Adaptation of
Point- and Line-Based
Visualization

Nils Rodrigues

Dissertation





Adaptation of Point- and Line-Based
Visualization

Von der Fakultät Informatik, Elektrotechnik und
Informationstechnik der Universität Stuttgart

zur Erlangung der Würde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigte Abhandlung

Vorgelegt von

Nils Rodrigues
aus Stuttgart

Hauptberichter: Prof. Dr. Daniel Weiskopf
Mitberichter: Prof. Dr. Lars Linsen
Tag der mündlichen Prüfung: 13. Dezember 2023

Visualisierungsinstitut
der Universität Stuttgart

2024





Acknowledgments
First, I want to thank Daniel Weiskopf for giving me the opportunity to
do research into visualization, advising me, providing guidance, feedback,
and enough liberty to explore various ideas. Daniel, I am also grateful for
you taking an active role and all the work you put in—even during the late
hours—that helped to get the research published! Thank you also for your
patience and motivation.

Thank you, Lars Linsen, for taking the time to review this dissertation
and participating in its defense. Thank you, Tobias Schreck, for welcoming
me to your institute for the research stay, for the continued collaboration,
and for suggestions on what to do in Graz.

I thank all coauthors I met over the years and with whom I had the joy
of working on various projects. In alphabetical order: Albrecht Schmidt,
Alexander Schultz, Andreas Bulling, Andreas Henicke, Andrés Bruhn, Anja
Haug, Antoine Lhuillier, Arman Mielke, Benedikt V. Ehinger, Bruno Burger,
Christian Baumann, Christoph Schulz, Cristina Morariu, Daniel A. Keim,
Daniel Baumgartner, Francesco Chiossi, Frederik L. Dennig, Guido Reina,
Harald Reiterer, Jakob Karolus, Jia Jun Yan, Johannes Zagermann, Katrin
Angerbauer, Kazi Riaz Ullah, Krishna Damarla, Kuno Kurzhals, Lewis L.
Chuang, Lin Shao, Lorenzo Di Silvestro, Marc O. Ernst, Marco Amann,
Martin Raubal, Maurice Koch, Michael Becher, Michael Burch, Michael
Sedlmair, Michael Stoll, Nelson de Jesus Silvério da Silva, Nelusa Path-
manathan, Peter Schäfer, Priscilla Balestrucci, Radu Jianu, René Cutura,
Rudolf Netzel, Sabine Storandt, Seyda Zerife Öney, Sven Mayer, Sören
Döring, Tanja Blascheck, Thomas Ertl, Tiare M. Feuchtner, Tim Krake,
Tobias Schreck, and Vincent Brandt. From the long list of coauthors, I want
to single out Christoph Schulz for the pleasant and fruitful discussions and
for always working hard in our collaborations.

I am also very grateful to Anabela, Christoph, Jessica, Kuno, Moataz, and
Tycho (in alphabetical order) for reading this thesis and for their valuable
feedback. Very heartfelt thanks to my family and friends who endured me
and stood by my side. I would not be where I am now without your strong
support! Thank you for everything! I am especially thankful to Jessica for
helping me stay focused, supporting me over the years as well as during the
write-up, and taking burdens off my back when she herself was swamped.

My work at the Visualization Research Center of the University of
Stuttgart was mostly funded by the German Research Foundation DFG
within project B01 of SFB-TRR 161.

iii





Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Abbreviations and Acronyms . . . . . . . . . . . . . . . . . . . xv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 Introduction 1
1.1 Goals and Research Questions . . . . . . . . . . . . . . . . . . . 2
1.2 Outline and Research Contributions . . . . . . . . . . . . . . . . 3

2 Data Types and Their Visualizations 9
2.1 Low-Dimensional Data . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Point-Based Visualization . . . . . . . . . . . . . . . . . . . . . 11
Line-Based Visualization . . . . . . . . . . . . . . . . . . . . . 12

2.3 Spatial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I Dot Plots 15

3 Nonlinear Dot Plots 21
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Original One-Way Sweep . . . . . . . . . . . . . . . . . . . . . 22
Two-Way Sweep Algorithm . . . . . . . . . . . . . . . . . . . . 24
Combining Sweeps . . . . . . . . . . . . . . . . . . . . . . . . 25
Dot Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Anti-Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Variants, Extensions, and Hybrid Visualizations . . . . . . . . . 32

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Distribution of Air Temperature . . . . . . . . . . . . . . . . . 33

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Contents

Citation Statistics . . . . . . . . . . . . . . . . . . . . . . . . 36
Renewable Energy . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Expert Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Relaxed Dot Plots 43
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Goals and Overview . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Plotting Space, Shape, and Dot Size . . . . . . . . . . . . . . . . 46

Kernel Frequency Estimation . . . . . . . . . . . . . . . . . . . 46
Nonlinear Frequency Scaling . . . . . . . . . . . . . . . . . . . 49
Individual Dot Diameter . . . . . . . . . . . . . . . . . . . . . 50

4.4 Dot Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Alternating Vertical Order . . . . . . . . . . . . . . . . . . . . 51
Centroidal Voronoi Tessellation (CVT) . . . . . . . . . . . . . . 52
Placement Correction . . . . . . . . . . . . . . . . . . . . . . 53
Swaps for Tunneling . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Termination Criterion . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Approaches Without KFE and Scaled Envelope . . . . . . . . . . . 56
4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Crowdsourced User Study . . . . . . . . . . . . . . . . . . . . 61
Blue Noise Property and Moiré Effect . . . . . . . . . . . . . . 65

4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Excursion: Mathematical Transition from Columns to Fre-
quencies 71
5.1 Recap, Definitions, and Observations . . . . . . . . . . . . . . . 71
5.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Generic Observations . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Linear Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Root Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Logarithmic Scaling . . . . . . . . . . . . . . . . . . . . . . . . 75
5.7 Generic Dot Diameter . . . . . . . . . . . . . . . . . . . . . . . 78
5.8 Conslusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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Contents

II Scatter Plots 81

6 Comparative Evaluation of Animated Scatter Plot Transi-
tions 87
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Animated Visualization in Other Contexts . . . . . . . . . . . . 89
Animations for Scatter Plot Visualizations . . . . . . . . . . . . 89
User Studies on Animated Visualizations . . . . . . . . . . . . . 91

6.3 Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Dimensions and Views in a Scatter Plot Matrix . . . . . . . . . . 92
Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Non-Evaluated Alternatives . . . . . . . . . . . . . . . . . . . 96

6.4 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Animation Speed . . . . . . . . . . . . . . . . . . . . . . . . . 101
Data Set Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Power Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Participant Recruitment . . . . . . . . . . . . . . . . . . . . . 103

6.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Exclusion Criterion . . . . . . . . . . . . . . . . . . . . . . . . 104
Point Task Performance . . . . . . . . . . . . . . . . . . . . . . 104
Cluster Task Performance . . . . . . . . . . . . . . . . . . . . 106
Animation Direction . . . . . . . . . . . . . . . . . . . . . . . 107
Subjective Rating . . . . . . . . . . . . . . . . . . . . . . . . 109

6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.8 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.9 Verdict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 Gaze-Based Recommendations for the Exploration of Scat-
ter Plot Matrices 117
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Gaze as Indicator for Data of Interest . . . . . . . . . . . . . . 119
From Gaze to Clusters . . . . . . . . . . . . . . . . . . . . . . 120
Data Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
(Dis)similarity for Recommendations . . . . . . . . . . . . . . . 121

7.3 Software System . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4 Use Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

vii



7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Excursion: Eye Tracking for Immersive and Situated Ana-
lytics 129
8.1 Hypothetical Use Case in Virtual Reality: FiberClay . . . . . . . . . 130

The Original Unchanged System . . . . . . . . . . . . . . . . . 130
Benefits and Challenges of Eye Tracking Support . . . . . . . . 130

8.2 Accuracy of Gaze Depth for Augmented Reality . . . . . . . . . . 132
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

III Parallel Coordinates Plots 141

9 Cluster-Flow Parallel Coordinates Plots 147
9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2 Model and Overview . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Axis Duplication . . . . . . . . . . . . . . . . . . . . . . . . . 151
Dimension Ordering . . . . . . . . . . . . . . . . . . . . . . . 153
Cluster Ordering . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.4 Line Mapping and Rendering . . . . . . . . . . . . . . . . . . . . 157
Curve Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 157
Density Rendering . . . . . . . . . . . . . . . . . . . . . . . . 159
Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.5 Visual Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Escherichia Coli . . . . . . . . . . . . . . . . . . . . . . . . . 165
NetPerf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Energy Production . . . . . . . . . . . . . . . . . . . . . . . . 167

9.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

10 Conclusion 171
10.1 Answers to Research Questions . . . . . . . . . . . . . . . . . . 171
10.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Bibliography 177



List of Figures

1.1 The visualization pipeline . . . . . . . . . . . . . . . . . . . . . . 2

I-1 A dot plot, according to Wilkinson . . . . . . . . . . . . . . . . . 17
I-2 Elements of the visualization pipeline that are affected by the

proposed dot plot techniques . . . . . . . . . . . . . . . . . . . . 18
I-3 A “dot plot,” according to Cleveland . . . . . . . . . . . . . . . . 19

3.1 Illustration of Wilkinson’s sweep layout algorithm . . . . . . . . 22
3.2 Dot overlap from varying value density . . . . . . . . . . . . . . 26
3.3 Low-resolution plots of normal distributions . . . . . . . . . . . 29
3.4 Comparison of visualizations for more than 9,000 air temperature

values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Number of citations referencing publications . . . . . . . . . . 36
3.6 Percentage of electricity produced from renewable energy sources 38

4.1 Illustration of various dot plot layouts . . . . . . . . . . . . . . . 44
4.2 Overview of the relaxed dot plot method . . . . . . . . . . . . . 46
4.3 Kernels for frequency estimation . . . . . . . . . . . . . . . . . . 48
4.4 Comparison between unbounded and bounded kernel frequency

estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Leaning columns of dots from relaxation . . . . . . . . . . . . . 51
4.6 A narrow bandwidth can lead to split Voronoi cells . . . . . . . 51
4.7 Analysis of positional error with varying settings for relaxation 53
4.8 Mean dot movement between relaxation iterations . . . . . . . 56
4.9 Different dot-based visualizations of temperature data . . . . . 58
4.10 Relaxed dot plot of the percentage of renewables in electricity

production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.11 Comparison of displayed variance in dot plots layouts . . . . . 60
4.12 Examples of study tasks for each hypothesis . . . . . . . . . . . 63
4.13 The independent variables of each experiment . . . . . . . . . . 63
4.14 Percentage of correct responses . . . . . . . . . . . . . . . . . . 65
4.15 Dot plots with overlap marked in red . . . . . . . . . . . . . . . 67

5.1 Plots of hf,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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List of Figures

5.2 Plot of the functions for envelope height and dot size from fre-
quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Plot of the piecewise functions for logarithmic dot plots (ĥf,l, d̂f,l) 79
II-1 Single scatter plot and matrix of the iris data set . . . . . . . . 83
II-2 Visualization pipeline with proposed additions for scatter plots 84

6.1 Generic concept for rotation animation between scatter plots . 90
6.2 1D and 2D transitions in a SPLOM . . . . . . . . . . . . . . . . . 93
6.3 Examples of spline-based animation paths . . . . . . . . . . . . 94
6.4 Concept of staged rotation . . . . . . . . . . . . . . . . . . . . . 95
6.5 Concept of perspective rotation . . . . . . . . . . . . . . . . . . 95
6.6 User interface for the point task . . . . . . . . . . . . . . . . . . 100
6.7 User interface for the cluster task . . . . . . . . . . . . . . . . . 100
6.8 Results of the pilot study with different animation speeds . . . 101
6.9 Results of the pilot study with different point counts . . . . . . 102
6.10 Distribution of errors in indicated dot position . . . . . . . . . . 105
6.11 Distribution of correctly identified cluster interactions . . . . . 106
6.12 Distribution of point task performance with 2D transitions . . . 108
6.13 Subjective feedback from study participants . . . . . . . . . . . 109

7.1 The three steps in our concept . . . . . . . . . . . . . . . . . . . 119
7.2 Main window of our visual analytics tool . . . . . . . . . . . . . 120
7.3 Initial view of our tool after loading a data set . . . . . . . . . . 122
7.4 GUI with options for subset generation . . . . . . . . . . . . . . 123
7.5 Comparative view with table and SPLOM . . . . . . . . . . . . . 124
7.6 Controls for the eye-tracking heatmap and threshold for data

subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.7 Analysis of surprising data in network performance data . . . . 125

8.1 Scene setup and real photograph. . . . . . . . . . . . . . . . . . 135
8.2 Augmented scene from the experiment. . . . . . . . . . . . . . . 135
8.3 Error in measured gaze depth . . . . . . . . . . . . . . . . . . . 138
III-1 Parallel coordinates plot of fictitious fruit data . . . . . . . . . . 143
III-2 Elements of the visualization pipeline affected in Part III . . . . 144
III-3 Comparison of rendering techniques for PCPs . . . . . . . . . . 145

9.1 Feature overview of CF-PCPs . . . . . . . . . . . . . . . . . . . . 149
9.2 Construction of CF-PCPs . . . . . . . . . . . . . . . . . . . . . . 150
9.3 Illustration of two clusters using different coordinate systems . 152
9.4 Concept of reading directions . . . . . . . . . . . . . . . . . . . 152
9.5 Tree representation of our model for dimension ordering . . . . 154
9.6 Simplified CF-PCP with crossing bundles . . . . . . . . . . . . . 156
9.7 Comparison of line geometry in CF-PCPs . . . . . . . . . . . . . 158

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List of Figures

9.8 Composite line geometry in CF-PCPs . . . . . . . . . . . . . . . 158
9.9 Varying opacity for data points with multiple soft labels . . . . 160
9.10 Visualizing uncertainty in the clustering results . . . . . . . . . 161
9.11 Inter-cluster patterns . . . . . . . . . . . . . . . . . . . . . . . . 162
9.12 Intra-cluster patterns . . . . . . . . . . . . . . . . . . . . . . . . 163
9.13 Classification of E. coli bacteria . . . . . . . . . . . . . . . . . . 164
9.14 Different clustering and PCP techniques applied to the NetPerf

data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.15 CF-PCP of electric power production . . . . . . . . . . . . . . . 168

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List of Tables

5.1 Conventions for function and variable names . . . . . . . . . . . 71

6.1 Comparison of point task performance . . . . . . . . . . . . . . 105
6.2 Comparison of cluster task performance . . . . . . . . . . . . . 106
6.3 Comparison of horizontal vs. vertical 1D transitions . . . . . . . 108
6.4 Comparison of subjective ease of the point task . . . . . . . . . 110
6.5 Comparison of reported preference for frequent use . . . . . . 111
6.6 Pairwise comparison of participant feedback on animation speed 111

List of Algorithms

3.1 Wilkinson’s original sweep algorithm . . . . . . . . . . . . . . 23
3.2 Adaptation for symmetric nonlinear dot plots. . . . . . . . . . 24

4.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 TunnelSwaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xiii





List of Abbreviations and Acronyms

2AFC two-alternative forced-choice
ACM Association for Computing Machinery
AR augmented reality
BNP blue noise plot
BSP bee swarm plot
CF-PCP cluster-flow parallel coordinates plot
CHCCS Canadian Human-Computer Communications Soci-

ety
CPU central processing unit
DBSCAN density-based spatial clustering of applications with

noise
DDA differential domain analysis
DR dimensionality reduction
EBL edge-bundling layout
FA Fourier analysis
GPU graphics processing unit
GUI graphical user interface
HMD head-mounted display
IEEE Institute of Electrical and Electronics Engineers
IPCs illustrative parallel coordinates
ISE Fraunhofer Institute for Solar Energy Systems
KDE kernel density estimation
KFE kernel frequency estimation
MDM mean dot movement
MOD mean dot overlap in display diameters
MSE mean squared error
NDP nonlinear dot plot
PCA principal component analysis
PCP parallel coordinates plot
RDP relaxed dot plot
RGB red, green, and blue color model
SP scatter plot

xv



List of Abbreviations and Acronyms

SPLOM scatter plot matrix
SSE sum of squared errors
TU Graz Graz University of Technology
UMAP uniform manifold approximation and projection
VAC vergence-accommodation conflict
VISUS Visualization Research Center of the University of

Stuttgart
VR virtual reality

xvi



Abstract
Visualization plays an important role in the lives of various heterogeneous
parts of society: from a voter looking for the latest results of an election, to
statisticians examining a distribution, to analysts trying to make sense of
multidimensional data sets. This thesis adapts existing point- and line-based
visualization methods to improve knowledge gain. The included contribu-
tions address three research questions: How to scale unit visualization for
1D data? How to improve navigation between 2D visualizations of multi-
variate data? How to combine the advantages of multiple 2D views in a
single static visualization for multivariate data?

The first part of the thesis focuses on unit visualization of 1D data with
dot plots. Compared to the previous state of the art, the developed visu-
alizations fit a wider range of data and expand the number of potential
users by requiring less prior knowledge for interpretation. They adapt the
definition of dot plots to scale nonlinearly with sample count, accurately
show value frequencies in high-dynamic-range data, reduce positional error
in displayed data points, and enhance the perception of subtle nuances
in the data while avoiding moiré effects. We provide evidence for claimed
improvements through evaluation with computational metrics and a crowd-
sourced user study.

The second part of the dissertation focuses on visualizing multivariate
data with scatter plots and scatter plot matrices. First, we evaluate six
animated transitions between plots of different 2D subspaces with respect
to task performance for tracking individual points and interactions between
clusters. The results of a quantitative study with 170 participants show
that orthographic rotation animation performs best and should be adopted
more widely. Next, we develop a novel concept for recommending views
in scatter plot matrices. It provides user- and task-specific suggestions
by focusing on the data of interest to the viewer. Together, animation and
recommendation adapt scatter plots to improve the user’s ability to analyze
more complex data effectively.

In the third part, we develop a new visualization technique that extends
parallel coordinate plots to provide a static alternative to scatter plots
with animated transitions. The approach does not require interaction to
display data flow between 2D subspace clusters. A custom density-based
rendering technique enables the visibility of individual lines and structures
within highly overdrawn regions. Our technique can communicate fuzzy
clustering results through binning and color mapping.

Finally, we discuss the presented contributions with respect to the
original main questions and show possible directions for future research.

xvii



Zusammenfassung
Visualisierung spielt im Leben verschiedener Teile der Gesellschaft eine
wichtige Rolle: Von einem Wähler, der nach den aktuellsten Wahlergeb-
nissen sucht, über Statistiker, die eine Verteilung untersuchen, bis hin
zu Analysten, die mehrdimensionale Datensätze verstehen wollen. Die
Beiträge dieser Arbeit passen bestehende punkt- und linienbasierte Visuali-
sierungsmethoden an, um den Wissenserwerb zu verbessern, und widmen
sich dabei drei Forschungsfragen: Wie kann man die Visualisierung von
1D-Daten mit einzelnen grafischen Elementen skalieren? Wie kann man
die Navigation zwischen 2D-Visualisierungen multivariater Daten verbes-
sern? Wie kann man die Vorteile mehrerer 2D-Ansichten in einer einzigen
statischen Visualisierung für multivariate Daten kombinieren?

Der erste Teil dieser Arbeit fokussiert sich auf die Visualisierung von
1D-Daten mit einzelnen grafischen Elementen in Punktdiagrammen. Im
Vergleich zum bisherigen Stand der Technik sind die entwickelten Visuali-
sierungen für ein breiteres Spektrum an Daten geeignet und erweitern den
Kreis potenzieller Nutzer, da sie weniger Vorkenntnisse erfordern. Sie pas-
sen die Definition von Punktdiagrammen an, um nichtlinear mit der Punk-
tezahl zu skalieren, Werteverteilungen mit hohem Dynamikumfang präzise
darzustellen, Positionsfehler der angezeigten Daten zu reduzieren und die
Wahrnehmung subtiler Nuancen zu verbessern, während Moiré-Effekte
vermieden werden. Wir belegen die Verbesserungen mit rechnerisch ermit-
telten Metriken und einer Crowdsourcing-basierten Nutzerstudie.

Der zweite Teil der Dissertation behandelt die Visualisierung multiva-
riater Daten mit Streudiagrammen und Streudiagrammmatrizen. Zunächst
untersuchen wir sechs Animationen zwischen Plots unterschiedlicher 2D-
Unterräume im Hinblick auf die Verfolgbarkeit einzelner Punkte und die
Interaktion zwischen Clustern. Die Ergebnisse einer quantitativen Stu-
die mit 170 Teilnehmern zeigen, dass orthografische Rotationsanimation
am besten abschneidet und breiter angewendet werden sollte. Anschlie-
ßend entwickeln wir ein neuartiges Konzept für die Empfehlung weiterer
Ansichten in einer Streudiagrammmatrix. Es bietet nutzer- und aufgaben-
spezifische Vorschläge, indem es den Fokus auf die für den Betrachter
interessanten Daten legt. Zusammen passen Animation und Empfehlung
Streudiagramme an, um die Fähigkeit des Benutzers zu verbessern, kom-
plexere Daten effektiv zu analysieren.

Im dritten Teil entwickeln wir eine neue Visualisierungstechnik, welche
parallele Koordinaten erweitert und eine statische Alternative zu Streudia-
grammen mit animierten Übergängen darstellt. Der Ansatz erfordert keine
Interaktion, um den Datenfluss zwischen 2D-Subraum-Clustern anzuzei-
gen. Angepasstes, dichtebasiertes Rendering ermöglicht die Sichtbarkeit

xviii



Zusammenfassung

einzelner Linien sowie Strukturen innerhalb stark überlagerter Bereiche.
Die Ergebnisse von unscharfem Clustering werden durch Intervallbildung
und Farbcodierung dargestellt.

Abschließend diskutieren wir die Beiträge im Hinblick auf die ursprüng-
lichen Fragen und zeigen mögliche Richtungen für zukünftige Forschung
auf.

xix





1

Introduction

Visualization is ubiquitous. Young pupils use them in school when they first
draw charts of mathematical functions to gain intuitive insight into the
characteristics of equations. Viewers are confronted with visualization as
part of the news when bar charts show a recent development. Infographics
help to understand complex systems and how individual components are
interconnected. Data analysts and visualization experts compose figures
to gain insight into large data and use them to communicate information
with other people. All these kinds of visualizations, simple or complex, can
be created using an underlying pipeline. Haber and McNabb created a
formal definition for such a pipeline in the context of scientific visualiza-
tion [HM90]. We adapted Card et al.’s derived pipeline [CMS99] and will
use it throughout this document to put proposed changes into context (see
Figure 1.1). The first stage of the pipeline takes a potentially large input
data set and extracts a subset of data that is relevant for visualization. The
second stage maps the data to a description of individual visual primitives.
The rendering stage creates a visible representation from the description
without referencing the original data. Visualization serves human beings.
Hence, the final stage is the perception by a viewer. In the case of inter-
active visualization, the viewer can adapt parts of the pipeline, creating a
feedback loop for visual analytics [Kei+08].

Some visualizations are dynamic or interactive, giving their users, e.g.,
data analysts, the ability to adjust parameters to gain new insight from
images. Other visualizations are created by a designer and remain static.
They can then serve the same person—this time in the role of an analyst—or
be distributed among others, who then use them for their own analysis or
as confirmation of communicated results. The latter is often the case in
print media, e.g., newspapers, where readers are purely “consumers” and
cannot change the presented graphics.

The scientific field of visualization research has matured, and publi-

1



1. Introduction

Transformed
data

Displayable
representation

View
user

Processed
data

Manual interaction and feedback

Rendering PerceptionFiltering Mapping

Abstraction

Figure 1.1: The visualization pipeline, adapted from [CMS99]. Humans can
use existing knowledge to adjust parameters and interact at various steps
to adapt the output and gain new insights.

cations include increasing numbers of visual representations [Rei+20;
Che+21]. Researchers have creative freedom for the design of new visual-
ization techniques. The range of used visual elements spans from text and
simple dots, over shapes composed of lines and polygons, to continuous
fields of color and further. The complexity of employed and developed
software for data visualization has increased over time [Rei+20]. On the
one hand, and if used correctly, this growing complexity of tools along the
visualization pipeline might reduce the workload for human users of the
resulting images by preprocessing data, using more elaborate algorithms
to find information of interest, optimizing perceptual variables, etc. On the
other hand, it might also lead to the creation of visualizations that encode
more and more data, contain evermore visual primitives, and become too
complex or overwhelming for humans to analyze and extract information.

1.1 Goals and Research Questions
Contrary to the underlying trend of growing possibilities, this thesis fo-
cuses on known visualization techniques that only use dots and lines. They
are typically used for relatively small data sets with a limited number of
dimensions. The overarching goal of this thesis is not the creation of radical
new visualizations but the adaptation and extension of proven methods to
improve the knowledge gain. The presented plots and techniques target
data analysts with prior experience but also the general public. To achieve
our overarching goal, we identify three central research questions:

RQ1 – How to scale unit visualization for 1D data?
Visualizing 1D data with a graphical element for each data point
helps to make it relatively simple to understand. It is, therefore,
well suited to target large audiences, including the general public.
The most straightforward visualization techniques for 1D data
might be limited to low sample counts in the orders of tens or
hundreds. We develop these plots further and make them suitable

2



1.2. Outline and Research Contributions

for high-dynamic-range data sets with several thousands of data
points. As screen space is limited, we do not expect the resulting
techniques to show each and every sample but to adapt to the
underlying data and retain the advantages of unit visualization
where possible. In the context of recent work on scalability in
visualization [Ric+23], the result should be more scalable in the
sense of shape because it requires nonlinear functions to improve
the visibility of outliers in high-dynamic-range data. Additionally,
the answer to this question should increase the suitable problem
size for a given threshold to allow for larger sample counts in the
same screen space.

RQ2 – How to improve navigation between 2D visualizations
of multivariate data?
Most visualization is 2D and presented on digital screens. It seems
logical to select only two attributes from a multivariate data set
and to only display this subspace. It results in comprehensible
visualizations that allow for multiple types of insight. To gain more
knowledge from the underlying data set, it is then necessary to
select and analyze multiple such subspaces. This question aims
to help the analysts with their analysis tasks by facilitating the
discovery of and switching between fitting 2D visualizations from
multivariate data.

RQ3 – How to combine the advantages of multiple 2D views in
a single static visualization for multivariate data?
Visualization is not always interactive, e.g., when printed on paper.
This question targets the creation of a plotting technique that can
provide insights from multiple 2D visualizations in a single static
image. To this end, we extend the range of included approaches
beyond simple dots and points to also encompass polylines, giving
us more freedom to design a new visualization technique.

1.2 Outline and Research Contributions
This thesis contains chapters that are based on published or submitted
papers and reused with consent from the coauthors. I am the main author
in most cases. For publications where I only contributed to distinct parts, I
will limit reuse to the sections that I wrote. Daniel Weiskopf is a coauthor
of all papers, contributing his experience and valuable feedback, helping
to polish the content for publication, and editing the manuscripts. He
supervised me in all projects behind the papers.

3



1. Introduction

After the introduction, Chapter 2 provides a coarse background on
data types and how they can be visualized. On this basis, it helps motivate
the main research questions and explains the selection of techniques that
follow. The main content of this document is structured around these
main research questions, which themselves are arranged according to
the dimensionality of numerical data and main drawing primitives for
visualization. Each of the following major parts of this dissertation creates
a link back to the research questions and provides an overview of how the
presented content ties into the general visualization pipeline.

Part I (Dot Plots) answers RQ1 and focuses on 1D data and its fre-
quency. At the beginning, it gives a short overview of the concepts behind
multiple visualizations that are called “dot plots” and clarifies which of
these techniques we use as a basis for further development.

Chapter 3 – Nonlinear Dot Plots This chapter introduces nonlinear
scaling to dot plots in order to allow the visualization of larger data sets
with highly dynamic data frequencies. It is based on the results of the first
project I started to work on at the Visualization Research Center of the
University of Stuttgart (VISUS) [RW18].
Daniel Weiskopf had the initial idea of nonlinear scaling for dot plots. I
implemented the adapted algorithm and extended it into a two-way-sweep
layout with two scaling functions. I developed a software suite for testing
the layouts and introduced a vertical blur in dense areas.
Chapter 4 – Relaxed Dot Plots In this chapter, we improve on the display
of data frequency and increased positional correctness in dot plots while, at
the same time, providing better aesthetics with little to no moiré effect. It
is based on a project that led to the publication of dot plots without straight
columns [Rod+23b; Rod+22b].
Christoph Schulz and I both identified parts of the Linde-Buzo-Gray stip-
pling algorithm [DSZ17] that might be applicable to dot plots and intro-
duced new constraints. As a result, we arrived at Lloyd relaxation as a
technique that was useful for redistributing circles inside an envelope
shape. I performed the transfer from dot-count-based nonlinear scaling
from Chapter 3 to a frequency-based approach in order to compute the
envelope shape from kernel frequency estimation of the underlying visu-
alization data. Tim Krake proved mathematical properties and helped to
create consistent equations.
Having a container shape and an algorithm for the distribution of circles, I
designed the initial adapted dot layout that serves as a starting point for
relaxation and avoids visual artifacts. I also developed an approach with
linear complexity for tunneling swaps between dots that increases posi-
tional correctness and decreases runtime. Two student assistants—Sören

4



1.2. Outline and Research Contributions

Döring and Daniel Baumgartner—implemented all algorithms. Christoph
Schulz and I analyzed the code and provided feedback for improvements. I
designed and conducted a crowdsourced study to evaluate the relaxed vi-
sualization against regular nonlinear dot plots. Sören Döring implemented
a generator for study stimuli and set up a template for the online survey.
Chapter 5 – Excursion: Mathematical Transition from Columns to
Frequencies The equations for the mathematical transformations from
column-based to frequency-based scaling are contained in this chapter. The
content is based on a document that is part of the supplemental material
to relaxed dot plots [Rod+22b]. Tim Krake provided a first draft for a clean
write-up of the math behind the transformations, which I then edited and
extended with other material for the completed document. I present this
work as an additional excursion to be able to better explain the ideas behind
the transformations and to not interrupt the flow of text in Chapter 4 with
too many equations.

Part II (Scatter Plots) answers RQ2 and focuses on exploring and
navigating pairs of two dimensions from multivariate data. It starts with
an introduction to the application of scatter plots—as individuals and in
matrices of small multiples—for the visualization of multidimensional data.

Chapter 6 – Comparative Evaluation of Animated Scatter Plot Transi-
tions This chapter compares multiple animation techniques for switching
between scatter plots. It evaluates them with respect to traceability of
individual points and interactions between data clusters. We published
the work behind this chapter in IEEE Transactions on Visualization and
Computer Graphics [Rod+24].
This project started with the desire to visualize flow between clusters in
different scatter plots of the same underlying data set. Under my super-
vision, Vincent Brandt created a software library for animated transitions
between scatter plots and made a preliminary small-scale comparison of a
wide range of animations that sparked interest in a more thorough evalu-
ation. To this end, I designed and conducted a crowdsourced study with
170 participants and two pilot studies. Frederik Dennig gave feedback on
the design of the study and provided a first partial draft for the write-up
of related work. Daniel Keim supervised Frederik Dennig and provided
feedback on the ideas of the paper. I performed the statistical analysis of
the study results and wrote the remainder of the publication.
Chapter 7 – Gaze-Based Recommendations for the Exploration of
Scatter Plot Matrices This chapter follows logically from the previous
one, and both together provide an answer to RQ2: since we now know
what animation between scatter plots works best, we want to recommend
specific dimension pairs that suit the analyst’s task. The main content of

5



1. Introduction

this chapter is based on a joint publication with researchers from Aus-
tria [Rod+22c].
The project started with a research stay at Graz University of Technology
(TU Graz). Tobias Schreck, Lin Shao, and I jointly supervised the student
Jia Jun Yan, who implemented a proof of concept for a visual analytics
application. I proposed possible ways of using gaze to implicitly select
data and helped by suggesting techniques to solve specific problems. I
took the lead role in writing the paper. Lin Shao and Tobias Schreck gave
feedback for the paper and helped with the related work based on their
prior knowledge of multiple recommender systems.
Chapter 8 – Excursion: Eye Tracking for Immersive and Situated
Analytics We based the implicit data selection from the previous chapter
on gaze data. Within this excursion, we extend the display dimension and
discuss the use of eye tracking in the context of immersive and situated
analytics. First, the chapter presents hypothetical benefits and challenges
of adding gaze support to an existing virtual reality application (Fiber-
Clay [Hur+19]). The work is based on my contributed sections about the
analytics tool that appear within a publication with many authors from
institutions in multiple countries [Sil+19].
The second part of the chapter contains an empirical evaluation of mea-
sured gaze depth on real and virtual objects in augmented reality. It is
based on a publication from 2020 [Öne+20]. Seyda Öney implemented all
necessary software as part of her bachelor thesis. Michael Becher advised
her on topics related to HoloLens, while I focused on eye tracking and
the comparative study. Seyda Öney conducted the study and analyzed the
results. She summarized them in a first draft that I used as inspiration to
write the publication. Michael Becher helped to edit the paper and was
supervised by Guido Reina and Thomas Ertl. Michael Sedlmair was the
examiner of the student work.

Part III (Parallel Coordinates Plots) answers RQ3 and presents a
visualization technique that aims at combining advantages from animated
scatter plot transitions and static parallel coordinates plots (PCPs). It starts
with a short introduction to rendering techniques that tackle the problem
of overdraw in PCPs.

Chapter 9 – Cluster-Flow Parallel Coordinates Plots This chapter is
based on published work [Rod+20] and describes a visualization technique
that we developed to answer RQ3. During conversations with Christoph
Schulz about fuzzy clusters, I came up with the initial idea of replicating
the axes of parallel coordinates to show cluster assignments. We involved
Antoine Lhuillier in the discussions due to his experience in visualizations
with bundled lines. All three of us regularly discussed ideas and came up

6



1.2. Outline and Research Contributions

with the final design. Thus direct attribution is difficult. I implemented
the web-based visualization, metrics, and an algorithm for ordering axes
(horizontal) and clusters (vertical). Christoph Schulz contributed fuzzy
clustering and his knowledge in the field of uncertainty. Antoine Lhuillier
performed initial analyses of the NetPerf data set. While I was involved in
writing and editing all sections of the paper, the other authors contributed
text segments and provided proofreading over the entire document.

Chapter 10 concludes this thesis with a discussion of the results with
respect to the research questions and an outline of future research direc-
tions.

Material from the publication [RW18] is copyrighted by IEEE and used
with permission under the agreement for reuse in a thesis. The material
from the publication [Rod+23b] is public domain and licensed to IEEE
under the Creative Commons Attribution 4.0 (CCBY 4.0). Material from
the publications [Rod+22c; Sil+19; Öne+20] is copyrighted by ACM and
used with their permission under the agreement for reuse in a dissertation.
The copyright for material from the publication [Rod+20] is retained by its
authors, while CHCCS is granted non-exclusive rights of publication.

I had the joy of collaborating with many good, intelligent, and motivated
researchers from VISUS and other fine institutions. To honor their contri-
butions to the work presented in this dissertation, I use the editorial “we”
for most of the content and only revert to “I” where appropriate. I do not
explicitly cite extracts from coauthored publications to improve readability.

During my time at VISUS, I also coauthored other publications that
do not fit the general theme of this thesis and from which I do not use
excerpts. A collaboration with the Fraunhofer Institute for Solar Energy
Systems (ISE) led to a paper that presented a map interface for data
from power plants that was integrated with existing charts on a website
(https://www.energy-charts.info, accessed 2024-01-27) [Rod+17b].
Collaboration with IBM resulted in a paper about the application of PCPs
for data analysis at the same company [Sch+17].

Eye tracking was a topic in multiple publications. We created visual-
izations and tools for gaze analysis on static stimuli [Rod+18; Sch+22;
Sch+23] and videos [Kur+20]. We used eye tracking as input for interactive
applications in augmented reality [Pat+20] and showed how it could be
used with adaptive user interfaces [Chi+22].

The human visual system and perception have also played a role in mul-
tiple publications. For example, we modeled the strength of simultaneous
orientation contrast and compensated it through counter-rotation [Net+19].
We developed a simulator for various physiological disorders of the visual
system [Sch+19] and analyzed the effects of color vision deficiency on the

7

https://www.energy-charts.info


1. Introduction

accessibility of figures that were published over the years at the IEEE VIS
conference [Ang+22].

Based on prior work from my diploma thesis and together with my
former supervisor Michael Burch, I published a visual analytics approach
for the exploration of texts from multiple corpora that won a �� best paper
award [Rod+17a]. I also generated figures for a previous publication about
the visualization technique that I implemented during the work on my
diploma thesis [Bur+14].

8



2

Data Types and Their Visualizations

All visualizations have one thing in common: they represent the underlying
data that forms the starting point of the pipeline in Figure 1.1. However,
not all visualization techniques are well-suited for all kinds of data. This
chapter discusses different types of data and provides a background for cor-
responding visualization approaches. It shows how the detailed techniques
proposed in the following parts of the thesis relate to the overarching goal
and research questions.

2.1 Low-Dimensional Data
Simple 1D data and its visualizations are essential for humans, even in a
world where big data and data retention are increasingly relevant to artifi-
cial intelligence [Jai+20] and further evolving quantum computing [Tow22;
UK 21]. For example, large portions of populations are affected by the
results of elections for public offices. The number of samples, i.e., votes, in
such a case is typically high while the dimensionality is low because each
vote is a single choice from a small range of values, i.e., parties. Visualiza-
tions of this low-dimensional data are sufficient to gain information and
educate the public about the outcome of the election: the total number of
votes, the ratio of abstentions, the ratio between parties, etc. Most people
do not know the details of the volatile energy market, how much electricity
is generated, where it comes from, or where it is currently needed. In the
context of climate change, however, they might be interested in knowing
simple, 1D data. For example, the percentage of renewable energy, the
amount of carbon dioxide emissions, or the mix of primary energy sources
of their consumed electricity. Single-dimensional data even arises when
friends and family enjoy their free time and play games: the heights of the

9



2. Data Types and Their Visualizations

bar in limbo, the number of goals in football, or the score of each round in
a card game.

Dot plots visualize 1D data. They only use a simple number line and
place dots along it. This works well for small data sets and is very accessible
to a broad range of viewers. Previously, this kind of visualization did not
scale well for larger high-dynamic-range data, i.e., containing outliers and
regions with high sample frequency alike. Addressing RQ1, we will in-
vestigate methods of applying nonlinear scaling to dot plots. Although they
require more experience to interpret correctly, nonlinear scales already
work well with bar charts. For the nonlinear plots proposed in Part I, we
will apply the scaling to the dots that represent individual data points. Such
unit visualization provides an accessible layout where the data frequency
is low because viewers are able to recognize and count the individual dots.
On the one hand, the simplicity of dot plots provides visualization for a
broad range of audiences. On the other hand, it is only suited to display
a single dimension. We can enrich the visualization by encoding a second
attribute as color or icons within the dots. Such an approach works best
for few and quantized values.

When the data contains two numerical dimensions, it is intuitive to also
use two display axes. Many viewers are used to this from school, where they
plotted equations with x- and y-variables in Cartesian coordinate systems.
But there is also a need to analyze data where there is no previously known
equation that governs the relationship between the two dimensions. In this
case, we can use the x-axis and y-axis of the display and draw unconnected
individual data points. The resulting scatter plot allows viewers to gain
deeper insight into possible connections between the data dimensions: It
allows them to find correlations, to identify trends and clusters.

2.2 Multivariate Data
Plots and graphs up to 2D are helpful in cases where the data is of the
same low dimensionality. Nevertheless, data sets can include more than
two attributes for the same record. For example, a table with champions
in motorsports might include the year of a race, name of the driver, name
of the team, various measurements of the vehicle, length of the track,
total race time, etc. A different data set might include additional attributes
forming a network graph that shows how different racing teams interact.
With the abundance and variety of recorded information, it is inevitable for
some analysts to work with multivariate data. For example, when trying to
optimize engine power output, it is necessary to analyze multiple variable
settings, i.e., dimensions, and find their effect on the final output but also

10



2.2. Multivariate Data

interactions between multiple settings. Increasing the amount of injected
fuel requires a different setting for the air intake valve and might influence
the combustion temperature.

What visualizations can analysts use when the number of dimensions
increases beyond two? There is a plethora of techniques that fit specific
kinds of data. To keep the length of this thesis within reasonable limits,
we will only discuss generic visualizations for numerical data on rational,
ordinal, or nominal scales.

2.2.1 Point-Based Visualization
Scatter plots are a suitable approach for 2D data, and many people have
used them, but their perpendicular axis layout does not scale with a higher
number of dimensions. 3D scatter plots on 2D paper or computer screens
are prone to overdraw, and the introduced virtual depth is not easily
perceivable for the user. It is, however, possible to create a grid with scatter
plots of all dimension pairs: the scatter plot matrix (SPLOM) [Eme+12]. As
the number D of data dimensions increases, each individual cell shrinks to
fit the increasing number of D2 cells in the matrix. Therefore, this approach
is often combined with a regular scatter plot. The matrix provides only
coarse thumbnails and allows the user to select a dimension pair for the
larger plot.

This technique allows analysts to explore the data in detail. But how
can they trace individual points between the different plots? Previous work
has suggested animating the dots when switching between data axes in the
main scatter plot [EDF08]. In accordance with RQ2, this thesis presents
a selection of animation techniques and compares them with respect to
user performance for following individual data points and for identifying
cluster interactions (see Chapter 6).

With animation, it is easier to maintain the mental model while switching
between different views. But how should analysts know what cells of the
SPLOM to select? Which cells are the most relevant and have the least
redundancy? Previous work has resulted in recommender systems that
use a manual selection of known relevant cells to suggest new dimension
pairs [Beh+14]. In line with RQ2, this dissertation presents a technique
that uses the analyst’s gaze to identify data of interest. The proof of concept
in Chapter 7 combines this implicit data selection with multiple metrics to
provide information that can help the viewer select the next interesting
cell in the SPLOM. The concept of using gaze to select data is reasonably
well suited for 2D visualization. Beyond the original scope of RQ2, we
also explored the use of 3D gaze with stereo eye-tracking hardware in
Chapter 8.

11



2. Data Types and Their Visualizations

2.2.2 Line-Based Visualization
A SPLOM scales with the square of the number of dimensions (D2), re-
quires much space, and the individual cells become increasingly smaller
to fit on the screen. Additionally, it requires interaction to switch between
different detailed views. What happens when the number of data dimen-
sions increases too far, or there is no way of interacting with the plotting
software? For this reason, we formulated RQ3 and investigated ways to
combine advantages from multiple 2D views in a single static visualization.
One could use dimensionality reduction techniques to arrive at a 2D scatter
plot from an arbitrary number of data attributes. However, the resulting
display dimensions are often not intuitive to grasp as the connection to the
underlying data dimensions is obfuscated.

A possible answer to RQ3 is not based on visualization with points
but rather with lines. A PCP shows all data dimensions with parallel—most
often vertical—axes. Each data point becomes a polyline, and the crossings
with the axes represent the data value in each dimension. The results of
an expert review suggest that they are helpful for analysts that want to
explore previously unknown multivariate data [Sch+17]. In Chapter 9, we
propose a technique that extends on the original PCPs and allows reading
the data values of individual samples but also shows soft clusters in 2D
subspaces (advantage of scatter plots). Our proposed visualization shows
how individual data points move between clusters and how the clusters of
adjacent dimension pairs interact with each other (advantage of animation
between scatter plots). The technique is, however, targeted at experienced
analysts.

As with all PCPs in general, the axis sequence on the screen influences
heavily what patterns of the data are visible. There is previous work on
quantifying the patterns and optimizing the sequence [DK10]. As our work
in the context of RQ3 focuses on transferring the advantages from
multiple 2D visualizations, we also developed new metrics to optimize the
dimension order for cluster interactions.

2.3 Spatial Data
Spatial data can be encoded with two and three dimensions. It is often well-
suited for display with individual dots marking a position on a map and can
also be useful to draw lines that show trajectories. We created a point-based
map visualization of data on power plants for the general public [Rod+17b].
We also worked on the visualization of eye-tracking scanpaths. They are
well suited for display on a stimulus by drawing lines for saccades and,

12



2.3. Spatial Data

optionally, dots for fixations. We also worked with eye-tracking trajectories
from multiple study participants on the same stimulus and found ways of
reducing the visual clutter of scanpaths. We used one approach based on
quad-trees [Rod+18] and one with group diagrams [Sch+22; Sch+23].

While work on spatial data is important, it is often targeted toward the
visualization of data sets from specific contexts and does not fit the broader
and more generic theme of this dissertation. Hence, we will not include
these applications in further discussions.

13





Part I

Dot Plots

15





Part I – Dot Plots

-2
Value

420

Figure I-1: A dot plot, according to Wilkinson [Wil99]. Each dot represents
a value along the number line at the bottom. Dots are stacked into straight
columns if they cannot be placed next to each other without overlap.

Visualization is not limited to complex interactive systems for the par-
ticular analysis of large data sets. It can be simple, show only a single
data dimension with few samples, and yet be helpful. Dot plots are a case
of such relatively simple visualization. Even school children without prior
knowledge can use them to extract characteristics of the data distribution [I
G15]. The basic algorithm behind dot plots does not require computing
hardware and can be used for hand-drawn visualizations.

Dot plots are composed of a single number line and a circular dot
for each data point. When neighboring dots would overlap, the layout
algorithm stacks them into straight columns (see Figure I-1). As a result,
while the input data only has a single attribute, dot plots use 2D output to
convey information: One axis—typically horizontal—displays the value of
the underlying items, and a perpendicular axis shows data frequency.

The attentive reader might now note how the goals of histograms and
dot plots overlap. There are, however, decisive differences. First, dot plots
are a kind of unit visualization that maps each data point to an individual
graphical element. This allows for a clearly readable representation with
countable items and tends to be more intuitive to understand [Bak04].
The second major difference concerns the bins in regular histograms. All
points along the visible data axis belong to a bin. There are no unassigned
ranges. Additionally, the bins themselves are uniform. Contrary to this, the
dot plot positions the circles where there are data items. This can create
empty spaces between areas of higher data frequency and outliers that are
positioned accurately with regard to their underlying value.

Unfortunately, dot plots also have disadvantages with regard to other
visualization techniques. Unit visualization is better for an intuitive un-
derstanding, but it does not scale well with increasing sample count and
high-dynamic-range data. Stacking dots into straight columns moves the
display points away from their underlying value. Finally, showing many

17



Part I – Dot Plots

Transformed
data

Displayable
representation

View
user

Processed
data

Manual interaction and feedback

Filtering

nonlinear dot size

Mapping

Abstraction

PerceptionRendering

directed blur
blue noise property

Figure I-2: The visualization pipeline (adapted from [CMS99]) with additions
from the proposed dot plot techniques. Affected elements are highlighted
in red. High-dynamic-range data is supported through dot size scaling.
Directed blur and blue noise reduce unwanted perceptual effects.

dots in close proximity tends to create moiré effects. Following the theme
of RQ1, we adapted and extended various parts of the visualization
pipeline of dot plots to tackle the disadvantages and make this intuitive and
accessible visualization suitable for a wider range of data (see Figure I-2).
In Chapter 3, we change the mapping from data points to differently sized
dots and include a data-agnostic low-pass filter to reduce moiré patterns. In
Chapter 4, we adapt the location of dots to decrease positional error and en-
sure the blue noise property, which is better suited for visual perception in
the later stage of the pipeline. With these latest changes, remaining visual
patterns are not an unwanted side effect but actually convey information
about the underlying data.

Definition of Dot Plots
Before continuing with details about the proposed techniques, we have
to define what we mean by “dot plots.” There is ambiguity because, over
time, there have been multiple visualizations that share the same name.
Already in 1884, Jevons is supposed to have included a dot plot to show the
distribution of coin weights [Wil99]. More than a century later, Cleveland
also employs the name “dot plot” for a visualization that functions like a
bar chart [Cle85]. The difference is that each bar is replaced by a single
dot at the far end to avoid meaninglessly covering an entire area. He
also proposed ordering the data items to enhance the perception of the
distribution of values, as in the example in Figure I-3. The same author
also composed the plots in matrices of small multiples to create “multiway
dot plots” [Cle93]. In 1996, Sasieni and Royston [SR96] presented dot
plots that use the regular binning of histograms but replace the bars with
columns of same-sized dots. Depending on the bin width, the individual
dots are hard to recognize.

So far, the mentioned techniques have a common theme: they are based

18



Part I – Dot Plots

-2
Value

420

Apples

Strawberries
Raspberries

Cherries
Pears

Bananas
Oranges

Figure I-3: A “dot plot,” according to Cleveland [Cle85]. The dots in each
row encode a value. Sorted rows facilitate the perception of distribution.
The underlying data is the same as in Figure I-1

on and related to bar charts but use circles instead of rectangles. For
our work, we will focus on the definition and algorithm introduced by
Wilkinson [Wil99]. His technique yields Figure I-1 when run on the same
source data as Figure I-3. We distilled four rules that apply to regular linear
dot plots:

DP1 There must be a one-to-one mapping between each data value and
each rendered dot.

DP2 All dots are of the same size.
DP3 Lone dots are placed at the exact position of their data value along

the displayed scale.
DP4 Colliding dots are stacked into straight columns.

These rules result in properties that distinguish dot plots from histograms
and other bar-chart-based visualizations.

More recently, Dang et al. [DWA10] generalized the concept of stacking
and produced symmetrical dot plots. They further extended the previously
existing technique to create 3D visualizations of 2D data. Their latest
advances, however, go beyond the scope of this thesis. The 3D visualization
is harder to read and more complex to create in hand-drawn plots, shifting
the focus toward a different target audience.

In this part of the dissertation and the included chapters, we often talk
about differences, advantages, and disadvantages. Note that we need to
stress differences to distinguish between the techniques but do not intend
to discredit or look down upon any of the mentioned plot types. They all
fit their own specific use cases—often in statistical contexts. Even in other
applications, such as landscape visualizations, dot-based visualization can
be more memorable and outperform alternative approaches [TSD09].

19





3

Nonlinear Dot Plots

This chapter is based on previous work and will use extracts thereof without
explicit quotation:
N. Rodrigues and D. Weiskopf. “Nonlinear dot plots.” In: IEEE Transactions
on Visualization and Computer Graphics 24.1 (2018), pp. 616–625. DOI:
10.1109/tvcg.2017.2744018.

3.1 Motivation
So far, we have established that regular dot plots provide a simple and
intuitive visualization for the distribution of data items. They show clusters,
gaps, and outliers and are most useful for smaller sample counts. However,
they do not scale well for high dynamic frequency ranges in the distribution
of underlying data items. The reasons lie within the rules and assumptions
behind their layout algorithm. Rule DP1 is responsible for the intuitive unit
visualization. There are, however, multiple implications from rule DP2 that
have negative implications that we will now discuss.

Imagine increasing the number of items for visualization by replicating
each point in the source data set. As a result, the sample count rises,
but the range of values remains unchanged, just as in the real-world
scenario of performing repeated measures to get a robust representation of
a distribution. Assuming a constant circle diameter, the columns in the plot
will grow linearly in height as the data set size increases. But the width is
only affected by the range that the data points cover on the x-axis. The plot
will need to be rescaled to fit the computer display or sheet of paper, and
the aspect ratio will shift toward a narrow visualization. To address this
issue, we need to reduce the diameter of the used circles. This will reduce
overlap and produce a more fitting aspect ratio.

21

https://doi.org/10.1109/tvcg.2017.2744018


3. Nonlinear Dot Plots

X1 X2 X3 X4 X5X1 X2 X3 X4 X5

Figure 3.1: Illustration of Wilkinson’s sweep layout algorithm [Wil99].
Adding dots for X2, X3, and X4 would create overlap (marked red). There-
fore, dots for these values are stacked. X1 and X5 can be plotted without
any issues.

With a shrinking diameter, the overall shape of dense areas in the dis-
tribution remains visible, but outliers can become too small to perceive.
Therefore, the selection of a suitable dot size is key to creating a usable
visualization. Wilkinson observed an analogy between circle size in dot
plots and the bin width in histograms. Similarly to data-dependent sugges-
tions for the latter, he presented a dot size that depends on the number
of samples in the source data and assumes a normal distribution with
none to moderate skewness. His approach yields a steady aspect ratio for
normal distribution, but outliers remain too small. The newer method from
generalized stacking does not solve this issue either [DWA10]. However,
depending on specific parameter settings and the variability in data density,
it introduces more overlap or larger empty spaces.

Directly addressing RQ1, our goal within this chapter is to expand on
the intuitive original dot plots and adapt them for higher dynamic ranges of
data density while preserving the visibility of outliers. A useful aspect ratio
requires small dots in dense areas of the plot, while outliers must remain
large enough for perception. Therefore, our approach scales the circle
diameters as the number of dots in a stack increases, yielding nonlinear
dot plots. While the concept of nonlinear scaling has already been used
with other visualizations [LA94], it is new for dot plots.

3.2 Technique
3.2.1 Original One-Way Sweep

We use a simplified version of Wilkinson’s original algorithm for linear dot
plots [Wil99] as a starting point for our extension to nonlinear dot plots. For

22



3.2. Technique

Algorithm 3.1: Wilkinson’s original sweep algorithm
Input: Data set X. Dot diameter d.

1 Sort X ascending
2 while X ̸= ∅ do
3 Xi ← lowest non-placed value from X
4 c← 1
5 while |Xi −Xi+c | ≤ d do
6 c←c +1

7 Place c dots with diameter d in a column above (Xi −Xi+c )/2
8 Mark Xi to Xi+c as placed

a self-contained description, we briefly summarize the original algorithm
without smoothing and lateral offsets in this section.

Algorithm 3.1 performs an upward sweep through the data to place
the dots from left to right. It assumes that the data samples are given as
data points with corresponding data value Xi, with index i = 1, 2, . . . , n for
n samples in the entire data set. It depends on the distance d, which is
the (constant) diameter of the dots for the creation of columns. The key
point is that the algorithm keeps incrementing the number of dots, c, for
the current column as long as further dots still overlap with the column
(lines 5 and 6).

To accommodate large, high dynamic range data sets and preserve the
advantages of dot plots, the diameter has to be varied in the same plot:
the higher the dot column, the smaller the rendered symbols. This leads
to a nonlinear modification in column height and width, which is not very
intuitive to interpret. In a histogram with nonlinear scaling, we would only
have to measure the height and know the transformation function in order
to calculate the represented value. With nonlinear dot plots, we also have to
factor in varying column widths, making the computation of displayed value
densities more difficult (see Section 3.2.5). However, the total size of the
plot can be reduced while retaining large dots for outliers. The decrease
in height also provides enough flexibility to bring the output closer to the
optimal aspect ratio.

We first describe an extended two-way sweep algorithm that allows
us to work with varying dot sizes (Section 3.2.2). Then, we discuss the
merging of intermediate results of the two-way sweep (Section 3.2.3),
models for the data-driven adaptation of the dot diameter (Section 3.2.4),
resulting envelopes (Section 3.2.5), and aliasing from rendering dots (Sec-
tion 3.2.6). The section closes with extensions and variants of the visualiza-
tion, including color coding and overlays with other diagram components
(Section 3.2.7).

23



3. Nonlinear Dot Plots

Algorithm 3.2: Adaptation for symmetric nonlinear dot plots.

Input: Data set X. Scaling function dc(c) for stacked dots.

1 Sort X ascending
2 while X ̸= ∅ do
3 Xi ← lowest / highest non-placed value from X

4 c← 1

5 while |Xi −Xi +− c
| ≤ dc(c) do

6 c← c +1
7 Place c dots with diameter d1 in a column above (Xi −Xi +− c

)/2

8 Mark Xi to X
i +− c

as placed

9 Do one upward and one downward pass

10 Use average of upward and downward pass

3.2.2 Two-Way Sweep Algorithm
We present a summary of our new two-way sweep in Algorithm 3.2. The
modifications with respect to the original Algorithm 3.1 are highlighted.

To adapt the dot size, we have to replace the constant diameter from
Wilkinson’s original algorithm (line 5) with a data-dependent variant:

dc(c) = d1 · gc(c) , (3.1)

where the index c denotes a function that performs the mapping of a
dot count within a column. The variable c (not indexed!) represents the
current number of data points in a column, and d1 is a given start diameter
that facilitates an overall scaling of all dots and remains constant during
the entire layout pass. The function gc(c) should be weakly monotonically
decreasing, i.e., dots should become smaller for columns with more data
points. Also, we should have gc(1) = 1 so that we obtain the diameter d1
if just a single dot is placed. Section 3.2.4 discusses concrete examples
of gc(c). With our approach, the original linear dot plots are included as
a special case when gc(c) = 1 for all c. Therefore, all improvements from
the new algorithm that are not related to scaling carry over to the linear
variant.

In general, scaling compresses the height range but also increases the
cognitive load for reading exact values [Hla+13; Tuf83; CM85; BDJ14;
Arb+17]. Please note that—as per Equation (3.1)—we keep diameters
within each stack constant. As a result, our technique provides a partially
linear representation that preserves rule DP1 locally to aid comprehension.

24



3.2. Technique

As indicated in Algorithm 3.2, we can use the data-dependent dot diam-
eter (Equation (3.1)) within Wilkinson’s original single-sweep algorithm.
However, while this single-sweep approach works very well for a constant
dot size, it exhibits some issues when the data value density decreases
along the sweep direction. The underlying reason is that shrinking diame-
ters affect column width and, in consequence, they change which dots are
to be stacked together.

To illustrate this problem, we will assume that the data consists of a
dense, sorted group of values X. The first and last values are within the
initial dot diameter, which would lead to rendering a single column for
linear plots: Xn −X1 ⪅ d1. During the upward sweep from the smallest to
the largest value Xi, the dot count c1 in the first stack increases, and the
column becomes narrower: dc(c1)≪ d1. Only a few values from X remain
and create their own column with a small number of c2 dots with a size
close to the initial diameter: dc(c2) ⪅ d1. Since all values have less than d1
distance, there will be overlap between the two columns’ dots.

The inverted case, however, presents no problems: When the density
increases along the sweep direction, the columns become narrower. There-
fore, a second column will not be as wide as the first one, and there will
not be any overlap between them.

This observation leads to a solution to the problem: we use two sweeps—
one in each direction—and combine their results by averaging. Algo-
rithm 3.2 indicates the two sweeps as upward and downward passes.
Please note that c remains a positive number of dots in the current column,
regardless of the sweep direction. Section 3.2.3 describes in more detail
how the two sweeps are combined.

The original sweep algorithm only needs one pass over the source
data. Since none of our alterations are of higher complexity than O(n), our
proposed algorithm still has the same linear runtime complexity (with a
factor of roughly 2 for the number of computations).

3.2.3 Combining Sweeps
Dang et al.’s greedy version of a dot plot algorithm [DWA10] was designed
to achieve symmetrical diagrams. While it is preferable to create sym-
metrical visualizations when using symmetrical data, the algorithm either
introduces severe overlaps or gaps between dot stacks (depending on the
chosen factor for h in step 1 of their algorithm). As already indicated in
Section 3.2.2, a single sweep direction creates overlap problems for non-
linear dot plots, too. In addition, it also causes asymmetry under certain
conditions.

25



3. Nonlinear Dot Plots

(a) Single left-to-right pass

(b) Two opposed passes combined

Figure 3.2: Two dot plots of the same data with varying value density: rising
from 1–2, falling from 3–4, and constant between 5–6. Vertical lines on the
bottom axis show the individual data value positions. Merging two passes
results in a plot with less overlap and more symmetry (b).

To address these problems, we decrease the sweep direction’s influ-
ence on the final layout by combining both directions, as described in
Algorithm 3.2. The open question is: how are the results of two opposing
sweeps adequately combined?

As we will explain later in this section, the rules we imposed on the dot
diameters (Section 3.2.2) guarantee that both sweeps return the same num-
ber of stacks. We can, therefore, define a one-to-one mapping of columns
from one layout pass to the other. Columns resulting from a single sweep
are defined by three values: a position in the data dimension as well as the
number and diameter of the dots. For the merging of stacks from opposite
sweep directions, their positions can be readily averaged by using their
arithmetic mean. However, it is not straightforward to merge the actual
dots if the two columns contain a different number of them: the arithmetic

26



3.2. Technique

mean of the number of included dots might contain “half” points, but we
can only draw entire dots.

We choose to only draw entire dots by rounding off the number of
dots in the current column and then carry the remainder to the following
column. In total, this will result in the same number of dots as from a
single pass, and the remainder is only distributed within a neighborhood
of columns with less than d1 distance. If the neighboring column—that
received the remainder—were farther apart than d1, it would belong to a
different cluster of columns. A distance between data points greater than d1
would not allow for overlap, thus, would not require stacking of dots that,
consequently, cannot interfere with each other in the sweep algorithm. In
a nutshell, dots that are farther than d1 apart can never appear in the same
column!

The arithmetic could be applied to the column’s positions directly and
their height (number of dots) with the slight modification above. However,
the arithmetic mean is not suitable for the diameter of rendered dots.
This is an implication of the nonlinear dependency between c and dc(c) in
Equation (3.1). Instead, we calculate the correct diameter from the actual
number of dots in each column by applying Equation (3.1) to c after the
number of dots is averaged.

There remains one prerequisite for the two-pass method to work: both
passes must return the same number of columns. This is important because
the first column of the upward pass has to be averaged with the last column
of the downward pass. If they do not match, the data values of different
neighborhoods will be merged. Now we analyze the scenarios encountered
during the layout process to check whether they meet the mentioned
prerequisite. Our first observation is that data values that are further apart
than the start diameter d1 divide the data samples into clusters that can
be treated separately. Therefore, we can restrict the discussion to a single
cluster of data samples.

Clusters that produce a single column in one direction will also produce
a single column in the opposite direction, i.e., this is a trivial case. Let
us assume the upward pass A resulted in two columns and look at the
following two scenarios.

Scenario 1: If the downward pass B returned a single column, it would
mean that the total number of values in the cluster was so small that dc(c)
was still larger than the distance between the first and last data value.
This would, in turn, also mean that pass A would have returned a single
column containing the complete cluster, which is in direct contradiction to
our premise.

Scenario 2: Can pass B return three columns? If it started at the same
data point, where pass A finished, then its first column would cover at least

27



3. Nonlinear Dot Plots

the same amount of data values as the last column of A. That would leave
the same data that pass A used for its first column, which in turn would also
only result in a single column of pass B. Therefore, three columns could
only be generated if pass B encountered additional data values outside
the cluster. However, we only look at the data inside the cluster because
different clusters are so far apart that they do not interfere with each other.

According to this logic (similarly to mathematical induction), the two
opposed passes cannot result in a different dot stack count, which makes
our averaging method applicable to arbitrary source data.

As shown in Figure 3.2b, the two-way layout leads to columns that are
better centered on contained data and have less severe overlaps than the
single-pass algorithm (see Figure 3.2a). This makes our algorithm well
suited not only for nonlinear dot plots but also improves the layout of
traditional, linear plots. Figure 3.2b also shows that a constant sample
density does not necessarily lead to a constant dot stack height, which
makes the visualization inconsistent with the underlying data. This is due
to rendering individual dots, which leads to quantized column heights. In
order to get a constant height among all columns of the third cluster, we
have to fit the initial dot diameter and the nonlinear transformation to
the specific samples. While these adapted parameters will fit the targeted
cluster, they might be wrong for the clusters at 1 and 3. Therefore, it is not
trivial to find a globally optimal solution.

It is not even clear how to completely define the optimality of a solu-
tion. We might define an optimal solution as one in which the distance of
rendered dots from their input values (layout error) is minimal. If we then
selected a sufficiently small dot diameter, we would render each dot at
the exact data value position and have an optimal solution according to
that metric. However, the resulting plot would not be readable because the
dots would be too small to perceive. Therefore, an objective function for
an optimal solution would have to consider the layout error but also the
dot sizes, display resolution, viewing distance, and aspect ratio. We leave
the definition of such an objective function as an open question for future
research.

3.2.4 Dot Diameter
The above two-way sweep algorithm makes heavy use of the dynamic
adaption of dot size according to Equation (3.1). We will now discuss useful
choices for the adaptation model, as formalized by g(c). For this, we assume
that we have a rather dense packing of dots so that the dot plot resembles
the corresponding histogram.

28



3.2. Technique

(a) 1,000 dots without blur (b) 10,000 dots without blur

(c) 1,000 dots with blur (d) 10,000 dots with blur

Figure 3.3: Low-resolution plots of normal distributions. Column-oriented
anti-aliasing dampens moiré patterns in visualizations (c) and (d). When
used with large dots, this blurring introduces unwanted optical effects (c).

Although gc(c) controls the nonlinearity of the dot plot, it is not identical
to the nonlinear mapping known from histograms or other function plots.
It is important to note that gc(c) does not map the original height of a
stack to the nonlinear modification. For example, we cannot simply use
gc(c) = log(c) to obtain the analog of a log-scale dot plot. In fact, the height
of a column with c dots is:

hc(c) = c · dc(c) = c · d1 · gc(c) . (3.2)

As noted earlier, we require that gc(1) = 1 and that gc(c) decreases with
increasing c. In addition, we want to guarantee weak monotonicity of the
plot: a column with more data points should never be smaller than a column
with fewer points, i.e.,

hc(c1) ≤ hc(c2) if c1 < c2 . (3.3)

The extreme case of constant height is obtained for:

gc(c) =
1

c
, (3.4)

i.e., this choice leads to hc(c1) = hc(c2) = d1. As shown later in Figure 3.4c,
there are some applications for this extreme model. The corresponding

29



3. Nonlinear Dot Plots

visualization resembles a combination of jittered strip charts and stripe
charts [Cha+83] (see Figure 3.4c). Typical nonlinear mappings target
strong monotonicity. To this end, the dot size needs to shrink less quickly
than in Equation (3.4). A corresponding mapping is achieved by:

gc,r(c) =
1

cs
, (3.5)

with an additional parameter s that controls the shrink rate. The useful
range for the shrink rate s is between 0 and 1. Selecting a shrink rate
of s = 0 will create traditional linear dot plots; s = 1 yields the earlier
model of Equation (3.4). For in-between shrink rates, the column height
is proportional to c1−s, with the fractional exponent 1 − s corresponding
to a root. We mark functions related to this root scaling with index r. For
images within this chapter, we select a default value of 0.4, which leads to
a column height of c0.6. This choice leads to good results for the upcoming
example plots but could certainly be replaced with other values. We added
a table with varying d1 and s as supplemental material to the original
publication [RW18] to illustrate the effect of these parameters on the final
layout.

The root mapping in Equation (3.5) might be good for many data sets,
but it will not fit most processes in nature that exhibit exponential growth,
where the growth rate is proportional to the current function value. Exam-
ples of such can be found in nuclear decay, the Weber-Fechner law [Fec60],
and population growth. The existence of exponential processes is also
reflected in the way we represent floating point numbers: mantissa and
exponent. To get a log-scale mapping, we naively try to approximate

hc(c) = log(c) . (3.6)

Based on Equation (3.2), we would calculate

gc(c) =
logb(c)

d1 · c
(3.7)

for any desired base b, but this would violate the requirements for single
dot columns, as log(1) = 0. To get gc(1) = 1, we could change the numerator
in Equation (3.7) to

logb(c) + d1 .

This, in turn, would lead to other problems. For instance, with b = 2 and
d1 = 0.5, the diameters in columns with two dots would be larger than
a single dot (gc(2) = 1.5). Instead, for logarithmic dot plots (index l), we
propose

gc,l(c) =
logb(c+ b− 1)

c
, (3.8)

30



3.2. Technique

as it fits all requirements when used with b ≥
√
5+1
2

(the golden ratio). The
supplemental material of the underlying publication [RW18] includes a
table with varying d1 and b to illustrate the effect of these parameters on
the layout.

The final issue is the choice of d1. This parameter should be chosen
according to the number of data samples, their distribution, the size of
the plot, and—very importantly—the targeted aspect ratio. We iteratively
optimize for d1 with nested intervals: first, we compute a plot with the
current value of d1; then increase it in the next iteration if the current
aspect ratio is wider than desired and vice versa until we reach a ratio that
is sufficiently close to the target.

3.2.5 Envelope
The above discussion holds for the mapping of height, i.e., a single dimen-
sion. Now, the dot plots intrinsically link column height and width because
both dimensions are determined by the dot diameter. If the diameter is
scaled by a factor a, the covered area (i.e., the area of the column) is scaled
by a factor a2. Therefore, a square root computation needs to be included
in all mappings if the adjustment is meant for areas, not just height.

Based on these observations, one can consider the limit case of a very
large number of dots and the envelope of the nonlinear dot plot. For the
case of root dot plots according to Equation (3.5), the height of the envelope
scales with α(1−s)/(1+s) if the number of dots is multiplied by α. Thus, the
exponent s from the diameter scaling matches an exponent of (1− s)/(1+ s)

in the corresponding nonlinear histogram.

3.2.6 Anti-Aliasing
In general, dot plots can come with high demands regarding rendering
quality because they consist of clearly defined dots with sharp boundaries.
Such boundaries, in combination with some rather regular placement of
dots, can lead to aliasing and moiré artifacts [Gla69; SW82]; see Figure 3.3.
Small differences in dot sizes are the main cause of moiré effects. These
accumulate along the height of neighboring columns and create virtual
tilted lines.

Typical anti-aliasing approaches from computer graphics work with
supersampling on the image plane, followed by low-pass filtering and down-
sampling. Our solution adopts the same strategy but exploits the special
characteristics of dot plots. In the rendering stage of the visualization
pipeline, low-pass filtering blurs the image, i.e., the individual dots would
eventually disappear, and only a solid colored area would appear. Other

31



3. Nonlinear Dot Plots

than generic anti-aliasing techniques, we limit blurring to the vertical
direction because individual columns should still be distinguishable. Fur-
thermore, a few dots at the top and bottom are left unchanged, as they
play a key role in estimating the size of individual dots and in comparing
column heights.

We also decided on not blurring those columns that are not surrounded
by others because they do not add to the moiré effect. Finally, we only
start blurring after the dot count inside a column exceeds a certain lower
threshold (in our examples: 12) because otherwise, the rendering does
not create an area that is big enough for the effect to be perceivable.
Figures 3.3c and 3.3d show examples of anti-aliasing. Dots in a blurred line
are not countable anymore. However, as the height of columns increases,
and the dot diameter decreases, it becomes more and more difficult to make
out individual dots for counting, anyway. We use our anti-aliasing method
with moderation in order to balance advantages and side effects. The
vertical lines in Figure 3.3c are an example of overly aggressive blurring
for the low dot density that trades the moiré pattern for even worse optical
effects, especially when rotating the image. Therefore, we recommend
anti-aliasing only for plots with very small dots, as in Figure 3.3d.

3.2.7 Variants, Extensions, and Hybrid Visualizations
Just like conventional dot plots, our nonlinear generalization can be widely
applied to depict any kind of data distribution. Similarly, it can be combined
with other visual mappings to include further information or emphasize
certain aspects of the data.

One example is additional color mapping. In general, color plays an im-
portant role in visualization because it can show additional data attributes
on top of the positional variables of the diagram. We argue that color map-
ping is especially useful in the context of dot plots because each single
data sample generates exactly one dot, i.e., we can have a direct mapping
between sample and color. To make use of perceptual grouping by color
(hue), similar colors should be spatially grouped in the dot plot. We cannot
change the layout between the columns in the dot plot because they are
driven by the distribution of data values. However, we may modify the
order of dots within a column. Therefore, the dots in each column should
be ordered vertically according to the additional data attribute mapped
to color. A typical example is a chronological data distribution, i.e., a data
set with samples that not only carry some data value but also a timestamp.
Such time-series distributions are best ordered chronologically in each dot
column. Another possible application of color is the comparative visualiza-

32



3.3. Examples

tion of several data sets integrated into one dot plot: the color indicates
the data sources. The figures in Section 3.3 use such colored dot plots.

We mostly render our dot plots with the stacks aligned to the x-axis.
It is possible, however, to center the columns vertically and create dia-
grams with a horizontal symmetry axis (like Wilkinson’s symmetrical dot
plots [Wil99]). Further variations can align dot stacks to the y-axis. This
is especially useful when dealing with nominal data as it improves the
layout of labels and creates an output similar to Cleveland’s multiway
dot plots [Cle93]. In addition to layout and rendering variations, we can
combine different visualization methods. For instance, Figure 3.6b shows a
symmetrical dot plot overlaid by a box plot, whereas Figure 3.2 adds strip
charts to the x-axis. Such hybrid diagrams can help users classify and iden-
tify data in meaningful ways, providing more insights. Tukey’s suspended
rootograms plot data in relation to a known density distribution [Tuk72].
The same technique could be applied to the vertical positioning of the dot
stacks to show deviations but would require special care to adjust for areal
distortions (see Section 3.2.5).

3.3 Examples
We demonstrate nonlinear dot plots for typical examples of real-world data.
These contain frequencies of varying ranges. We include comparisons to
histograms and linear dot plots to examine different characteristics.

3.3.1 Distribution of Air Temperature
Air temperature has a direct impact on our daily life, but it is also related to
issues of global climate change. Therefore, we pick this application as our
first example. In 2017, we downloaded a data set from the “Climate Data
Centre” of the German meteorological service (Deutscher Wetterdienst). It
contains monthly averages of maximum daily air temperatures by weather
station for the years 1961 to 1990. The stations are labeled using the
identifiers supplied by the World Meteorological Organization (WMO). This
data set provides a total of 9685 data samples from 875 weather stations
worldwide; temperatures are in degrees Celsius. The data is still available
online, albeit in a different format and with more samples [Deu].

Visualizing this data with the plots in Figure 3.4, one can immediately
see that the distribution is unimodal and has its maximum between 31 to
32 degrees. While mean temperatures down to around -30 degrees are
still quite common, only single instances of temperatures at or below -
31 degrees can be observed. Temperatures between 16 and 29 degrees

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3. Nonlinear Dot Plots

(a) Root dot plot

(b) Linear dot plot

(c) Strip dot plot (root plot with s = 1)

(d) Logarithmic histogram

Figure 3.4: More than 9,000 air temperature values (in degrees Celsius).
The data set shows the monthly average of daily maxima measured by
weather stations all around the world. Individual dots in (a) and (b) are
colored according to the month: J F M A M J J A S O N D . We use black
dots in plot (c) to maximize the perceivable range of brightness.

34



3.3. Examples

are almost evenly distributed, forming a plateau in the linear dot plot.
This plateau is also visible in the logarithmic histogram and the root dot
plot, albeit less noticeable. The nonlinear plots outperform their linear
counterparts when we focus on less dense areas of the plots. Figures 3.4a
and 3.4d both show minimum and maximum values (outliers) clearly, but
only the dot plot allows counting them. The histogram would need a fine-
grained vertical axis with ticks at intervals of 1, which would lead to
overplotting. The extreme, strip chart-like visualization in Figure 3.4c
only shows the outliers clearly but turns the dense regions into light-gray
areas. Cross-referencing the temperature data with the WMO database
yields additional geographic information. Picking data points in nonlinear
dot plots for cross-referencing is very simple because of the one-to-one
relationship between dots and data points and since the low-frequency
points are rendered as quite large dots.

While the two nonlinear visualizations look similar at first glance, the
dot plot provides more detail in dense areas. The histogram cannot provide
such a view, as all its bins are of equal width. Figures 3.4a and 3.4b also
exhibit some gaps in the stacks (for instance, near =4 °C), which contrast
the tightly packed neighboring areas. This is due to the characteristics of
the data source: the temperatures are rounded to a single decimal place,
and there are many data points with the same value. The value density in
this area is too low to create high and narrow columns, but at the same
time, it is too high to place a wider column at each decimal of a degree.
Therefore, the layout algorithm cannot create a tightly packed field of
uniform columns, which leads to visible gaps.

To encode more information, we colored the individual dots in Fig-
ures 3.4a and 3.4b according to the four seasons. We compensated the
phase shift between the Earth’s northern and southern hemispheres by
adding six months to data from weather stations below the equator. Using
this colorization with the nonlinear dot plot, it is immediately noticeable
that higher temperatures tend to occur in summer, while the lower ones
are predominantly measured in winter. That is no surprise; however, the
tendency is not so obvious when looking at the linear dot plot. This effect
can be explained by the unimodal sample distribution that has a lower value
frequency at the outer edges and the bias toward outliers in Figure 3.4. It
would be possible to color the histograms analogously to the dot plots by
subdividing the bars. However, there is no appropriate color coding for the
logarithmic histogram because there is a conflict between the linear (pro-
portional) splitting of individual bars versus the overall logarithmic scaling.
There is no such ambiguity with the colored dot plots, as the individual
stacks are always linear, giving the users an impression of the distribution
of sample categories within the stacks.

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3. Nonlinear Dot Plots

(a) Log2 dot plot using d1 = 795 (Shnei-
derman only)

(b) Root dot plot using s = 0.6 and d1 =
526

(c) Log2 dot plot using d1 = 795 (d) Linear dot plot using d1 ≈ 8.4

(e) Logarithmic histogram (f) Linear histogram

Figure 3.5: Number of citations for papers by Ben Shneiderman ,
William S. Cleveland , Leland Wilkinson , and William E. Lorensen . Each
dot represents one of 1,463 publications from these authors.

3.3.2 Citation Statistics

Our next example shows bibliometric data in the form of citation statistics.
The h index [Hir05] is a popular indicator of publication impact by an
author, heavily aggregating data about all papers into a single number. In
contrast, visualizing the complete citation data presents a challenge, as the
number of citations may vary extremely per author and paper: there tend
to be few publications with a big impact (i.e., many references) and many
papers that hardly anyone notices (i.e., little to no references). Therefore,
we obtain a high concentration in the frequency plot near zero and some
outliers with many citations, forming a long tail. Since these outliers are
the most relevant publications, the visualization should represent them

36



3.3. Examples

accordingly. However, even the bulk of low-citation papers is interesting
because it indicates publication productivity.

For illustration purposes, we use citation data of four well-known re-
searchers (Shneiderman, Cleveland, Wilkinson, and Lorensen) obtained
from Google Scholar1 through the Publish or Perish software [Har16] on
June 22, 2017 (without any data cleansing). Figure 3.5 shows the results.
In 3.5a, we plotted 1,045 publications by Ben Shneiderman. His most
cited work is “Designing the user interface: strategies for effective human-
computer interaction” (14,309). There are six additional publications that
are clearly distinguishable, but most of the other papers seem to have one
thousand or fewer citations.

To compare his citation data to that of the other three authors, we
add their data and use color-mapped dots. By the dominant dark blue
color in the nonlinear dot plots in (b) or (c), we can immediately see
that Shneiderman has the largest number of publications. Dots below
about 1,000 citations become too small to distinguish individually, but
their nonlinearly scaled column heights can still be perceived; therefore,
we can obtain the approximate frequencies and compare them between
authors. The much fewer papers with high citation counts are large and
clearly visible. From these dot plots, we can see that all four authors have
publications with 9,000 or more references. Bill Lorensen even has two
papers with very high citation counts: “Marching cubes: A high resolution
3D surface construction algorithm” (13,495) and “Object-oriented modeling
and design” (11,147). Leland Wilkinson’s most cited work is “SYSTAT for
Windows: statistics, graphics, data, getting started, version 5” (9,914). The
red dot at 9,017 represents William Cleveland’s “Robust locally weighted
regression and smoothing scatter plots.”

In contrast, linear plots (Figures 3.5d and 3.5f) are not well suited for
such high-dynamic-range data, as they cannot show any useful information
about the important long tail. The logarithmic histogram in (e) renders
the highly referenced publications as bars. This allows for a relatively
accurate estimation of the citation counts but does not provide any method
of showing and comparing the authors.

3.3.3 Renewable Energy
Our third example combines dot plots with box plots to address the way in
which electricity is produced. The European Commission provides the gen-
eral public with access to statistical data through its website2. It includes

1https://scholar.google.com, accessed 2024-01-27.
2https://ec.europa.eu/eurostat/web/main/home, accessed 2024-01-27.

37

https://scholar.google.com
https://ec.europa.eu/eurostat/web/main/home


3. Nonlinear Dot Plots

2 31
PT SEAT IS NO

(a) Root dot plot with annotations for clusters and country names

(b) Symmetric root dot plot with box plot overlay

(c) Linear histogram

Figure 3.6: Percentage of electricity produced from renewable energy
sources versus total consumed electricity of 30 European countries from
2004 to 2014. There is a value for each country and year. Dot plots use a
color scale to represent the value’s year: ’04 ’05 ’06 ’07 ’08 ’09 ’10 ’11 ’12 ’13 ’14 .

information on the amount and type of energy produced in each country
of the European Union (EU), as well as Norway and Iceland. The data set
“tsdcc330”3 compares the generated renewable electricity with the total
consumed amount on a yearly basis from 2004 to 2014.

Visualizing this data set with dot plots and histograms, three groups
of values stick out that can be interpreted by combining the use of plots
with the tabular representation of the source data. As Figure 3.6 and,
more specifically, cluster 1 in Figure 3.6a show, there is a relatively high
data density near 0 %. Temporal information from the dot plots shows
that it must have been from countries that have only recent