Contrast Driven Color-Group Assignment in Categorical Data

Visualization

Eric Languenou

LS2N, Nantes Universit

e, 2, Rue de la Houssini

ere BP 92208 44322 Nantes Cedex 03, France

Keywords:

Categorical Data Visualization, Class Color Assignment, Color Contrast, Streamgraph, Chord Diagram.

Abstract:

Ubiquitous digital technology has facilitated the collect of multi-dimensional numerical data that are analyzed

by specialists. Their need to explore and to explain this data to non-specialists is important. With categorical

data, we construct various diagrams on a color-coded paradigm associating colors with data classes. Depend-

ing on the number of classes or the geometry of diagrams, the class-color assignment choice can become a

complicated task, with the number of permutations growing in a factorial way with the number of categories.

The goal of this research is to develop an algorithm aiming at assigning the best color, among a user given

color palette, for each class of objects of a categorical data visualization. We optimize the ability, for a viewer,

to distinguish classes’ geometrical objects one from another using a concept of contrast importance factors

expressing the need to get for a pair of objects classes a high color contrast. The method relies on a ﬁtness

function separation between palette color distances and geometrical contrast need. We indicate applications of

the concept to two kinds of categorical visualizations: streamgraphs and chord diagrams for which optimized

color assignment has never been published so far.

1 INTRODUCTION

The current pandemic has increased the data visual-

ization presence in the public everyday world. Up-

dated time series are often depicted on various me-

dia, sometimes demonstrating general audience lesser

comprehensible visualization types, like heat maps.

The goal of the presented research is, for a user set

categorical colormap and a categorical visualization

diagram of which the number of categories is equal

to the number of colors of the colormap, to aim at

obtaining the best color-class assignment while op-

timizing the ability to distinguish class depicted ob-

jects one from another. Obviously, this assignment

can be set manually by data scientists with data con-

taining a few categories and when the neighboring of

graphical metaphors is simple enough. We can also

choose a colormap with a large hue interval, easing

therefore legibility, but degrading the visualization di-

agram aesthetic.

In the presented method, we estimate the distin-

guishability between two categories using color dis-

tance and graphical properties. In fact, color distance

formulas have been continually improved to evaluate

https://orcid.org/0000-0003-2327-5464

human perception of color difference and similarity.

The recent developments have come with validity for

the whole cube of colors and the last DE2000 (Luo

et al., 2001) possesses almost all the mathematical

properties of a mathematical distance, though being

not exactly symmetrical.

To optimize category legibility while assigning

colors to categories, we express an overall ﬁtness

function using for each category pair a contrast need

function providing a scalar value, which represents,

given the class object geometries, the need to assign

to class objects a large color difference to distinguish

one from the other. We denote this value in the fol-

lowing contrast importance factor or in a shorter way

importance coefﬁcient. Such contrast need functions

are enumerated in the chapter 6 for streamgraphs and

for chord-diagram visualizations.

As detailed in the following, the ﬁnal ﬁtness scalar

for a given color assignment is expressed using a

contrast importance matrix, containing the contrast

importance coefﬁcients, and a color distance matrix,

containing the color distance between colors of the

colormap.

The paper is organized as follows. After a review

of published methods about class-color assignment in

case of categorical visualization, we explain the pro-

Languenou, É.

Contrast Driven Color-Group Assignment in Categorical Data Visualization.

DOI: 10.5220/0011615000003417

In Proceedings of the 18th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theor y and Applications (VISIGRAPP 2023) - Volume 3: IVAPP, pages 53-64

ISBN: 978-989-758-634-7; ISSN: 2184-4321

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

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Number of tracks played by hour grouped by artist (21)

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Random Assignment

Optimized Assignment

Figure 1: Streamgraph: optimized (top) and randomized (bottom) color-layer assignment (Viridis reduced colormap).

posed approach. Then, two applications of the con-

cept, to streamgraph and to chord diagram, are de-

tailed. A discussion chapter follows before a conclu-

sion and future work section, which completes the pa-

per.

2 RELATED WORKS

In image understanding, Bertin (Bertin, 1983) wrote

that color is associative, selective and ordered. Var-

ious works in psychology explore color human re-

sponse, especially that color precedes size, shape and

orientation. From empirical studies in visualization to

crowd sourced color aesthetics experiments and dis-

tance perception between visual objects, color human

reception has been explored.

2.1 Colormap Choice and Color

Assignment

Research papers provide guidance on colormap de-

sign (Ware, 2012) and emphasis on optimizing for

color harmony and aesthetics (Wang et al., 2008)

while others propose an interactive tool (Meier et al.,

2004) to mix, organize colors and explore color com-

binations. Zhou and Hansen have published a sur-

vey on colormap design in 2016 (Zhou and Hansen,

2016).

Tasks that could be performed about visualization

have been systematically studied (Brehmer and Mun-

zner, 2013). Given a dataset, a kind of visualization

diagram and a task, there is a lot of research about

deﬁning rules to determine which colormaps should

be applied (Mittelstadt et al., 2015). In practice, an

appropriate color mapping scheme is often obtained

by a two-step procedure, selecting a categorical color

palette (Harrower and Brewer, 2003; Gramazio et al.,

2017), assigning colors to the classes through a trial-

and-error process.

Aupetit et al. (Aupetit and Sedlmair, 2016) pub-

lished in 2016 a state-of-the-art on visual class sepa-

ration measures, applied mostly to scatterplots. Sedl-

mair et al. (Sedlmair et al., 2012) developed a tax-

onomy of factors that inﬂuences the human percep-

tion of visual class separation on scatterplots, where

most factors are derived from the positions of the data

points. Researches are also concerned with factors

like scatterplots class visibility (Lee et al., 2013), and

IVAPP 2023 - 14th International Conference on Information Visualization Theory and Applications

perceptual distance. Fang et al.(Fang et al., 2016)

proposed, for scatterplots, to maximize the percep-

tual distance among a set of given colors while in-

corporating a set of user-deﬁned constraints. They

further compared three optimization algorithms solv-

ing this problem and found that a Genetic Algorithm

(GA) can ease the effect of sticking to a local maxi-

mum. Wang et al.(Wang et al., 2018) also developed

an approach in class-color assignment to optimize vi-

sual class separability in scatterplot diagrams. Based

on forms, distinctness and contrast with background,

it considers user colormap and optimizes through ge-

netic algorithm. They provide a parameter study with

numerical measure.

Line colors assignment is explored by Hurter et al.

(Hurter et al., 2010) who proposed an optimization

method to choose lines colors in a metro map, assign-

ing close routes with the most distinguishable colors.

Kim et al. (Kim et al., 2014) proposed a perception-

driven color assignment method for assigning colors

to segmented image, emphasizing on color aesthetics

and where contrast is incorporated.

2.2 Various Approaches

Some categorical visualization researches use data

graphical properties to optimize color assignment, in

which some are pixel-based (Lee et al., 2013; Zeng

et al., 2019) and others are segmented-based or else

mark-based (Fang et al., 2016; Hurter et al., 2010;

Lu et al., 2021; Kim et al., 2014; Wang et al., 2018).

To optimize color assignment, researchers estimated

various factors like aesthetics, harmony, class visi-

bility, legibility, color distance, semantics, density,

overlapping or ﬁdelity to user preferences. Factors

are summed using weights to get an energy formula.

Among the encountered optimization methods, au-

thors take advantage of simulated annealing, genetic

algorithm, stochastic or gradient-based optimization.

We can ﬁnd another partition among these works.

Some are optimizing for both colormap and color

assignment (Fang et al., 2016; Hurter et al., 2010;

Zeng et al., 2019) and others are based on a user

set colormap (Lu et al., 2021; Wang et al., 2018).

Wang et al. present a general view of existing tech-

niques and introduced their approach using point dis-

tinctness, non-separability, and genetic algorithm to

optimize scatterplots diagrams. In 2021(Lu et al.,

2021), the algorithm was extended to line and bar

charts. Another paper (Hurter et al., 2010) close to

our approach is about air-trafﬁc and metro map dia-

grams optimization. Their method, besides creating

a colormap, and changing the diagram object’s lo-

cation, computes distances between lines to optimize

color differences through a result quality estimation.

Lee et al. (Lee et al., 2013) introduced the concept

of class visibility and a color optimization algorithm

based on their class visibility metric. Via color sat-

uration, their pixel-based algorithm tries to increase

the saturation difference for small regions. The class

visibility factor is calculated through integrating point

saliency, which measures how much a particular point

color differs from its surrounding color.

Our method includes neither semantic color pref-

erences nor object geometry modiﬁcations. The pa-

per contribution is twofold, ﬁrst, introducing a con-

trast oriented general class-color assignment frame-

work using a user set colormap, demonstrating its ver-

satility and opening the way to express visualization

type specialized contrast importance matrix calcula-

tions and second, detailing novel assignment color to

stacked-graphs/streamgraph and chord diagrams, by

applying the previous concept. The author has al-

ready developed concept applications to line charts

and polygon-based map (Languenou, 2022). It can be

noted that some notions, expressed in the above men-

tioned papers, like point distinctness, point saliency

and class visibility, could be integrated into our con-

trast importance matrix by re-organizing the corre-

sponding mathematical expressions.

3 THE APPROACH

This section explicates the principles of the energy ex-

pression necessary in our class-color assignment op-

timization process. Let us remind that our contrast

importance concept relies on an energy function sep-

arated in two parts, a color part, independent of the

data and a geometrical one, depending only on the vi-

sualization type and the data.

Let us consider a categorical visualization display-

ing n categories/groups/classes G = {g

,...,g

Visualization specialists have colormap prefer-

ences which leads the author to choose an approach

allowing the user to select a colormap, say C =

,...,C

}. The optimization process tries to com-

pute the best assignment between n object classes and

n color samples to maximize contrast while preserv-

ing distinguishability.

The expression of a ﬁtness function, which sep-

arates color distance values from the need to get a

color contrast between categories, gives the ﬂexibility

to adapt the calculation of this ”contrast need” factor

calculation to multiple categorical visualization dia-

grams. In the following, we denote this contrast need

evaluation factor as ”importance factor”.

Contrast Driven Color-Group Assignment in Categorical Data Visualization

3.1 Importance Matrix and Color

Matrix

The optimization ﬁtness function taking as input a

color-class association (i.e. a permutation) is based

on two symmetrical matrices:

• D is the color matrix, made of n×n elements, that

contains D(i, j) the color distance between col-

ormap samples C

and C

. We use, for this ma-

trix D, the most recent color distance DE2000 (or

CIEDE2000) formula (Luo et al., 2001).

• M, which we call importance matrix, also made of

n × n elements, that codes M(i, j) the need to as-

sign a high color contrast between classes g

and

. The associated calculations involve graphical

object’s size, neighborhood, frontier, distance be-

tween objects and therefore depend on depicted

data and diagram type. We display, in ﬁgure 2,

a representation of the importance matrix corre-

sponding to the streamgraph examples in ﬁgure 1

displayed using viridis colormap

The importance matrix, as we deﬁne it, is sym-

metrical and possesses null diagonal values.

Depending on the diagram kind, we may associate

a null importance value to a class pair that does not

share a frontier. This shortcut is, for example, used in

our streamgraph importance matrix calculus. We pro-

pose several diagram type specialized importance ma-

trix evaluation formulas in section 6, naming stream-

graphs and chord diagrams.

Once the user has chosen the colormap, the color

distance matrix remains constant, and for a visualiza-

tion type displaying some given data, the importance

matrix is constant as well. We then express the ﬁtness

function for a color permutation p using an inner ma-

trix product of the two matrices D and M where we

permute D using the permutation p. The next section

details this energy function.

3.2 Energy Evaluation

If we assume that the adequacy A(i → k, j → l) of a

single assignment of two color samples C

to two

data classes g

is given by the product of the inter-

color distance D(i, j) and the class pair importance

value M(k, l), then the overall adequacy for a permu-

tation p represented by (p

,.., p

) can be expressed

by the sum of the individual products, where the color

distance is taken from D(p

, p

) to handle the color

permutation:

Nathaniel Smith and St

efan van der Walt, 2015

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Frankmusik

Is_Tropical

Junior_Senior

Justice

M83

MGMT

Metric

Metronomy

Mr_Oizo

Passion_Pit

Phoenix

Ratatat

The_Naked_and_Famous

The_Postal_Service

The_xx

Air

Ellie_Goulding

Empire_of_the_Sun

Fever_Ray

Foals

Foster_the_People

Frankmusik

Is_Tropical

Junior_Senior

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MGMT

Metric

Metronomy

Mr_Oizo

Passion_Pit

Phoenix

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The_Naked_and_Famous

The_Postal_Service

The_xx

interval:[0,400]

0.00

0.25

0.50

0.75

1.00

Figure 2: Streamgraph: contrast importance matrix coefﬁ-

cients.

A(p

,.., p

) =

∑

i=1

∑

j=1

(D(p

, p

) × M(i, j)) (1)

Using matrix notation, we express the ﬁtness func-

tion A(p), for a class-color association p, by a simple

matrix inner product

(denoted < A, B > ) of the two

matrices where the color matrix D is permuted using

, the square matrix coding the permutation p map-

ping.

A(p) =< P

.D.P

,M > (2)

Because the matrices are symmetrical and the el-

ements i = j are of no importance, the complexity

of the ﬁtness function evaluation is then reduced to

n re-directions (via the permutation), n(n − 1)/2 mul-

tiplications, (n(n− 1)/2)−1 additions, where n is the

number of classes.

4 SOLVING FOR

PERMUTATIONS

Once we have computed the two matrices, trying to

get the best permutation is performed by optimiza-

tion. The class-color assignment optimization can

be a complicated problem (known as a NP-Complete

problem) when data visualization is complex and the

number of classes high. Let us remind that the cardi-

nal of permutations for n classes is equal to n factorial,

thus growing drastically fast with n.

Luckily, for a number of classes n, say n ≤ 8, the

number of permutations (≤ 40,320) is low enough to

perform an exhaustive evaluation in a reasonable time

and keep the best permutations. So far in our lan-

guage R implementation, the exhaustive evaluation is

also called Frobenius product

IVAPP 2023 - 14th International Conference on Information Visualization Theory and Applications

automatically performed for n ≤ 8, but as the ﬁtness

formula is only expressed in term of the permutation

values, the contrast importance matrix and the color

distance matrix, the evaluation could be calculated in

an implementation that could take advantage of cur-

rently available multi-core processors. This would al-

low the exhaustive search to be performed up to 12

or 13 categories. A comparison between the assign-

ments obtained by exhaustive search (top) and genetic

algorithm (bottom) is displayed in ﬁgure 3. Even if

the results are different, the two assignments can be

evaluated as contrasted.

It appears that the splitting in two matrices of

the energy expression proposed here before for color

assignment problem has already been published in

economics ﬁeld (Beckman and Koopmans, 1957).

Known as the Quadratic Assignment Problem (QAP),

it aims at achieving an economic objective of as-

signing facilities to locations to minimize the over-

all transport cost. Surveys (Zaied et al., 2015; Loiola

et al., 2007; Burkard et al., 1998) discuss QAP reso-

lution methods like dynamic programming, enumer-

ation, local search algorithms, metaheuristics, simu-

lated annealing, ant colony optimization, neural net-

works and markov chains, tabu search, greedy ran-

domized adaptive search procedure, variable neigh-

borhood search, evolutionary algorithms or else trans-

genetic heuristics.

4.1 Genetic Algorithm Optimization for

Permutation

To get an assignment in a reasonable amount of time,

heuristic approaches may be exploited. In the Data

Visualization literature, various methods, simulated

annealing, genetic, non-linear gradient and greedy,

has been experimented, but when the number of

quite satisfying solutions is high enough, genetic ap-

proaches gives an advantage as various solutions may

be obtained when conducting several optimizations

or keeping the best results among which the user

may choose. In case of permutation search, the clas-

sic genetic optimizer has to be adapted. Chromo-

somes are speciﬁcally coded using the bijection from

initial colors to the categories, which can be repre-

sented as (p

, p

,..., p

) where p

∈ [1,n] and p

̸=

∀i ̸= j. Here, crossover methods, inheriting from

two chromosomes, have to generate valid permuta-

tions and many crossover techniques have been devel-

oped for permutation specialized chromosomes (Um-

barkar and Sheth, 2015). Mutation must be adapted as

well by performing a simple swap between two ran-

dom indexes.

5 VISUAL PROPERTIES AND

COLOR SEPARATION

Our contrast importance factor is designed to in-

crease the visual discriminability of the classes com-

posing the categorical diagram. Identiﬁcation capa-

bility is let to the user colormap choice, or made pos-

sible by labeling the graphical objects. The visual

solid angle should be used to express the ability of

the viewer to separate visually one object from an-

other, but so far, we calculate the importance factors

through object size ratio, therefore the result is inde-

pendent of the future context of viewing (viewing dis-

tance and diagram display size).

Many graphical objects may compose the cate-

gory, and they have to be incorporated into the calcu-

lus. The eye ability to visually ”follow” a continuous

graphical object (a line, a layer, a thick arrow, etc.)

fosters the class visual separability. But the layers of

a streamgraph can be either continuous along the time

interval, or appearing only on a part of the diagram,

or else be somewhere reduced to a null thickness and

visible again later.

The object visual separation is critical for adja-

cent objects to avoid spreading (the visual grouping

of close objects caused by too much similar colors).

In consequence, identifying the frontier along adja-

cent objects is an important aspect.

In fact, to ﬁt in our framework, as the need for

contrast for each category pair is expressed through a

single scalar, i.e. an element of the importance matrix,

there must be a synthesis of the many local needs for

contrast, corresponding to various parts of the graphi-

cal metaphors. This aspect is detailed in the next sec-

tion.

5.1 Elementary Importance Coefﬁcients

Visualizations diagrams are usually composed of sev-

eral marks (scatterplots, bar-charts) or composed of

curves or polygons (stacked-graphs, streamgraphs,

polygon maps, chord diagrams). For example, in

scatterplots diagram, numerous discs may represent

a class, but in streamgraph visualization, a category

layer may be a single object. In this later case, we

decompose the layer into smaller parts to evaluate

local discriminability on parts of the class graphical

metaphor. Let us denote these local evaluations, ele-

mentary importance factors (or in a shorter way: ele-

mentary importances).

In the section 6, where we explain the impor-

tance matrix calculation, the diagram speciﬁc expres-

sions used to compile/synthesize the elementary im-

portance factors into a single factor will be detailed.

Contrast Driven Color-Group Assignment in Categorical Data Visualization

In case of adjacent class objects visualization, be-

sides the importance of the width and area of the

colored region, and when objects discrimination is

needed, Newhall (Newhall, 1955) noticed that the

length of the frontier does not affect the discriminabil-

ity. The perpendicular distance from the interface

frontier to the other side of the region (a perpendicular

thickness) seems to represent a better criterion. The

thinner the region, the higher the risk of spreading

(region visual aggregation) when using similar colors.

To the knowledge of the author, there are no published

formula, in term of color distance and mutual thick-

ness (or visual angle) that evaluate this risk.

The object’s visual angle is the main property to

be considered, but usually there is no information

on the ﬁnal display size or knowledge on the dis-

tance from the screen (or paper) to the viewer’s eye.

Though, there are published researches about discrim-

inability in visualization. For example, Healey and

Sawant published a paper (Healey and Sawant, 2012)

about the limits of pixel-based visualizations and vi-

sual angles while Maureen Stone examined the size

consequences on spreading effect and distinctness

when generating colormap in the software “tableau”

(Stone, 2012). To explore colors which are subjec-

tively considered as the same, the concept of Just No-

ticeable Difference (JND) has been expressed as well.

Unfortunately, no result of the previous publications

applies directly to our cases.

6 APPLICATIONS

This section details two segmented-based categorical

visualization applications of the presented concept:

streamgraph and chord diagram for which we propose

expressions to evaluate the need for color contrast cal-

culation (i.e. the importance matrix specialized calcu-

lations).

6.1 Streamgraphs and Stacked Graphs

Classical stacked-graphs displays polylines stacked

vertically on top of the horizontal axis. In a New York

Times article (Byron and Wattenberg, 2008), Lee By-

ron and Martin Wattenberg popularized streamgraph

diagrams (see ﬁgure 1) which were published before

by Havre et al. (Havre et al., 2002).

Since the ﬁrst scientiﬁc publication many im-

provements have been added, by categories aggrega-

tion, by optimizing the layer order, to correct the layer

thickness in case of signiﬁcant slope and developing

multi-resolution streamgraphs (Cuenca et al., 2018).

6.1.1 Visualization Principles and Graphical

Rules

Usually chosen to depict time varying categories,

streamgraph layers height is related to a speciﬁc di-

mension value. The category layers are stacked on y,

while respecting a constant y order along x axis (vari-

ous constant orders can be chosen, depending on pref-

erences or data criteria) but layers can present a null

thickness for some time intervals. Therefore, layers

may share a frontier with many other layers and may

be thin on a part of the diagram and thick on another

part. The number of layers, depending on the data,

can be large, causing complicated legibility and pre-

venting any category identiﬁcation by color. Interac-

tive diagrams or labeling the layers are then manda-

tory.

6.1.2 Graphical Criteria for Color Assignment

The main visual criterion is concerned with a visual

separation between layers, therefore, effortless detec-

tion of layer borders by the viewer is an important

property. As some layer parts may be thin, it is im-

portant that layers’ color choice demonstrates enough

color contrast to satisfy legibility. Therefore, in our

case, where colormap is provided by the user, a trade-

off has to be found to optimize legibility.

6.1.3 Importance Computation

Consequently, given a layer ordering and depicted

data, we must identify the frontier sharing between

layers and calculate corresponding contrast impor-

tance factors that rely on mutual layers thickness to

prevent from visual ”spreading”.

6.1.4 Elementary Importance Computation

As layer thickness and frontiers with other layers

evolve along the time dimension, evaluating elemen-

tary importance factors which code the local need to

have a large color contrast for two adjacent layers is

a ﬁrst step, before compiling a global importance fac-

tor along horizontal dimension interval. Let us denote

T the set of time samples t used to display a stream-

graph composed of n layers for which h

is the height

of layer i for time value t.

Let us also deﬁne γ

i, j

a neighborhood descriptor

which codes the fact that layers i and j share a frontier

on time sample t. This descriptor is a function of data

values and of layers chosen order and γ

i, j

is equal to

1 if the two layers share a frontier and 0 otherwise. If

we assume that the thinner the layer, the less visible

the layer frontier is, therefore a decreasing function

IVAPP 2023 - 14th International Conference on Information Visualization Theory and Applications

could be a starting point to get the desired behavior.

We consider in this research that the need to assign

high contrasted colors to the layer couple depends on

the minimum of the two layers thickness (which can

be discussed). In absence of knowledge on the human

response in such geometrical cases, we can evaluate

the i, j inter-layer elementary importance factor δ

i, j

for time t, using the inverse of layers height:

i, j

= γ

i, j

∗ max(1/h

,1/h

) (3)

Then, the expression for the whole interval T of t

values for layers i and j gives an array δ

i, j

which has

to be reduced to a single factor M(i, j), the importance

factor for the couple of layers i, j. So far, we do not

calculate the visual thickness as a layer perpendicular

thickness, but we use directly the data value corre-

sponding to y size, with the consequence of an under-

valuation of elementary importance with layer having

large slopes. This could be enhanced by generating

streamgraphs with a more correct thickness (Bu et al.,

2021).

6.1.5 Synthesizing to Obtain an Importance

Factor

We must, thus, compile the elementary importance

factor array values δ

i, j

to get M(i, j) the global (i, j)

inter-layers importance factor. We experimented the

synthesis process via several functions:

• maximum value

M(i, j) = max

t∈T

i, j

(4)

• average value

M(i, j) =

card(T )

∑

t∈T

i, j

(5)

The risk of averaging while synthesizing leads

in compensation would sometimes cancel problems.

Using maximum formula could, for some data, re-

sult in stating that all couples of layers share the

same importance value. Examining the data proper-

ties which make a speciﬁc synthesis more appropriate

would need further researches. Maximum synthesis

is applied in the top part of ﬁgure 1 and in ﬁgure 3.

6.1.6 Discontinuous Layers

The presented method aims to increase the contrast

between categories in which graphical metaphors are

neighboring. The diagram legibility and the category

identiﬁcation depend on the chosen colormap. Us-

ing a narrow hue range color palette involves category

identiﬁcation difﬁculties, which could sometimes be

solved by labeling graphic objects.

In streamgraph diagrams, a layer can be made of

a single thick curve or composed of several discon-

nected thick curves when streams become null and

expand later (called discontinuous layers in the fol-

lowing). In this case, they must be assigned with

colors that must be contrasted with potential other

layer colors. As our vision can follow a continuous

layer, only the category layers which are stacked on

the part of the diagram where a discontinuous layer is

re-appearing may be excluded from the need for con-

trast. Therefore, all the other layers must show a color

contrasted with the discontinuous layer color.

The corresponding contrast importance matrix

values could be increased to handle the special con-

trast need between these categories.

8 10 12 14 16 18 20 22

11 layers

39,916,800 permutations

Genetic Optimization

Using R GA: 3.6 secs

Exhaustive search

Using R: 24 mins

700 generations

100 individuals

Figure 3: Streamgraph showing 11 layers, with reduced

viridis colormap. Top: exhaustive search assignment, bot-

tom: genetic search assignment.

6.2 Chord Diagram

Martin Krzywinski proposed chord diagrams in a

2007 New York Times article called ”Close-ups of the

Genome, Spieces by Spieces by Spieces”. Chord dia-

grams depict ﬂuxes (directional type) and weighted

relationships between entities (un-directional type).

The left part of the ﬁgure 4 depicts

a directional

chord diagram (using simple direct viridis colormap

color assignment) which is visible on the data-to-viz

website

which presents migration between world re-

gions, using data provided by Abel (Abel, 2018).

The same chord diagram is displayed in the right

with permission

www.data-to-viz.com/graph/chord.html

Contrast Driven Color-Group Assignment in Categorical Data Visualization

part of ﬁgure 4 for which the assignment has been op-

timized using our method. The improvement is, for

example, noticeable in the lower left part of the dia-

gram (Oceania) where the very narrow arrows present

contrasting colors and are therefore more visually rec-

ognizable.

6.2.1 Visualization Graphical Rules

In a chord diagram, we display the n categories along

a circle perimeter with grouped arrows (ﬂuxes) ar-

riving on or starting from each category. Moreover,

arrows of the chord-diagram depict non-null ﬂuxes

F(i, j) coming from group i and arriving on group j

and the scale factor used to calculate each arrow width

is constant and depends only on the data.

There are various options while displaying chords

diagrams and the type examined here after depicts

arrows ﬁlled using the starting ﬂux category color.

Therefore, all the ﬂuxes starting from a group are

ﬁlled using the same color. For each group, incoming

arrows are located after outgoing arrows. Last rule

states that for a group, incoming ﬂuxes are displayed

in decreasing order. Therefore, within each group, the

neighboring of arriving arrows involves various col-

ors and various thicknesses depending on the depicted

data.

6.2.2 Graphical Criteria for Color Assignment

In every group, to allow the user to interpret the di-

agram, the arrows must be visually distinct one from

another. We then focus the importance factor calcu-

lation on the arrows neighborhood and on their visual

width. As, along the circle perimeter, a margin sep-

arates each category from its two neighboring cate-

gories, there is no need to insure inter-group visual

separability at this level.

On the contrary, the center of the chord diagram

can be visually confusing, and trying to optimize the

legibility of the center part of the diagram could also

be an objective which is not addressed here. But in

term of data ﬂux understanding, the thickness and the

depicted order of the arrows on each group are essen-

tial.

6.2.3 Aspects of Importance Computation

To get contrast need factors, several important aspects

have therefore to be computed for a group i which can

be expressed by the following interrogations:

• What are the incoming groups of the arrows arriv-

ing on group i?

• In which order are they depicted? What are these

arrows neighborhood?

• What are their thicknesses?

• What is the width of the grouped outgoing arrows

of group i?

6.2.4 Notation

To evaluate geometrical properties of the arrows, let

us ﬁrst denote O

as the decreasing ordered list of

the group indexes of incoming arrows/ﬂuxes for the

group i and nb

as the number of incoming ﬂuxes for

the group i.

= {x

} with:

• x

is the element at index k in O

• k ∈ [1,nb

• x

∈ [1,n]

As O

is ordered in decreasing order, we have

F(x

,i) ≥ F(x

k+1

,i) for k ∈ [1,nb

− 1].

To ease expression writing, let us call:

• o

the index of group p in O

6.2.5 Flux Neighborhood

Using previous notations, let us deﬁne γ

q,r

a direc-

tional chord diagram neighborhood descriptor which

codes the fact that the ﬂuxes coming from the groups

p and q share a frontier on the arrival part of group i.

q,r

is equal to 1 if the index of p and q are succes-

sors in O

q,r

(

1 if p ∈ O

and q ∈ O

and |o

− o

| = 1

0 otherwise

(6)

6.2.6 Elementary Importance Matrices

As there are n parts on the circle perimeter corre-

sponding to the n categories in the chord diagram,

group arrows may be neighbors in each of these n

parts. Therefore, for each category i, we construct a

local importance matrix that codes the need to assign

color contrast between (arriving and outgoing) arrows

on a single group i visual part.

We denote these n matrices as elementary impor-

tance matrices and the calculation of their elements

involves the frontier descriptor γ

q,r

and the ﬂux values

F(i, j). If a category possesses no incoming ﬂux from

other groups, we ﬁll its elementary importance matrix

with 0 values, as there are no neighboring groups to

be considered. We depict the elementary importance

matrix for the Europe category in ﬁgure 5.

Let M

be the elementary importance matrix for

group i, and m

(q,r) its elements. M

is a square

IVAPP 2023 - 14th International Conference on Information Visualization Theory and Applications

Same Colormap with Optimized Assignment

Viridis Colormap (data-to-viz)

Migration between

World Regions (Abel, 2018)

Figure 4: Chord Diagram: left: without optimization and right: optimized category-color assignment.

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

interval:[0,453]

0.00

0.25

0.50

0.75

1.00

Figure 5: Chord Diagram: Elementary importance matrix

for Europe group.

matrix containing n × n elements, like the chord di-

agram’s ﬁnal importance matrix.

We chose to relate the importance factor, as in

streamgraph diagram, on the width (i.e. thickness)

of arrows. Therefore, it is based on the inverse of

arrow width, which is related to the corresponding

ﬂuxes values. Considering neighboring information,

the contrast importance value associated with two in-

coming ﬂuxes groups (including ﬂux from the same

group i) is estimated using the maximum of the in-

verse ﬂuxes. As the scale used to depict arrows is

constant, we can omit it.

(q,r) = γ

q,r

× max(1/F(q,i), 1/F(r,i)) (7)

6.2.7 Final Frontier

A remaining neighboring possibility has to be consid-

ered, which corresponds to the frontier for a group i

between the overall outgoing arrows and the ﬁrst in-

coming arrow. This case can be observed on the right

part of ﬁgure 4, on West Asia group, purple outgoing

arrow is neighboring an arriving green arrow. The in-

dex of the ﬁrst incoming ﬂux is given by x

which is

also the greatest incoming ﬂux for group i.

The size of grouped outgoing ﬂuxes is expressed

∑

u∈[1,n]

F(i, u). Its neighboring ﬂux, i.e. the in-

coming greatest ﬂux, is provided by F(x

,i) (if x

ex-

ists and x

̸= i).

The corresponding in/out frontier importance, if

exists and x

̸= i, is given by:

(i,x

) = max(

∑

u∈O

F(i, u)

F(x

,i)

) (8)

Finally, we update the element m

(i,x

) of the group

i elementary importance matrix by getting the maxi-

mum of the importance calculated on the frontier by

equation 8 and the previous corresponding m

(i,x

)

about incoming ﬂux (see equation 7). Note that if

= i the formula has to be slightly adjusted as the

arriving ﬂux is adjoining the outgoing ﬂux increasing

thus the corresponding shape width.

6.2.8 Resulting Importance Matrix

To be used by the class-color assignment optimiza-

tion, we must produce a single importance matrix.

Synthesizing the n elementary matrices M

in a chord

Contrast Driven Color-Group Assignment in Categorical Data Visualization

diagram importance matrix M containing n × n el-

ements, is performed by getting, for each element

M(l,m), the maximum of the corresponding elements

(l, m) of the n elementary importance matrices.

M(l,m) = max

i∈[1,n]

(l, m)) (9)

The synthesized importance matrix for the migra-

tion data chord diagram (ﬁgure 6.2) is depicted in ﬁg-

ure 6.

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

interval:[0,532]

0.00

0.25

0.50

0.75

1.00

Figure 6: Chord Diagram: Synthesized importance matrix.

6.3 Multiple Diagrams

To compare data, we often create multiples ﬁgures.

Therefore, the color-category assignment must be the

same for the ﬁgures. Some categories may appear in

one ﬁgure and not be present in some other diagrams.

The method described in section 6.2.8 to syn-

thesize importance matrices for the chord-diagram is

used to get the color assignment for the multiple di-

agrams case. Therefore, an elementary contrast im-

portance matrix is generated for each diagram. We

construct then the synthetic category set as the union

of the categories appearing in the elementary contrast

matrices. Finally, the aggregated contrast matrix is

calculated by taking for every category pair the maxi-

mum value encountered in the elementary matrices. If

a category pair does not appear in any of the elemen-

tary matrices, the corresponding importance value is

set to zero.

Solving for the synthetic matrix and a colormap,

of size equal to the cardinal of the synthetic category

set, provides with a color-category assignment trade-

off.

7 EVALUATION

As, to the knowledge of the author, there is no previ-

ous work on class-color assignment algorithm about

streamgraph or chord diagram, a comparison with ex-

isting color-category assignment work is impossible.

Though, a controlled-user study is mandatory to

evaluate the proposed method. Even if random as-

signments examples are presented here before, com-

parison with random assignment would not be real-

istic as data scientists manually modify the color as-

signment and test various colormaps to achieve leg-

ibility. The main goal of the presented work lies

in the possibility to improve visualization aesthetic

(by avoiding rainbow colormap, when possible) while

maintaining legibility. Therefore, it would be possible

to ask specialists to create diagrams which would be

compared to contrast-driven assignment visualization

for aesthetic and ability to perform some given data

tasks in a user-controlled experiment.

However, we present dedicated diagrams showing

adequacy between color distances and contrast impor-

tance values in the case of the migration chord dia-

gram using viridis colormap in ﬁgures 7, 8 and 9. We

depict the color distances as green discs and the con-

trast importance values as red discs, discs for which

diameters are scaled to a same max value. The more

red in the ﬁgure, the less the assignment corresponds

to contrast asked values. These visualizations only

prove that the optimization process converges using

our ﬁtness function and does not prove the adequacy

between the legibility goal and our ﬁtness function.

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

imp[0.00,532.46]

colDist[0.00,100.63]

0.00

0.25

0.50

0.75

1.00

type

colorDist

importance

zinter

Figure 7: Chord Diagram: comparison : contrast driven

optimized assign.

IVAPP 2023 - 14th International Conference on Information Visualization Theory and Applications

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

imp[0.00,532.46]

colDist[0.00,100.63]

0.00

0.25

0.50

0.75

1.00

type

colorDist

importance

zinter

Figure 8: Chord Diagram: comparison : colormap sample

order assign.

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

Africa

East Asia

Europe

Latin Ame.

North Ame.

Oceania

South Asia

South East Asia

Soviet Union

West.Asia

imp[0.00,532.46]

colDist[0.00,100.63]

0.00

0.25

0.50

0.75

1.00

type

colorDist

importance

zinter

Figure 9: Chord Diagram: comparison : random assign.

8 DISCUSSION

The selection of a user color palette and the sepa-

ration within the ﬁtness function between geometric

aspects on the one hand, and palette color distances

on the other hand, reduces the cost of evaluating as-

signments. Even if the proposed diagram dedicated

functions (streamgraph and chord-diagram), calculat-

ing the category pair contrast need, have to be dis-

cussed and improved, the framework seems extensi-

ble to other diagram types like scatterplots for exam-

ple.

Though, the results highly depend on the user cho-

sen colormap. Thus a trial-error process remains to

get a trade-off between aesthetic (through a reduced

hue interval) and legibility. Another drawback lies in

the fact that the method does not warn the user about

a possible lack of contrast within the diagram using

the resulting color permutation.

9 CONCLUSION AND FUTURE

WORK

The article proposes a framework that calculates a

color category assignment in the case of categori-

cal visualization. It focuses on class color contrast

needs, using a user-selected color palette and sepa-

ration between diagram geometric information (con-

trast importance matrix) and color palette sample dis-

tances (color distance matrix using DE2000). Op-

timization is then reduced to the choice of a good

color permutation which is carried out using a genetic

algorithm. Two diagram-type applications, stream-

graph and chord-diagram, are detailed. The method

has been implemented in Java (streamgraph) and in

R language (streamgraph, chord-diagram) for which

a package will be available.

A full evaluation of the results, in term of legibil-

ity, aesthetic and tasks, via user-controlled testing will

have to be conducted.

More importantly, to improve the quality of color

category assignment, a perceptual model must be

carefully designed for both printed and on-screen vi-

sualizations. Therefore, since the current expression

of the need for contrast is based on the size ratio of

displayed graphic objects, which does not correspond

to the way the human eye reacts, the visual angle of

the objects (actual diagram size, distance visualiza-

tion) should be taken into account.

ACKNOWLEDGEMENTS

The author thanks Pascale Kuntz, Kadi Bouatouch

for their support, Yan Stoltz for the chord diagram R

source and the reviewers for their constructive advice.

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