W Wouter Meulemans

KelpFusion: A Hybrid Set Visualization Technique

We present KelpFusion: a method for depicting set membership of items on a map or other visualization using continuous boundaries. KelpFusion is a hybrid representation that bridges hull techniques such as Bubble Sets and Euler diagrams and line- and graph-based techniques such as LineSets and Kelp Diagrams. We describe an algorithm based on shortest-path graphs to compute KelpFusion visualizations. Based on a single parameter, the shortest-path graph varies from the minimal spanning tree to the convex hull of a point set. Shortest-path graphs aim to capture the shape of a point set and smoothly adapt to sets of varying densities. KelpFusion fills enclosed faces based on a set of simple legibility rules. We present the results of a controlled experiment comparing KelpFusion to Bubble Sets and LineSets. We conclude that KelpFusion outperforms Bubble Sets both in accuracy and completion time and outperforms LineSets in completion time.

Area-preserving subdivision schematization

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 rectilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff- and Frechet-distance) that show that better schematizations might result in worse shapes.

Topologically safe curved schematisation

Abstract Traditionally schematised maps make extensive use of curves. However, automated methods for schematisation are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematisation that allows a choice of quality measures and curve types. The framework fits a curve to every part of the input. It uses Voronoi diagrams to ensure that curves fitted to disjoint parts do not intersect. The framework then employs a dynamic program to find an optimal schematisation using the fitted curves. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.

Drawing planar cubic 3-connected graphs with few segments: algorithms & experiments

A drawing of a graph can be understood as an arrangement of geometric objects. In the most natural setting the arrangement is formed by straight-line segments. Every cubic planar 3-connected graph with n n vertices has such a drawing with only n/2+3 n/2+3 segments, matching the lower bound. This result is due to Mondal et al. [J. of Comb. Opt., 25], who gave an algorithm for constructing such drawings. We introduce two new algorithms that also produce drawings with n/2+3 n/2+3 segments. One algorithm is based on a sequence of dual edge contractions, the other is based on a recursion of nested cycles. We also show a flaw in the algorithm of Mondal et al. and present a fix for it. We then compare the performance of these three algorithms by measuring angular resolution, edge length and face aspect ratio of the constructed drawings. We observe that the corrected algorithm of Mondal et al. mostly outperforms the other algorithms, especially in terms of angular resolution. However, the new algorithms perform better in terms of edge length and minimal face aspect ratio.

Drawing planar graphs with few geometric primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n / 4 straight-line segments on a polynomial grid, and with n / 2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with \((8n\,-\,17)/3\) segments on an \(O(n)\,\times \,O(n^2)\) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n / 2 edges on an \(O(n)\,\times \,O(n^2)\) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only \((5n\,-\,11)/3\) arcs. This provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.

Exploring Curved Schematization of Territorial Outlines

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of “curviness”: we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents.

Experimental analysis of the accessibility of drawings with few segments

The visual complexity of a graph drawing is defined as the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges, e.g., one needs only one line segment to draw two collinear incident edges. We study the question if drawings with few segments have a better aesthetic appeal and help the user to asses the underlying graph. We design an experiment that investigates two different graph types (trees and sparse graphs), three different layout algorithms for trees, and two different layout algorithms for sparse graphs. We asked the users to give an aesthetic ranking on the layouts and to perform a furthest-pair or shortest-path task on the drawings.

Efficient trajectory queries under the Fréchet distance (GIS Cup)

Consider a set P of trajectories (polygonal lines in R2), and a query given by a trajectory Q and a threshold ϵ > 0. To answer the query we wish to find all trajectories P ∈ P such that δF(P, Q) ≤ ϵ, where δF denotes the Fréchet distance. We present an approach to efficiently answer a large number of queries for the same set P. Key ingredients are (a) precomputing a spatial hash that allows us to quickly find trajectories that have endpoints near Q; (b) precomputing simplifications on all trajectories in P; (c) using the simplifications and optimizations of the decision algorithm to efficiently decide δF(P, Q) ≤ ϵ for most P ∈ P.

A framework for algorithm stability and its application to kinetic Euclidean MSTs

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.

Drawing Planar Graphs with Few Geometric Primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n−17)/3 segments on an O(n)×O(n2) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/2 edges on an O(n)×O(n2) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n−11)/3 arcs. This is significantly smaller than the lower bound of 2n for line segments for a nontrivial graph class.