fréchet Matchings
[chapter]

2012
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Lecture Notes in Computer Science
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*Meulemans*and B. Speckmann are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707. ...

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Computing Stable Demers Cartograms
[article]

2019
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arXiv
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pre-print

Recently,

arXiv:1908.07291v2
fatcat:cxhsbjibnferpncxblyq5v4pqi
*Meulemans*[22] introduced a constraint program to compute optimal solutions under orthogonal order constraints for diamond-shaped symbols. ...
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