arXiv subjects

Wouter Meulemans

Publications and source records attributed to Wouter Meulemans.

At least 19 recordsLinked to original sources

arXiv2026

An Algorithmic Perspective on Information Visualization

Information visualization is inherently a field that brings together various research domains. Roughly speaking, we may identify two perspectives: the design perspective, revolving around how to ensure that a human can work effectively with the visual representations of data and the tools that offer them, and the algorithmic perspective, focusing on how to automatically create such visual representations. Munzner's model for visualization design places design choices before algorithmic considerations. It offers predominantly a design perspective; as a consequence, applications of this model may consider the algorithmic perspective as an afterthought, bypassing a step that translates the design into the formalism necessary for algorithmic study. As a result, the design may be entangled with the algorithms used to compute a visualization. Focusing on layout algorithms, we explore the ramifications of this entanglement: quality often goes undefined and unmeasured, and ad-hoc heuristics tend to be applied, reducing trustworthiness and potentially leading to incorrect conclusions. We look at how we may complement Munzner's model---the design perspective---with a clear model of the algorithmic perspective, to obtain a formal, measured understanding of the interplay between visualizations and the algorithms used to create them. Paradoxically, the solution lies in a clearer separation of concerns between design and algorithm. We argue that this model leads to better comparison between approaches, a more fine-grained understanding of their strengths and weaknesses, and allows for uncovering new opportunities, as to eventually lead to a better understanding of visualization.

cs.HC↗

arXiv2026

Automated Responsive Thematic Mapping with Layout Guides

Thematic maps visually communicate statistical information about spatial units such as countries or states. They must balance the individual readability of those map elements that carry the statistical information and the overall cartographic context. Nowadays, most maps are not static images, but must flexibly respond to a range of device types and display sizes. Current approaches to responsive thematic mapping are limited: they are labor-intensive for practitioners and often rely on combining disjointed visual encodings to cover different device types. In this paper we introduce the first algorithmic framework to efficiently compute responsive thematic maps that smoothly adapt to different display sizes. A key component of our framework is the layout guide: a combinatorial structure which encodes the two essential aspects of a thematic map. The first aspect are the visual requirements of each statistical map element (at least their desired width and height), the second aspect is the cartographic context in the form of relative positions of map elements. Our main algorithmic contribution is the map arranger which takes a visual container as input and returns a suitable layout guide. The map arranger does so in a stable and consistent manner: if the container changes only a little, then so does the layout guide, and the same input container always results in the same layout guide. To use our framework, one needs three ingredients: $(1)$ a reference layout, which corresponds to the ``ideal'' thematic map, $(2)$ a total vertical and horizontal order for all map elements (the desired layouts for containers with extreme aspect ratios), and $(3)$ a thematic mapping algorithm that can construct a thematic map from a layout guide. We demonstrate our framework on two types of thematic maps, namely rectangular and Demers cartograms.

cs.CG↗

arXiv2026

Unfolding Ordered Matrices into BioFabric Motifs

BioFabrics were introduced by Longabaugh in 2012 as a way to draw large graphs in a clear and uncluttered manner. The visual quality of BioFabrics crucially depends on the order of vertices and edges, which can be chosen independently. Effective orders can expose salient patterns, which in turn can be summarized by motifs, allowing users to take in complex networks at-a-glance. However, so far there is no efficient layout algorithm which automatically recognizes patterns and delivers both a vertex and an edge ordering that allows these patterns to be expressed as motifs. In this paper we show how to use well-ordered matrices as a tool to efficiently find good vertex and edge orders for BioFabrics. Specifically, we order the adjacency matrix of the input graph using Moran's $I$ and detect (noisy) patterns with our recent algorithm. In this note we show how to "unfold" the ordered matrix and its patterns into a high-quality BioFabric. Our pipelines easily handles graphs with up to 250 vertices.

cs.HC↗

arXiv2026

Noisy Graph Patterns via Ordered Matrices

The high-level structure of a graph is a crucial ingredient for the analysis and visualization of relational data. However, discovering the salient graph patterns that form this structure is notoriously difficult for two reasons. (1) Finding important patterns, such as cliques and bicliques, is computationally hard. (2) Real-world graphs contain noise, and therefore do not always exhibit patterns in their pure form. Defining meaningful noisy patterns and detecting them efficiently is a currently unsolved challenge. In this paper, we propose to use well-ordered matrices as a tool to both define and effectively detect noisy patterns. Specifically, we represent a graph as its adjacency matrix and optimally order it using Moran's $I$. Standard graph patterns (cliques, bicliques, and stars) now translate to rectangular submatrices. Using Moran's $I$, we define a permitted level of noise for such patterns. A combination of exact algorithms and heuristics allows us to efficiently decompose the matrix into noisy patterns. We also introduce a novel motif simplification that visualizes noisy patterns while explicitly encoding the level of noise. We showcase our techniques on several real-world data sets.

cs.HC↗

arXiv2025

Minimizing Vertical Length in Linked Bar Charts

A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with ``dependent'' links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in $O(nm)$ time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an $O(n^4m)$-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar.

cs.CG↗

arXiv2025

Summarizing Classed Region Maps with a Disk Choreme

Chorematic diagrams are highly reduced schematic maps of geospatial data and processes. They can visually summarize complex situations using only a few simple shapes (choremes) placed upon a simplified base map. Due to the extreme reduction of data in chorematic diagrams, they tend to be produced manually; few automated solutions exist. In this paper we consider the algorithmic problem of summarizing classed region maps, such as choropleth or land use maps, using a chorematic diagram with a single disk choreme. It is infeasible to solve this problem exactly for large maps. Hence, we propose several point sampling strategies and use algorithms for classed point sets to efficiently find the best disk that represents one of the classes. We implemented our algorithm and experimentally compared sampling strategies and densities. The results show that with the right sampling strategy, high-quality results can be obtained already with moderately sized point sets and within seconds of computation time.

cs.CG↗

arXiv2025

StorySets: Ordering Curves and Dimensions for Visualizing Uncertain Sets and Multi-Dimensional Discrete Data

We propose a method for visualizing uncertain set systems, which differs from previous set visualization approaches that are based on certainty (an element either belongs to a set or not). Our method is inspired by storyline visualizations and parallel coordinate plots: (a) each element is represented by a vertical glyph, subdivided into bins that represent different levels of uncertainty; (b) each set is represented by an x-monotone curve that traverses element glyphs through the bins representing the level of uncertainty of their membership. Our implementation also includes optimizations to reduce visual complexity captured by the number of turns for the set curves and the number of crossings. Although several of the natural underlying optimization problems are NP-hard in theory (e.g., optimal element order, optimal set order), in practice, we can compute near-optimal solutions with respect to curve crossings with the help of a new exact algorithm for optimally ordering set curves within each element's bins. With these optimizations, the proposed method makes it easy to see set containment (the smaller set's curve is strictly below the larger set's curve). A brief design-space exploration using uncertain set-membership data, as well as multi-dimensional discrete data, shows the flexibility of the proposed approach.

cs.GR↗

arXiv2024

Visual Complexity of Point Set Mappings

We study the visual complexity of animated transitions between point sets. Although there exist many metrics for point set similarity, these metrics are not adequate for this purpose, as they typically treat each point separately. Instead, we propose to look at translations of entire subsets/groups of points to measure the visual complexity of a transition between two point sets. Specifically, given two labeled point sets A and B in R^d, the goal is to compute the cheapest transformation that maps all points in A to their corresponding point in B, where the translation of a group of points counts as a single operation in terms of complexity. In this paper we identify several problem dimensions involving group translations that may be relevant to various applications, and study the algorithmic complexity of the resulting problems. Specifically, we consider different restrictions on the groups that can be translated, and different optimization functions. For most of the resulting problem variants we are able to provide polynomial time algorithms, or establish that they are NP-hard. For the remaining open problems we either provide an approximation algorithm or establish the NP-hardness of a restricted version of the problem. Furthermore, our problem classification can easily be extended with additional problem dimensions giving rise to new problem variants that can be studied in future work.

cs.CG↗

arXiv2024

SimpleSets: Capturing Categorical Point Patterns with Simple Shapes

Points of interest on a map such as restaurants, hotels, or subway stations, give rise to categorical point data: data that have a fixed location and one or more categorical attributes. Consequently, recent years have seen various set visualization approaches that visually connect points of the same category to support users in understanding the spatial distribution of categories. Existing methods use complex and often highly irregular shapes to connect points of the same category, leading to high cognitive load for the user. In this paper we introduce SimpleSets, which uses simple shapes to enclose categorical point patterns, thereby providing a clean overview of the data distribution. SimpleSets is designed to visualize sets of points with a single categorical attribute; as a result, the point patterns enclosed by SimpleSets form a partition of the data. We give formal definitions of point patterns that correspond to simple shapes and describe an algorithm that partitions categorical points into few such patterns. Our second contribution is a rendering algorithm that transforms a given partition into a clean set of shapes resulting in an aesthetically pleasing set visualization. Our algorithm pays particular attention to resolving intersections between nearby shapes in a consistent manner. We compare SimpleSets to the state-of-the-art set visualizations using standard datasets from the literature.

cs.CG↗

arXiv2024

Greedy Monochromatic Island Partitions

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k \geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of $S$ such that $\mathcal{CH}(I) \cap S = I$, where $\mathcal{CH}(I)$ denotes the convex hull of $I$. We identify an island with its convex hull; therefore, a partition into islands has the additional requirement that the convex hulls of the islands are pairwise-disjoint. We present three greedy algorithms for constructing island partitions and analyze their approximation ratios.

cs.CG↗

arXiv2023

Optimizing Symbol Visibility through Displacement

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper, we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider closed unit square symbols that need to be placed at specified $y$-coordinates. We optimize the drawing order of the symbols as well as their $x$-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most $2$, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in $O(n\log n)$ time. If the width is at most $2$, there is a vertical line that stabs all squares, and in this case, we design a 2-approximation algorithm (assuming fixed container height) that runs in $O(n\log n)$ time. As it turns out that a minimum visible perimeter of 2 is always achievable with a generic construction, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

cs.CG↗

arXiv2021

Simultaneous Matrix Orderings for Graph Collections

Undirected graphs are frequently used to model networks. The topology of an undirected graph G can be captured by an adjacency matrix; this matrix in turn can be visualized directly to give insight into the graph structure. Which visual patterns appear in such a matrix visualization depends on the ordering of its rows and columns. Formally defining the quality of an ordering and then automatically computing a high-quality ordering are both challenging problems; however, effective heuristics exist and are used in practice. Often, graphs exist as part of a collection of graphs on the same set of vertices. To visualize such graph collections, we need a single ordering that works well for all matrices simultaneously. The current state-of-the-art solves this problem by taking a (weighted) union over all graphs and applying existing heuristics. However, this union leads to a loss of information, specifically in those parts of the graphs which are different. We propose a collection-aware approach to avoid this loss of information and apply it to two popular heuristic methods: leaf order and barycenter. The de-facto standard computational quality metrics for matrix ordering capture only block-diagonal patterns (cliques). Instead, we propose to use Moran's I, a spatial auto-correlation metric, which captures the full range of established patterns. The popular leaf order method heuristically optimizes a similar measure which supports the use of Moran's I in this context. We evaluated our methods for simultaneous orderings on real-world datasets using Moran's I as the quality metric. Our results show that our collection-aware approach matches or improves performance compared to the union approach, depending on the similarity of the graphs in the collection. Specifically, our Moran's I-based collection-aware leaf order implementation consistently outperforms other implementations.

cs.HC↗

arXiv2021

Near-Delaunay Metrics

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 triangulation 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.

cs.CG↗

arXiv2021

Obstructing Classification via Projection

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 we 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.

cs.CG↗

arXiv2020

Route Reconstruction from Traffic Flow via Representative Trajectories

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 more 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.

cs.CG↗

arXiv2019

Stable Visual Summaries for Trajectory Collections

The availability of devices that track moving objects has led to an explosive growth in trajectory data. When exploring the resulting large trajectory collections, visual summaries are a useful tool to identify time intervals of interest. A typical approach is to represent the spatial positions of the tracked objects at each time step via a one-dimensional ordering; visualizations of such orderings can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality -- how well does the ordering capture the structure of the data at each time step, and stability -- how coherent are the orderings over consecutive time steps or temporal ranges? In this paper we introduce a new Stable Principal Component (SPC) method to compute such orderings, which is explicitly parameterized for stability, allowing a trade-off between the spatial quality and stability. We conduct extensive computational experiments that quantitatively compare the orderings produced by ours and other stable dimensionality-reduction methods to various state-of-the-art approaches using a set of well-established quality metrics that capture spatial quality and stability. We conclude that stable dimensionality reduction outperforms existing methods on stability, without sacrificing spatial quality or efficiency; in particular, our new SPC method does so at a fraction of the computational costs.

cs.HC↗

arXiv2019

Computing Stable Demers Cartograms

Cartograms are popular for visualizing numerical data for map regions. Maintaining correct adjacencies is a primary quality criterion for cartograms. When there are multiple data values per region (over time or different datasets) shown as animated or juxtaposed cartograms, preserving the viewer's mental-map in terms of stability between cartograms is another important criterion. We present a method to compute stable Demers cartograms, where each region is shown as a square and similar data yield similar cartograms. We enforce orthogonal separation constraints with linear programming, and measure quality in terms of keeping adjacent regions close (cartogram quality) and using similar positions for a region between the different data values (stability). Our method guarantees ability to connect most lost adjacencies with minimal leaders. Experiments show our method yields good quality and stability.

cs.CG↗

arXiv2019

Geometry and Generation of a New Graph Planarity Game

We introduce a new abstract graph game, Swap Planarity, where the goal is to reach a state without edge intersections and a move consists of swapping the locations of two vertices connected by an edge. We analyze this puzzle game using concepts from graph theory and graph drawing, computational geometry, and complexity. Furthermore, we specify quality criteria for puzzle instances, and describe a method to generate high-quality instances. We also report on experiments that show how well this generation process works.

cs.CG↗