The State of the Art in Cartograms

Cartograms combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income). Cartograms make it possible to gain insight into patterns and trends in the world around us and have been very popular visualizations for geo-referenced data for over a century. This work surveys cartogram research in visualization, cartography and geometry, covering a broad spectrum of different cartogram types: from the traditional rectangular and table cartograms, to Dorling and diffusion cartograms. A particular focus is the study of the major cartogram dimensions: statistical accuracy, geographical accuracy, and topological accuracy. We review the history of cartograms, describe the algorithms for generating them, and consider task taxonomies. We also review quantitative and qualitative evaluations, and we use these to arrive at design guidelines and research challenges

Evaluating Cartogram Effectiveness

Cartograms are maps in which areas of geographic regions (countries, states) appear in proportion to some variable of interest (population, income). Cartograms are popular visualizations for geo-referenced data that have been used for over a century and that make it possible to gain insight into patterns and trends in the world around us. Despite the popularity of cartograms and the large number of cartogram types, there are few studies evaluating the effectiveness of cartograms in conveying information. Based on a recent task taxonomy for cartograms, we evaluate four major different types of cartograms: contiguous, non-contiguous, rectangular, and Dorling cartograms. Specifically, we evaluate the effectiveness of these cartograms by quantitative performance analysis, as well as by subjective preferences. We analyze the results of our study in the context of some prevailing assumptions in the literature of cartography and cognitive science. Finally, we make recommendations for the use of different types of cartograms for different tasks and settings

Sea regions for rectangular cartograms

In a rectangular cartogram, each region of a map is represented by a rectangle whose area is proportional to some statistical data of interest. Current techniques for constructing rectangular cartograms partition a large rectangle (the map) into a set of smaller rectangles which correspond to land or sea regions. The position and size of sea rectangles determine the outline of land masses. Therefore, sea regions have a direct impact on the recognizability and, thus, on the visual quality of cartograms. In this paper, we describe the first algorithm for the automated creation of sea regions for rectangular cartograms and present results obtained with our method

Drawing graphs for cartographic applications

Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems

Visualizing data as objects by DC (difference of convex) optimization

In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the standard Multidimensional Scaling Analysis, is written as a global optimization problem whose objective is the difference of two convex functions (DC). Suitable DC decompositions allow us to use the Difference of Convex Algorithm (DCA) in a very efficient way. Our algorithmic approach is used to visualize two real-world datasets.Ministerio de Economía y CompetitividadJunta de AndalucíaUnión EuropeaUniversidad de Sevill

Computing Fast and Scalable Table Cartograms for Large Tables

Given an m x n table T of positive weights and a rectangle R with an area equal to the sum of the weights, a table cartogram computes a partition of R into m x n convex quadrilateral faces such that each face has the same adjacencies as its corresponding cell in T, and has an area equal to the cell's weight. In this thesis, we explored different table cartogram algorithms for a large table with thousands of cells and investigated the potential applications of large table cartograms. We implemented Evans et al.'s table cartogram algorithm that guarantees zero area error and adapted a diffusion-based cartographic transformation approach, FastFlow, to produce large table cartograms. We introduced a constraint optimization-based table cartogram generation technique, TCarto, leveraging the concept of force-directed layout. We implemented TCarto with column-based and quadtree-based parallelization to compute table cartograms for table with thousands of cells. We presented several potential applications of large table cartograms to create the diagrammatic representations in various real-life scenarios, e.g., for analyzing spatial correlations between geospatial variables, understanding clusters and densities in scatterplots, and creating visual effects in images (i.e., expanding illumination, mosaic art effect). We presented an empirical comparison among these three table cartogram techniques with two different real-life datasets: a meteorological weather dataset and a US State-to-State migration flow dataset. FastFlow and TCarto both performed well on the weather data table. However, for US State-to-State migration flow data, where the table contained many local optima with high value differences among adjacent cells, FastFlow generated concave quadrilateral faces. We also investigated some potential relationships among different measurement metrics such as cartographic error (accuracy), the average aspect ratio (the readability of the visualization), computational speed, and the grid size of the table. Furthermore, we augmented our proposed TCarto with angle constraint to enhance the readability of the visualization, conceding some cartographic error, and also inspected the potential relationship of the restricted angles with the accuracy and the readability of the visualization. In the output of the angle constrained TCarto algorithm on US State-to-State migration dataset, it was difficult to identify the rows and columns for a cell upto 20 degree angle constraint, but appeared to be identifiable for more than 40 degree angle constraint

On the number of regular edge labelings

We prove that any irreducible triangulation on n vertices has O (4:6807n ) regular edge labeling,s and that there are irreducible triangulations on n vertices with (3:0426n ) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O (1:87n ). Keywords: Counting; Regular edge labeling; Shearer's entropy lemm

전근대 토지대장과 지적도의 대화형 분석을 위한 시각화 설계

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2016. 2. 서진욱.We propose an interactive visualization design tool, called JigsawMap, for analyzing and mapping historical textual cadasters. A cadaster is an official register that records land properties (e.g., location, ownership, value and size) for land valuation and taxation. Such mapping of old and new cadasters can help historians understand the social and economic background of changes in land uses or ownership. JigsawMap can effectively connect the past land survey results to modern cadastral maps. In order to accomplish the connection process, three steps are performed: (1) segmentation of cadastral map, (2) visualization of textual cadastre, (3) and mapping interaction. We conducted usability studies and long term case studies to evaluate JigsawMap, and received positive responses. We summarize the evaluation results and present design guidelines for participatory design projects with historians. Followed by our study on JigsawMap, we further investigated on each components of our tool for more scalable map connection. First, we designed a hybrid algorithm to semi-automatically segment land pieces on cadastral map. The original JigsawMap provides interface for user to segment land pieces and the experiment result shows that segmentation algorithm accurately extracts the regions. Next, we reconsidered the visual encoding and simplified it to make textual cadastre more scalable. Since the former visual encoding relies on traditional map legend, the visual encoding can be selected based on user expert level. Finally, we redesigned layout algorithm to generate a better initial layout. We used evolution algorithm to articulate ambiguity problem of textual cadastre and the result less suffered from overlapping problem. Overall, our visualization design tool will provide an accurate segmentation result, give the user an option to select visual encoding that suits on their expert level, and generate more readable initial layout which gives an overview of cadastre layout.Chapter 1 Introduction 1 1.1 Background & Motivation 1 1.2 Main Contribution 7 1.3 Organization of the Dissertation 8 Chapter 2 Related Work 11 2.1 Map Data Visualization 11 2.2 Graph Layout Algorithms 13 2.3 Collaborative Map Editing Service 14 2.4 Map Image Segmentation 15 2.5 Premodern Cadastral Maps 17 2.6 Assessing Measures for Cartogram 18 Chapter 3 Visualizing and Mapping Premodern Textual Cadasters to Cadastral Maps 20 3.1 Textual Cadastre 21 3.2 Cadastral Maps 24 3.3 Paper-based Mapping Process and Obstacles 24 3.4 Task Flow in JigsawMap 26 3.5 Design Rationale 32 3.6 Evaluation 34 3.7 Discussion 40 3.8 Design Guidelines When Working with Historians 42 Chapter 4 Accurate Segmentation of Land Regions in Historical Cadastral Maps 44 4.1 Segmentation Pipeline 45 4.2 Preprocessing 46 4.3 Removal of Grid Line 48 4.4 Removal of Characters 52 4.5 Reconstruction of Land Boundaries 53 4.6 Generation of Polygons 55 4.7 Experimental Result 56 4.8 Discussion 59 Chapter 5 Approximating Rectangular Cartogram from Premodern Textual Cadastre 62 5.1 Challenges of the Textual Cadastre Layout 62 5.2 Quality Measures for Assessing Rectangular Cartogram 64 5.3 Quality Measures for Assessing Textual Cadastre 65 5.4 Graph Layout Algorithm 66 5.5 Results 72 5.6 Discussion 73 Chapter 6 Design of Scalable Node Representation for a Large Textual Cadastre 78 6.1 Motivation 78 6.2 Visual Encoding in JigsawMa 80 6.3 Challenges of Current Visual Encoding 81 6.4 Compact Visual Encoding 83 6.5 Results 84 6.6 Discussion 86 Chapter 7 Conclusion 88 Bibliography 90 Abstract in Korean 101Docto

Mathematical optimization for the visualization of complex datasets

This PhD dissertation focuses on developing new Mathematical Optimization models and solution approaches which help to gain insight into complex data structures arising in Information Visualization. The approaches developed in this thesis merge concepts from Multivariate Data Analysis and Mathematical Optimization, bridging theoretical mathematics with real life problems. The usefulness of Information Visualization lies with its power to improve interpretability and decision making from the unknown phenomena described by raw data, as fully discussed in Chapter 1. In particular, datasets involving frequency distributions and proximity relations, which even might vary over the time, are the ones studied in this thesis. Frameworks to visualize such enclosed information, which make use of Mixed Integer (Non)linear Programming and Difference of Convex tools, are formally proposed. Algorithmic approaches such as Large Neighborhood Search or Difference of Convex Algorithm enable us to develop matheuristics to handle such models. More specifically, Chapter 2 addresses the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. This information is represented using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. The visualization problem is formulated as a Mixed Integer Linear Programming model, and a matheuristic that has this model at its heart is proposed. Chapter 3 generalizes the model presented in the previous chapter by developing a visualization framework which handles simultaneously the representation of a frequency distribution and a dissimilarity relation. This framework consists of a partition of a given rectangle into piecewise rectangular portions so that the areas of the regions represent the frequencies and the distances between them represent the dissimilarities. This visualization problem is formally stated as a Mixed Integer Nonlinear Programming model, which is solved by means of a matheuristic based on Large Neighborhood Search. Contrary to previous chapters in which a partition of the visualization region is sought, Chapter 4 addresses the problem of visualizing a set of individuals, which has attached a dissimilarity measure and a frequency distribution, without necessarily cov-ering the visualization region. In this visualization problem individuals are depicted as convex bodies whose areas are proportional to the given frequencies. The aim is to determine the location of the convex bodies in the visualization region. In order to solve this problem, which generalizes the standard Multidimensional Scaling, Difference of Convex tools are used. In Chapter 5, the model stated in the previous chapter is extended to the dynamic case, namely considering that frequencies and dissimilarities are observed along a set of time periods. The solution approach combines Difference of Convex techniques with Nonconvex Quadratic Binary Optimization. All the approaches presented are tested in real datasets. Finally, Chapter 6 closes this thesis with general conclusions and future lines of research.Esta tesis se centra en desarrollar nuevos modelos y algoritmos basados en la Optimización Matemática que ayuden a comprender estructuras de datos complejas frecuentes en el área de Visualización de la Información. Las metodologías propuestas fusionan conceptos de Análisis de Datos Multivariantes y de Optimización Matemática, aunando las matemáticas teóricas con problemas reales. Como se analiza en el Capítulo 1, una adecuada visualización de los datos ayuda a mejorar la interpretabilidad de los fenómenos desconocidos que describen, así como la toma de decisiones. Concretamente, esta tesis se centra en visualizar datos que involucran distribuciones de frecuencias y relaciones de proximidad, pudiendo incluso ambas variar a lo largo del tiempo. Se proponen diferentes herramientas para visualizar dicha información, basadas tanto en la Optimización (No) Lineal Entera Mixta como en la optimización de funciones Diferencia de Convexas. Además, metodologías como la Búsqueda por Entornos Grandes y el Algoritmo DCA permiten el desarrollo de mateheurísticas para resolver dichos modelos. Concretamente, el Capítulo 2 trata el problema de visualizar simultáneamente una distribución de frequencias y una relación de adyacencias en un conjunto de individuos. Esta información se representa a través de un mapa rectangular, es decir, una subdivisión de un rectángulo en porciones rectangulares, de manera que las áreas de estas porciones representen las frecuencias y las adyacencias entre las porciones representen las adyacencias entre los individuos. Este problema de visualización se formula con la ayuda de la Optimización Lineal Entera Mixta. Además, se propone una mateheurística basada en este modelo como método de resolución. En el Capítulo 3 se generaliza el modelo presentado en el capítulo anterior, construyendo una herramienta que permite visualizar simultáneamente una distribución de frecuencias y una relación de disimilaridades. Dicha visualización se realiza mediante la partición de un rectángulo en porciones rectangulares a trozos de manera que el área de las porciones refleje la distribución de frecuencias y las distancias entre las mismas las disimilaridades. Se plantea un modelo No Lineal Entero Mixto para este problema de visualización, que es resuelto a través de una mateheurística basada en la Búsqueda por Entornos Grandes. En contraposición a los capítulos anteriores, en los que se busca una partición de la región de visualización, el Capítulo 4 trata el problema de representar una distribución de frecuencias y una relación de disimilaridad sobre un conjunto de individuos, sin forzar a que haya que recubrir dicha región de visualización. En este modelo de visualización los individuos son representados como cuerpos convexos cuyas áreas son proporcionales a las frecuencias dadas. El objetivo es determinar la localización de dichos cuerpos convexos dentro de la región de visualización. Para resolver este problema, que generaliza el tradicional Escalado Multidimensional, se utilizan técnicas de optimización basadas en funciones Diferencia de Convexas. En el Capítulo 5, se extiende el modelo desarrollado en el capítulo anterior para el caso en el que los datos son dinámicos, es decir, las frecuencias y disimilaridades se observan a lo largo de varios instantes de tiempo. Se emplean técnicas de optimización de funciones Diferencias de Convexas así como Optimización Cuadrática Binaria No Convexa para la resolución del modelo. Todas las metodologías propuestas han sido testadas en datos reales. Finalmente, el Capítulo 6 contiene las conclusiones a esta tesis, así como futuras líneas de investigación.Premio Extraordinario de Doctorado U

SYNTHESIS OF NON-REGULAR ARCHITECTURAL FORMS

Master'sMASTER OF SCIENC