A Physiologically-Based Model for Simulation of Color Vision Deficiency (vol 15, pg 1291, 2009)

Abstract

Color vision deficiency (CVD) affects approximately 200 million people worldwide, compromising the ability of these individuals to effectively perform color and visualization-related tasks. This has a significant impact on their private and professional lives. We present a physiologically-based model for simulating color vision. Our model is based on the stage theory of human color vision and is derived from data reported in electrophysiological studies. It is the first model to consistently handle normal color vision, anomalous trichromacy, and dichromacy in a unified way. We have validated the proposed model through an experimental evaluation involving groups of color vision deficient individuals and normal color vision ones. Our model can provide insights and feedback on how to improve visualization experiences for individuals with CVD. It also provides a framework for testing hypotheses about some aspects of the retinal photoreceptors in color vision deficient individuals.

Full text

A Physiologically-based Model for Simulation of Color Vision
Deficiency
Gustavo M. Machado, Manuel M. Oliveira, Member, IEEE, and Leandro A. F. Fernandes
Reference Protanomaly (6 nm) Protanomaly (14 nm) Protanopia
Fig. 1. Scientific visualization under color vision deficiency. Simulation, for a normal trichromat, of the color perception of individuals
with color vision deficiency (protanomaly) at different degrees of severity. The image on the left (turbulent flows) illustrates the
perception of a normal trichromat and is used for reference. The numbers in parenthesis indicate the amount of shift, in nanometers,
applied to the spectral response of the Lcones. The image on the right shows the simulated perception of a dichromat (protanope),
which is approximately equivalent to the perception of protanomalous with a spectral shift of 20 nm. Note the progressive loss of color
contrast as the degree of severity increases. Images simulated using our model.
Abstract—Color vision deficiency (CVD) affects approximately 200 million people worldwide, compromising the ability of these indi-
viduals to effectively perform color and visualization-related tasks. This has a significant impact on their private and professional lives.
We present a physiologically-based model for simulating color perception. Our model is based on the stage theory of human color
vision and is derived from data reported in electrophysiological studies. It is the first model to consistently handle normal color vision,
anomalous trichromacy, and dichromacy in a unified way. We have validated the proposed model through an experimental evaluation
involving groups of color vision deficient individuals and normal color vision ones. Our model can provide insights and feedback on
how to improve visualization experiences for individuals with CVD. It also provides a framework for testing hypotheses about some
aspects of the retinal photoreceptors in color vision deficient individuals.
Index Terms—Models of Color Vision, Color Perception, Simulation of Color Vision Deficiency, Anomalous Trichromacy, Dichromacy.
1 INTRODUCTION
Human normal color vision (also called normal trichromacy) requires
three kinds of retinal photoreceptors with peak sensitivity in the large,
medium and short wavelengths portions of the visible spectrum. Such
photoreceptors are called L,M, and S cones, respectively. The spec-
tral response of each kind of cone is defined by the specific type of
photopigment (opsin) it contains. Natural variations of some proteins
that constitute a given photopigment may shift its sensitivity to a dif-
ferent band of the spectrum [27]. In this case, the condition is called
anomalous trichromacy and can be further classified as protanomaly,
deuteranomaly, or tritanomaly, if the altered photopigment is the one
associated in normal color vision with the L,M, or Scones, respec-
tively. For a given color stimulus, the difference in color perception
between an anomalous trichromat and an average normal trichromat
will vary with the amount of spectral shift (measured in nanometers).
Dichromacy is caused by the absence of one of the photopigments and,
similarly, can be classified as protanopia,deuteranopia, or tritanopia.
A much rarer condition, called monochromacy, results from the exis-
tence of a single kind of photopigment (cone monochromacy) or no
•Gustavo M. Machado is with UFRGS, E-mail: gmmachado@inf.ufrgs.br.
•Manuel M. Oliveira is with UFRGS, E-mail: oliveira@inf.ufrgs.br.
•Leandro A. F. Fernandes is with UFRGS, E-mail:
laffernandes@inf.ufrgs.br.
Manuscript received 31 March 2009; accepted 27 July 2009; posted online
11 October 2009; mailed on 5 October 2009.
For information on obtaining reprints of this article, please send
email to: tvcg@computer.org .
photopigment at all (rod monochromacy).
Conditions involving the L cones (i.e., protanopia and protanomaly)
and the M cones (i.e., deuteranopia and deuteranomaly) are the most
common cases of color vision deficiency (CVD) and are also known
as red-green CVD. These are caused by alterations in the opsin gene
array in the X chromosome [27]. Given its recessive nature, the inci-
dence of red-green CVD is significantly higher among the male pop-
ulation. Table 1 shows the estimated incidences of red-green CVD
among different ethnic groups [25, 27]. These numbers suggest that
approximately 200 million individuals worldwide suffer from some
kind of color vision deficiency.
CVD compromises people’s ability to effectively perform color and
visualization-related tasks, impacting their private and professional
lives. Understanding how such individuals perceive colors has a sig-
nificant practical importance for using visualizations effectively. As a
result, some techniques have been developed for simulating the per-
ceptions of dichromats [4, 20] and anomalous trichromats [33]. How-
ever, no previous technique is sufficiently general to simulate all types
of red-green CVDs. Moreover, the simulated results obtained by the
technique described in [33] seem to disagree with the reports in the lit-
erature, as well as with the reports of color vision deficient individuals,
as we show in Section 5.
We present a model for simulating color perception based on the
stage theory [15] of human color vision. Our model is the first
to consistently handle normal color vision, anomalous trichromacy,
and dichromacy in a unified way. Unlike previous techniques that
are based on the reports of unilateral dichromats [4, 20] or on the
spectral response of the photoreceptors only [33], our approach uses
a two-stage model. It simulates color perception by combining a

photoreceptor-spectral-response stage and an opponent-color stage de-
fined according to data reported in electrophysiological studies [5].
This guarantees the generality of the proposed approach.
Fig. 1 illustrates the results produced by our model in the context of
scientific visualization. The image on the left shows a reference image
(i.e., the perception of a normal trichromat). The two images in the
middle show simulated views for two protanomalous individuals with
different degrees of severity. The numbers in parenthesis indicate the
amount of shift, in nanometers, applied to the spectral response of the
Lcones. The image on the right is a simulated view of a protanope
(a dichromat), which is approximately equivalent to the perception of
protanomalous with a spectral shift of 20 nm. Note the progressive
loss of color contrast as the degree of severity increases.
Ethnic Groups Incidence of red-green CVD (%)
Male Female
Caucasians 7.9 0.42
Asians 4.2 0.58
Africans 2.6 0.54
Table 1. Incidence of CVD among different ethnic groups [25, 27].
2 RE LATE D WORK
Despite the relevance of understanding how individuals with CVD per-
ceive colors, little work has been done in simulating their perception
for normal trichromats. In particular, none of the previous approaches
is capable of handling both dichromacy and anomalous trichromacy.
One should also note that the simulation process is not symmetrical:
in general, it is not possible to simulate a normal trichromatic color
experience for individuals with CVD.
The techniques designed for simulating dichromacy [4, 20] are
based on the reports of unilateral dichromats (i.e., individuals with
one dichromatic eye and one normal trichromatic eye). According to
these reports [10, 13], such individuals perceive achromatic colors as
well as a few other hues similarly with both eyes (475 nm and 575
nm, for protanopes and deuteranopes, and 485 nm and 660 nm for tri-
tanopes). This information was then used to define two half planes in
color space. Color simulation is obtained by projecting the color to
be simulated onto the surface defined by the two half planes. While
the technique introduced by Meyer and Greenberg [20] works in the
XYZ color space, the most popular one is the technique presented by
Brettel et al. [4], which performs the computation in the LMS color
space. These techniques produce good results, but they are specific for
dichromats and cannot be generalized for anomalous trichromats.
Kondo [16] proposed a model to simulate anomalous trichromacy
based on dichromatic vision. The results of this model, however, do
not preserve achromatic colors, which are known to be perceived by
by both dichromats and anomalous trichromats.
Yang et al. [33] proposed an algorithm for simulating anomalous
trichromacy that consists of mapping RGB colors to an anomalous
LMS color space, followed by a transformation from a normal trichro-
mat LMS color space back to RGB. By limiting the computation to the
photoreceptor level, the algorithm does not comply with the opponent-
color processing that takes place in the human visual system. As a re-
sult, the simulated images tend to contain colors that are not perceived
by color vision deficient individuals (Fig. 9).
Kuhn et al. [17] introduced an optimization-based contrast-
enhancing technique for dichromats. The authors demonstrated the
effectiveness of their technique for recoloring both information and
scientific visualization images. Their experiments indicated that the
technique is also effective for anomalous trichromats. Kuhn et al.’s
technique, however, does not provide a model for color vision simula-
tion. Like in a related recoloring technique by Rasche et al. [23], they
use Brettel et al.’s algorithm [4] for simulating dichromacy.
The mapping of data content to some color scale is a fundamental
operation in visualization and several techniques have been proposed
for building or guiding the construction of color maps [3, 11, 18, 30].
None of these techniques, however, has addressed the issue of color
vision deficiency.
3 STAGE THEORIES OF HUMAN COLOR VISION
The trichromatic theory of color vision assumes the existence of three
kinds of photoreceptors (cone cells) with different spectral sensitiv-
ities. The responses produced by these photoreceptors would then
be sent to the central nervous system and perceived as color sensa-
tions [32]. Also known as the Young-Helmholtz three-component the-
ory, it is based on the analysis of the stimuli required to evoke color
sensations and provides satisfactory explanation for additive color-
matching experiments. Unfortunately, the theory cannot explain some
perceptual issues, such as the opponent nature of visual afterimages,
as well as why some hues are never perceived together while others
(e.g., green and yellow, green and blue, red and yellow, and red and
blue) are easily found [8]. All these effects can be satisfactorily ex-
plained by Hering’s opponent-color theory, which assumes the exis-
tence of six basic colors (white, black, red, green, yellow, and blue).
According to Hering, light is absorbed by photopigments but, instead
of having six separate channels, the visual system uses only three op-
posing channels: white-black (W S), red-green (RG), and yellow-blue
(YB). While equal amounts of black and white produce a gray sensa-
tion, equal amounts of yellow and blue cancel to zero. Likewise, equal
amounts of red and green also cancel out. Zero in this context means
that the spectral response functions for the opponent channels become
zero at the points where opponent colors take equal values (see the
zero crossings in Fig. 3 right)
Considered separately, neither the trichromatic theory nor the
opponent-color theory satisfactorily explains several important color-
vision phenomena. When combined, however, they could explain and
predict many color vision phenomena involving color matching, color
discrimination, color appearance, and chromatic adaptation, among
others, for both normal color vision and color vision deficient ob-
servers [32]. von Kries suggested that the trichromatic theory should
be valid at the photoreceptor level, but the resulting signals should be
further processed in a later stage according to the opponent-color the-
ory [15]. This so-called stage theory (also known as zone theory) pro-
vides the best models for human color vision. Besides the two-stage
theory suggested by von Kries, other two- and three-stage theories of
color vision have been proposed, including M¨
uller three-stage theory.
A discussion of some of these theories can be found in [15].
A stage theory can qualitatively explain human color vision. How-
ever, before one can use it to define a model for rendering images
that simulate color perception, we need to describe both stages us-
ing equations. While curves describing the spectral sensitivity of the
cones can be measured in vivo and are available for an average indi-
vidual [28], one still needs the coefficients that define how the signals
generated by the cones are combined to form the achromatic (WS ) as
well as the two chromatic channels (RG and Y B). Such coefficients
cannot be easily obtained, but fortunately Ingling and Tsou [5] pro-
vided transformations for mapping cone responses (in the LMS color
space) to an opponent-color space. The suprathreshold form of their
transformation presents advantages over the threshold one, as it tries to
take into account reports by psychophysical and electrophysiological
studies regarding light adaptation. Eq. 1 describes Ingling and Tsou’s
suprathreshold transformation:


Vλ
y−b
r−g

=

0.600 0.400 0.000
0.240 0.105 −0.700
1.200 −1.600 0.400



L
M
S

(1)
where Vλrepresents the luminance channel W S, and r−gand y−b
represent the two opponent chromatic channels RG and Y B, respec-
tively. Fig. 2 illustrates how the cones’ output signals are combined
into the spectral response functions of the opponent channels WS,Y B,
and RG. Fig. 3 (left) shows the spectral sensitivity functions for the
cones of an average normal trichromat according to Smith and Poko-
rny [28]. The resulting spectral response functions of the opponent
channels for this average normal trichromat according to Ingling and
Tsou’s model are shown on Fig. 3 (right).

Fig. 2. Ingling and Tsou’s [5] two-stage model of human color vision.
The output of the photoreceptor stage (L, M and S cones) is linearly
combined in the opponent stage (Vλ,y−b, and r−gnodes).
Fig. 3. (left) Cone spectral sensitivity functions for an average normal
trichromat (after Smith and Pokorny [28]). (right) Spectral response
functions for the opponent channels of the average normal trichromat
according to Ingling and Tsou’s model [5]. These functions are obtained
by evaluating Eq. 1 for the LMS triples resulting from the cone spectral
sensitivity functions at all wavelengths in the visible range.
4 SIMULATING COLO R VISION DEFICIENCY
Except for the cases resulting from trauma, the causes of color vision
deficiency are genetic and result from alterations in the cones’ pho-
topigment spectral sensitivity [2, 27]. The conditions involving the L
and Mcones are hereditary and associated with a gene array in the X
chromosome [27]. The conditions involving the Scones (tritanomaly
and tritanopia) are considerably less frequent [25, 27] and are believed
to be acquired [27].
Our physiologically-based model treats CVD as changes in the
spectral absorption of the cones’ photopigments. While CVD is essen-
tially modeled at the retinal photopigment stage, the opponent-color
stage is crucial for producing the correct results and cannot be under-
estimated. For this, we use Ingling and Tsou’s model (Fig. 2) that, de-
spite its simplicity, is useful for estimating the results of several color
vision experiments, even though it is limited by insufficient knowl-
edge [5]. One should note, however, that our approach is not tied to
any particular stage model. For instance, we have also used it with
a three-stage model based on M¨
uller’s theory using the parameters
derived by Judd [14]. According to our experience, however, the pa-
rameters provided by Ingling and Tsou’s model produce better results.
Moreover, M¨
uller’s theory explanation for the occurrence of deutera-
nopia [14] does not seem to be in accordance with evidence reported
in the literature [2, 6, 27, 31].
4.1 Simulating Anomalous Trichromacy
Anomalous trichromacy is explained by a shift in the spectral sensitiv-
ity function of the anomalous cones [7, 21, 22, 27, 32]. Arrangements
of DNA bases called exons are involved in producing proteins which
are responsible to define specific characteristics. The Land Mpho-
topigment characteristics in humans are defined by sequences of six
exons from which the first and the last are invariant. The four interme-
diary exons in the sequence are responsible for the variability between
the spectral responses of normal and anomalous photopigments [27].
Hybrid genes contain exons from both Land Mpigments as illustrated
in Fig. 4. The squares indicate gene-specific for L and M pigments. All
hybrid genes produce photopigments with peak sensitivity between the
peaks of normal L and M photopigments. Each exon contributes to the
spectral shift of the produced hybrid photopigment, but exon five is de-
terminant of the basic type of photopigment.
Fig. 4. Exon arrangements of the L, M, and hybrid photopigment genes.
X-linked anomalous photopigment spectral sensitivity are interpreted as
interpolations of the normal L and M photopigment spectra. Adapted
from Sharpe et al. [27].
We model anomalous trichromacy by shifting the spectral sensitiv-
ity function of the anomalous cone according to the degree of severity
of the anomaly. A shift of approximately 20 nm represents a severe
case of protanomaly or deuteranomaly [19, 27], causing the spectral
sensitivity functions of the anomalous L(or M) cones to almost com-
pletely overlap with the normal M(or L) cones. As a result, the per-
ception of a severe protanomalous (deuteranomalous) is very similar
to the perception of a protanope (deuteranope). The much rarer case
of tritanomaly can also be simulated by shifting the spectral sensitiv-
ity function of the Scones. The spectral sensitivity functions of the
anomalous cones are represented as
La(λ) = L(λ+∆λL),(2)
Ma(λ) = M(λ+∆λM),(3)
Sa(λ) = S(λ+∆λS)(4)
where L(λ),M(λ), and S(λ)are the cone spectral sensitivity functions
for an average normal trichromat [28]. ∆λL,∆λM, and ∆λSrepresent
the amount of shift applied to the L,M, and Sanomalous cone, respec-
tively. Since these curves represent the outcome of the photoreceptor
level in our two-stage model, they still need to be processed by the
opponent-color stage. As previously noted, we use the opponent-color
space defined by Ingling and Tsou [5], whose transformation from
LMS to opponent space is represented by the 3 ×3 matrix shown in
Eq. 1, which will be referred to as TLMS2Opp.
As CVD results from changes in the spectral properties of the pho-
topigments, which happens at the retinal level, our model assumes that
the neural connections that link the photoreceptors themselves to the
rest of the visual system are not affected. Thus, we use the transforma-
tion TLMS2Opp to obtain anomalous spectral response functions for the
opponent channels, as shown by Eqs. 5 to 7. In those equations, pa,
da, and t a stand for protanomalous, deuteranomalous, and tritanoma-
lous, respectively. Fig. 5 shows examples of the resulting spectral op-
ponent functions for protanomaly and deuteranomaly instantiated for
∆λL=15 nm, and ∆λM=−19 nm. Note that the transformation for
normal trichromats is represented by Eq. 1.


W S(λ)
Y B(λ)
RG(λ)

pa
=TLMS2Opp 

La(λ)
M(λ)
S(λ)

(5)


W S(λ)
Y B(λ)
RG(λ)

da
=TLMS2Opp 

L(λ)
Ma(λ)
S(λ)

(6)


W S(λ)
Y B(λ)
RG(λ)

ta
=TLMS2Opp 

L(λ)
M(λ)
Sa(λ)

(7)

Fig. 5. Spectral opponent functions for anomalous trichromats. (left)
Protanomaly (∆λL=15 nm). (right) Deuteranomaly (∆λM=−19 nm).
We obtain a transformation from an RGB color space to an
opponent-color space simply by projecting the spectral power distribu-
tions ϕR(λ),ϕG(λ), and ϕB(λ)of the RGB primaries onto the set of
basis functions W S(λ),Y B(λ), and RG(λ)that define the opponent-
color space, as shown in Eq. 8. By using the appropriate set of basis
functions, Eq. 8 transforms RGB triples to opponent colors for either
normal trichromats, for anomalous trichromats, or for dichromats (dis-
cussed in Section 4.2). For instance, using the functions shown on the
left-hand side of Eq. 5 as basis functions, Eq. 8 will produce the el-
ements of a matrix that maps RGB to the opponent-color space of
protanomalous with a spectral sensitivity shift of ∆λL.
W SR=ρWS RϕR(λ)WS (λ)dλ,
W SG=ρWS RϕG(λ)WS (λ)dλ,
W SB=ρWS RϕB(λ)WS (λ)dλ,
Y BR=ρYB RϕR(λ)YB(λ)dλ,
Y BG=ρYB RϕG(λ)YB(λ)dλ,
Y BB=ρYB RϕB(λ)YB(λ)dλ,
RGR=ρRG RϕR(λ)RG(λ)dλ,
RGG=ρRG RϕG(λ)RG(λ)dλ,
RGB=ρRG RϕB(λ)RG(λ)dλ
(8)
The normalization factors ρWS ,ρYB , and ρRG are chosen to sat-
isfy the restrictions in Eq. 9. They guarantee that the achromatic
colors (gray shades) have the exact same coordinates ranging from
(0,0,0)to (1,1,1)both in RGB as well as in all possible versions of
the opponent-color spaces (normal trichromatic, all anomalous trichro-
matic, and all dichromatic). This is key for the simulation algorithm.
W SR+W SG+W SB=1,
Y BR+Y BG+Y BB=1,
RGR+RGG+RGB=1
(9)
Thus, the general class of transformation matrices Γthat map the RGB
color space to various instances of the opponent-color space can be
expressed as:
Γ=

W SRW SGW SB
Y BRY BGY BB
RGRRGGRGB

(10)
Let Γnormal be the matrix that maps RGB to the opponent-color space
of a normal trichromat. Γnormal is obtained by using the functions
shown on Fig. 3 (right) as basis functions for the projection operations
represented by Eq. 8. Thus, the simulation for a normal trichromat
of the color perception of an anomalous trichromat is obtained with
Eq. 11. As we will show next, the same general solution applies to the
simulation of dichromatic vision.


Rs
Gs
Bs

=Γ−1
normal Γ

R
G
B

(11)
4.2 Simulating Dichromacy
Measurements of visual pigment absorption using retinal densitometry
showed that dichromats lack one type of photopigment [1, 26]. Cur-
rently, researchers work with three possible alternatives for explaining
the lack of one kind of cone photopigment [2]: (i) the empty spaces
model, which states that a given class of cones and its correspond-
ing photopigment are lost, producing empty spaces in the cone mo-
saic. This hypothesis, however, is not supported by the findings of
Wesner et al. [31] who verified that the foveal cone photoreceptor mo-
saics of dichromats are similar in structure to the ones normal trichro-
mats. (ii) The replacement model suggests that the cones are still there,
but filled with one of the remaining kinds of photopigments. Finally,
(iii) the empty cones model suggests that a given class of cones con-
tains no photopigment. While the work of Vos and Walraven [29] may
support models (i) or (iii), evidence supporting the replacement model
can be found in the results of several researchers [2, 6, 31]. This makes
the photopigment substitution the most accepted model for explaining
dichromacy, with genetic arguments for protanopia and deuteranopia.
Our model makes its easy to test these hypotheses. For instance,
for simulating color appearance according to the empty space or to
the empty cone models, all one needs to do is to zero the outcome
of the corresponding cone type (either L,M, or S) before transforming
these signals into opponent color space functions. Given such curves, a
transformation matrix from RGB to opponent-color space is obtained
using Eqs. 8 and 9, and a simulation of color perception is obtained
using Eq. 11. The cases of deuteranopia and tritanopia are similar. The
first column of Fig. 8 (Empty) shows the simulated results obtained for
the flower image shown in Fig. 7 (a) using the empty space and empty
cone models, for protanopia (top row), and deuteranopia (bottom row).
These results are incorrect. For reference, we show in the last column
of this figure the results produced by Brettel et al.’s algorithm [4].
4.2.1 The Replacement Model
The replacement model seems to be the most plausible hypothesis for
explaining dichromacy [2, 6, 31]. The occurrence of a given photopig-
ment in a “wrong” type of cone seems more plausible between the L
and Mcones, and less plausible when it involves the Scones. For
instance, L- and M-cone photopigment genes show 96% mutual iden-
tity [27]. Moreover, the genes encoding the L- and M-cone photopig-
ments reside in the X-chromosome (at location Xq28) and have similar
exon arrangement coding. S-cone photopigments, on the other hand,
reside in chromosome 7, and its coding is given by five exons, one
less than L- and M-cone photopigment genes [27]. Thus, there is no
genetic basis for a photopigment substitution model of tritanopia. Tri-
tanopia is generally considered an acquired, as opposed to inherited,
condition [27]. For this reason, our model is not intended to handle tri-
tanopia (which is expected to affect about 0.003% of the population,
according to the data available for the Caucasian population [25, 27]).
Our model of dichromacy uses three cone types, but only two kinds
of photopigments. The replacement model could be simulated simply
by replacing the spectral sensitivity function of the Lcones with the M
cones for the case of protanopia, and the other way around for the case
of deuteranopia. Eq. 12 illustrates this for the case of protanopia. The
second column of Fig. 8 (Photop. Subst.) illustrates the results ob-
tained with this technique, which, again, are clearly incorrect. This re-
sulted from the fact that, even though the spectral sensitivity functions
of all three types of cones had their peak sensitivity independently nor-
malized to 1.0, the areas under these curves are sufficiently different
(Fig. 3 left) and need to be taken into account.


W S(λ)
Y B(λ)
RG(λ)

protanopia
=TLMS2Opp 

M(λ)
M(λ)
S(λ)

(12)
The replacement of the L-cone spectral sensitivity curve by the M-
cone spectral sensitivity curve, which has a smaller area than L’s,
causes the restrictions defined in Eq. 9 to only be satisfied for ρRG <0.
In this case, the resulting coefficients RGR,RGG, and RGBhave their
signs reversed (with respect to the corresponding coefficients for a nor-
mal trichromat), making it impossible to preserve the achromatic col-
ors in the range from (0,0,0)to (1,1,1)in the opponent-color space.
A similar phenomenon happens when the spectral sensitivity curve of
the Mcone is replaced by the Lcone one. The solution to this problem
lies in rescaling the replaced curves.

(a) (b) (c) (d)
Fig. 6. Comparison of the plausible dichromacy models considering the
entire RGB space (protanopia case). A surface obtained using Brettel et
al.’s [4] algorithm is shown for reference in all images. (a) Empty Space
/ Empty Cones model. (b) Replacement model. (c) Replacement model
using Eq. 15. (d) Same as (c) but also scaled by 0.96.
The rescaling of the curves is performed so that the new curves
preserve the areas under the curves of the host cones (Eqs. 15 and 16).
The third column of Fig. 8 (Scale Ratio) illustrates the results obtained
with this technique. Note that while the colors of the leaves and petals
are approximately correct, they still contain some excessive redness.
AreaL=RL(λ)dλ,(13)
AreaM=RM(λ)dλ,(14)
Lprotanope (λ) = AreaL
AreaMM(λ),(15)
Mdeuterano pe(λ) = AreaM
AreaLL(λ)(16)
A small adjustment in the area ratios in Eqs. 15 and 16 produces
a significant improvement in image quality. For instance, the results
shown in the fourth column of Fig. 8 were obtained after scaling the
ratio (AreaL/AreaM)in Eqs. 15 and 16 by 0.96. Such a scaling fac-
tor was used to improve the matching between the surfaces obtained
when the entire RGB color space is simulated for dichromatic vision
using Brettel et al.’s and our model (Fig. 6d). This seems to support
our model’s prediction that the correctness of the replacement model
requires normalization by the ratio of the areas under the original spec-
tral sensitivity curves of the Land Mcones. Such a prediction still
needs to be verified experimentally. While the prediction is off by a
0.04 factor, one should consider two important points: (i) the coeffi-
cients of the TLMS2Opp matrix (Eq. 1) used in the current implemen-
tation of our model are expected to contain some inaccuracies; and
(ii) Brettel’s model, used for reference, provides an approximation to
the actual dichromat’s perception and cannot be taken as ground truth.
For a qualitative comparison of Brettel et al.’s and our results, please
see the last two columns of Fig. 8.
Fig. 6 compares the simulations of the protanoptic vision for the
entire RBG color space performed by the several models discussed in
this section. A surface obtained using Brettel et al.’s [4] algorithm is
shown for reference (the blue-and-yellow surface). Fig. 6 (a) shows
a simulation of the empty space / empty cones models. (b) illustrates
the result obtained with the use of a replacement model without nor-
malization (photopigment substitution). The image in (c) shows the
resulting surface after the original ratio (AreaL/AreaM)has been pre-
served using Eq. 15. Fig. 6 (d) shows the resulting surface obtained
after the original ratio (AreaL/AreaM)has been scaled by 0.96. The
two surfaces are now considerably closer, although not coincident.
(a) (b) (c) (d) (e)
Fig. 7. Reference images. (a) Flower. (b) Brain. (c) Cat’s Eye nebula.
(d) Scatter plot. (e) Slice of the HSV color space (V=1).
Empty Photop. Subst. Scale Ratio 0.96*Ratio Brettel
ProtanopiaDeuteranopia
Fig. 8. Simulation of dichromatic perception for the flower shown in
Fig. 7(a) according to four different models. From left to right: empty
space / empty cones, photopigment substitution (replacement model),
photopigment substitution with scaling according to Eq. 15, same as
previous but also scaled by 0.96, Brettel et al.’s (for reference).
4.3 The Algorithm for Simulating CVD
As discussed in Section 4.1, anomalous trichromacy can vary from
mild to severe depending the amount of shift found in the peak sen-
sitivity of the photopigments. Such shifts are caused by the exon ar-
rangements (Fig. 4) and the color perception of a severe anomalous
trichromat is similar to the perception of a dichromat of the same class
(i.e., protan or deutan). Thus, it is reasonable to speculate that as the
peak sensitivity of the L(or M) cones gets shifted toward the peak
sensitivity of the M(or L) cone, the rescaling of the associated curve
required for the case of protanopia and deuteranopia, also needs to be
performed, in proportion to the amount of shift. This is justified con-
sidering that as one transitions between the spectral sensitivity curves
of the Land Mcones, there should also be a corresponding transition in
the bandwidths and areas under the intermediate curves. Such a grad-
ual rescaling guarantees a smooth transition between the various de-
grees of protanomaly (deuteranomaly) and protanopia (deuteranopia).
Eqs. 17 and 18 model this smooth transition:
La(λ) = αL(λ)+(1−α)0.96 AreaL
AreaMM(λ)(17)
Ma(λ) = αM(λ)+(1−α)1
0.96
AreaM
AreaLL(λ)(18)
Sa(λ) = S(λ+∆λS)(19)
where α= (20 −∆λ)/20, for ∆λ∈[0,20]. Note that the 0.96 factor
has been added to try to compensate for the inaccuracies of the avail-
able data, assuming Bretell et al.’s model as reference. As more accu-
rate data describing the mapping from cones responses into opponent
channels and/or a more accurate reference model become available,
the need for such a factor might be eliminated.
Since there are no strong biological explanations yet to justify the
causes of tritanopia and tritanomaly, we simulate tritanomaly based on
the shift paradigm only (Eq. 19) as an approximation to the actual phe-
nomenon and restrain our model from trying to model tritanopia. Like-
wise, we do not try to simulate monochromacy (either rod or cone).
One should note, however, that the conditions covered by our model
(i.e., anomalous trichromacy and dichromacy) correspond to approxi-
mately 99.38% of all CVD cases [25, 27].
The actual algorithm for simulating anomalous trichromatic vision
uses Eqs. 17, 18, and 19 in conjunction with Eqs. 5 to 11. Likewise, the
simulation of protanoptic and deuteranoptic vision is obtained using
Eqs. 20 and 21, respectively, in conjunction with Eqs. 8 to 11.


W S(λ)
Y B(λ)
RG(λ)

protanopia
=TLMS2Opp 

Lprotanope (λ)
M(λ)
S(λ)

(20)


W S(λ)
Y B(λ)
RG(λ)

deuterano pia
=TLMS2Opp 

L(λ)
Mdeuterano pe(λ)
S(λ)

(21)

where
Lprotanope (λ) = 0.96 AreaL
AreaMM(λ),(22)
Mdeuterano pe(λ) = 1
0.96
AreaM
AreaLL(λ)(23)
Thus, the simulation of any protan or deutan anomaly can be obtained
using Eq. 24, where ΓCVD should be instantiated with the Γmatrix
(Eq. 10) specific for the kind of target CVD.


Rs
Gs
Bs

=Γ−1
normal ΓCVD 

R
G
B

(24)
5 RE SU LTS
We have incorporated our model in a visualization system and imple-
mented it in MATLAB. It has been used it to simulate the perception
of both dichromats and anomalous trichromats (at different degrees of
severity). The simulation operator (Eq. 24) only requires one matrix
multiplication per pixel and can be efficiently implemented on GPUs.
Fig. 9 compares the results produced by our technique with the ones
produced by the technique of Yang et al. [33] for the image shown in
Fig. 7 (e). Note that the images simulated using Yang et al.’s technique
show some green, red and purple shades, for both protanomalous and
deuteranomalous at all degrees of severity. This is inconsistent with
the perception of these individuals, who after some degree of severity
have trouble distinguishing red from green. The last column in Fig. 9
(Brettel) shows the results simulated using Brettel et al.’s algorithm
for comparison with the severe case (20 nm).
Fig. 1 shows a reference image (left) and the simulated perception
obtained with our model for different degrees of protanomaly (6 nm
and 14 nm) and for protanopia. Note the progressive loss of color
contrast as the degree of severity increases. Fig. 10 shows examples of
simulation of anomalous trichromatic vision in scientific visualization.
For all examples, we provide simulations for both protanomalous and
deuteranomalous vision at severity levels corresponding to 2 nm, 8 nm,
14 nm, and 20 nm. Note how the ability to perceive red and green
vanishes with the increase of the anomaly severity.
(2 nm) (10 nm) (20 nm) Brettel
P (Our)
P (Yang)
D (Our)
D (Yang)
Fig. 9. Simulation of protanomalous and deuteranomalous vi-
sion for several degrees of severity (expressed in nm). Last col-
umn: result of Brettel et al.’s algorithm for reference. P/D (Our):
Protanomaly/Deuteranomaly simulated with our technique. P/D (Yang):
Protanomaly/Deuteranomaly simulated with Yang et al.’s technique.
5.1 Experimental Validation
To validate the proposed model, we performed some experiments in-
volving both normal trichromats and color vision deficient individu-
als. All subjects performed two rounds of the color discrimination
Farnsworth-Munsell 100-Hue (FM100H) test [9]. The original test
consists of 85 movable color caps and four wooden boxes (trays). One
(2 nm) (8 nm) (14 nm) (20 nm) Brettel
PnomalyDnomalyPnomalyDnomalyPnomalyDnomaly
Fig. 10. Simulation of protanomalous and deuteranomalous vision in
scientific visualization. From top to bottom: brain dataset, Scatter plot,
and Cat’s Eye nebula. The degrees of severity are expressed in nm. Last
column: Brettel et al.’s dichromatic simulation for reference. Pnomaly :
Protanomaly. Dnomaly : Deuteranomaly.
box holds 22 caps, while the other three hold 21 caps each. The tested
subject must take one box at a time and arrange its color caps in a con-
tinuous color sequence using two extra fixed caps at opposite ends of
the box as reference. The color sequences range from red to yellow,
yellow to blue-green, blue-green to blue, and blue to purple-red, re-
spectively. The 85 colors are sampled from the Munsell color system
(Hue-Value-Chroma space) with equally spaced hue values (starting
at red, 5R in Munsell’s notation), and equal saturation and brightness
control (chroma 6 and value 6 in Munsell’s notation).
Each color cap is identified by an indexing number (ranging from 1
to 85) printed on its back. After the arrangement of the caps, the color
discrimination aptitude of the subject is verified by computing an error
score for each color cap as the sum of the absolute difference between
its index and the index of its two adjacent neighbors. The total error
score is the sum of the individual error scores less the minimum error
score of 170. A plot like the one shown in Fig. 11 is used to simplify
the analysis. In such a plot, the caps are numbered counterclockwise
and the individual error scores are plotted radially outward from the
circle with an error score of two on the inner circle. The color dis-
crimination aptitude of a subject is analised over the average scores
for all color caps computed from at least two trials, a test and retest.
We implemented a computerized version of the FM100H using C++
and OpenGL. Our implementation is based on the Meyer and Green-
berg’s work [20]. For the tests, we used 17-inch CRT flat screen mon-
itors (model LG Flatron E701S, 1024 ×768 pixels, 32-bit color, at
85 Hertz). The monitors were calibrated using a ColorVision Spyder
2 colorimeter (Gamma 2.2 and White Point 6500K). Both calibration
and tests were performed with the room lights off. Unlike Meyer and
Greenberg, we presented one sequence of color caps at a time (like in
the original test). The user interface consisted of a black screen hav-
ing the color caps placed horizontally in the central row. Each cap was
rendered as a circle with 1 cm of diameter and could be moved using
the mouse (except the reference color caps, which are fixed). Both the

presentation order of the trays of caps, and the initial arrangement of
the caps in each tray were random.
The group of normal trichromats (NTggroup) consisted of 17 male
subjects (ages 19 to 29). The group of the color vision deficient in-
dividuals (CV Dggroup) consisted of 13 male subjects classified as
follows: 4 protanomalous (ages 23 to 53), 4 protanopes (ages 22 to
59), 3 deuteranomalous (ages 22 to 28), and 2 deuteranopes (ages 18
to 44). The classification of the subjects in the CV Dggroup was done
after the application of an Ishihara test [12].
Each subject in the CVDggroup performed two trials of the
FM100H test using the original colors. We then averaged the 16 results
of the 8 protans (protanomalous and protanopes) and did the same for
the 10 results of the 5 deutans (deuteranomalous and deuteranopes).
The results of these averaged tests are shown in Fig. 12 (b) and (d), for
the protans and deutans, respectively.
The subjects in the NTggroup were divided into two subgroups:
NTgp (8 individuals, ages 19 to 26) and NTgd (9 individuals, ages 21
to 29). Both subgroups performed two trials of the FM100H test us-
ing the original colors. We then averaged these 34 results, which are
depicted in the plot shown in Fig. 11. We then used our model to
simulate how protanomalous with shifts of 12 nm, 16 nm, and 19 nm,
would perceive the original colors of the FM100H test. We called these
sets of simulated colors CP
12nm,CP
16nm, and CP
19nm, respectively. We
then performed a similar simulation for the case of deuteranomalous,
obtaining sets of simulated colors CD12nm,CD16nm , and CD19nm .
The subgroup NTgp then performed two trials of the FM100H test
using the sets CP
12nm,CP
16nm, and CP
19nm, one at a time, instead of
the original colors. This should simulate for the normal trichromat the
perception of the corresponding degrees of protanomaly. Fig. 12 (a)
shows a plot of averaged 48 results (8×3×2). A similar procedure
was applied to the 9 members of the NTgd subgroup using the sets
CD12nm,CD16nm, and CD19nm of simulated colors. Fig. 12 (c) shows
a plot of averaged 54 results (9×3×2).
A comparison of the plots corresponding to the averaged results of
the NTgp subgroup (Fig. 12 a) and the averaged results of the protans
(Fig. 12 b) reveals great similarity. The small blue segment next to the
yellow hue (see the larger versions of these plots in the supplementary
materials) represents the eigenvector with largest absolute eigenvalue
computed for the covariance matrix of the error scores. A comparison
of the averaged results of the NTgp subgroup (Fig. 12 c) and the aver-
aged results of the deutans (Fig. 12 d) reveals an even better agreement
of the corresponding eigenvectors. Note how these plots are signifi-
cantly different from the one shown in Fig. 11. These results indicate
that the proposed model provides good simulations for the color per-
ception by individuals with color vision deficiency.
Fig. 11. Averaged results of the Farnsworth-Munsell 100H test per-
formed by 17 normal trichromats using the test original colors.
5.2 Discussion
Once incorporated into visualization systems, our model can provide
immediate feedback to visualization designers. As such, it can be a
valuable tool for the design of visualizations that are meaningful both
for individuals with normal and deficient visual color systems. Note
that although one could consider the simpler solution of just adopting
color scales that completely fit in the color gamut of dichromats, this
might be unnecessarily restrictive in many situations. In such cases,
a larger color gamut could be exploited to obtain more effective re-
sults, especially when dealing with multidimensional visualizations.
The top row of Fig. 13 compares the color gamut of normal trichro-
mats, protanomalous with a shift of 10 nm, and protanopes, for two
slices of the HSV color space (V=1.0 and V=0.75). Although the 10-
nm protanomalous only perceive a fraction of the HSV disks, they
still have a much larger gamut than the protanopes, who only per-
ceive a line across each disk (from blue to yellow). The bottom row
of Fig. 13 shows visualizations of the Visible Male’s head using the
same transfer function defined over the color gamut of these individu-
als. Note that the protanomalous’ larger color gamut in comparison to
the protanope’s lends to better color contrast. Such images were cap-
tured using some visualization software that incorporates our model
and allows switching between normal color vision and the simulation
of any degree of anomaly in real-time (see video). The supplemen-
tary materials show the matrices used to incorporate our model in the
system.
V = 1.0 V = 0.75 V = 1.0 V = 0.75 V = 1.0 V = 0.75
Normal Trichromat Protanomalous (10 nm) Protanope
Fig. 13. Visualization of the Visible Male’s head using the same transfer
function (bottom) defined over the color gamut (top row) of a normal
trichromat, a protanomalous (10 nm), and a protanope.
The choice of an appropriate color scale for a given visualization
should take into account different factors, such as characteristics of
the dataset, questions that one would like to answer about the data,
the intended viewers and their cultural backgrounds [24]. Thus, our
model is not intended to automatically build or guide the construction
of color maps, even though it can be used to provide information for
approaches that do so [3, 11, 18, 30]. Instead, it gives visualization
designers an understanding of the perceptual limitations of each class
of CVD. As such, our model can help designers to refine their visual-
izations, making them more effective to a wider range of viewers.
More research is needed to develop better color selection method-
ologies that take into account the limitations of specific groups of ob-
servers. Healey [11] described a technique for choosing multiple col-
ors for use in visualization. He accomplished this by measuring and
controlling perceptual metrics like color distance,linear separation
and color category during color selection. Healey’s technique could be
used to define sets of colors that provide good differentiation among
data elements for both normal trichromats and any given class of CVD.
6 CONCLUSION
We have presented a physiologically-based model for simulating color
perception, and have shown how it can be used to help designers to
produce more effective visualizations. Our model is the first to consis-
tently handle normal color vision, anomalous trichromacy, and dichro-
macy using a unified framework. By means of a controlled user exper-
iment, we have demonstrated that the results produced by our model
seem to closely match the perception of individuals with color vi-
sion deficiency. We have also compared our results to the ones ob-
tained with existing models for simulating the perception of anoma-

(a) (b) (c) (d)
Fig. 12. Averaged results of the Farnsworth-Munsell 100H test. (a) Normal trichromats simulating protan vision. (b) Protan results for the original
colors. (c) Normal trichromats simulating deutan vision. (d) Deutan results for the original colors.
lous trichromats [33] and of dichromats [4]. Such comparisons indi-
cate that our results are superior to the ones of Yang et al. [33] for
anomalous trichromacy, and equivalent to the ones of Brettel et al. [4]
for the case of dichromacy. Currently, we are working on the devel-
opment of automatic techniques for making optimal use of anomalous
trichromat color gamut for visualization.
Our model also provides a flexible framework for allowing scien-
tists to test different hypotheses about color vision models. We have
shown how we tested the plausibility of the three most accepted hy-
potheses for the causes of dichromatic vision. While it is difficult to
verify such hypotheses in vivo, our model suggests that pigment sub-
stitution is the most plausible one. Moreover, it indicates that pigment
substitution would require a renormalization of the spectral sensitivity
curve of the affected cones. Such an observation, not yet reported in
the vision literature, if verified, would provide some strong evidence
in favor of the correctness of our model.
ACK NOW LE DG ME NT S
We deeply thank our volunteers, and Francisco Pinto for pro-
viding the visualization software. We also thank the anony-
mous reviewers for their insightful comments. This work was
sponsored by CNPq-Brazil (grants 200284/2009-6, 131327/2008-9,
476954/2008-8, 305613/2007-3 and 142627/2007-0). Fig. 1 (left),
and Fig. 7 (a) have been kindly provided by CCSE at LBNL,
and Karl Rasche [23], respectively. Figs. 7 (c) and (d) are from
http://commons.wikimedia.org.
REFERENCES
[1] M. Alpern and T. Wake. Cone pigments in human deutan colour vision
defects. Journal of Physiology, 266(3):595–612, 1977.
[2] T. T. Berendschot, J. van de Kraats, and D. van Norren. Foveal cone
mosaic and visual pigment density in dichromats. Journal of Physiology,
492.1:307–314, 1996.
[3] L. Bergman, B. Rogowitz, and L. Treinish. A rule-based tool for assisting
colormap selection. In Proc. Visualization ’95, pages 118–125, 1995.
[4] H. Brettel, F. Vi´
enot, and J. D. Mollon. Computerized simulation of color
appearance for dichromats. J. Opt. Soc. Am., 14(10):2647–2655, 1997.
[5] J. Carl R. Ingling and B. H.-P. Tsou. Orthogonal combination of the three
visual channels. Vision Res., 17(9):1075–1082, 1977.
[6] C. M. Cicerone and J. L. Nerger. The density of cones in the fovea cen-
tralis of the human dichromat. Vision Res., 29:1587–1595, 1989.
[7] P. DeMarco, J. Pokorny, and V. C. Smith. Full-spectrum cone sensitivity
functions for x-chromosome-linked anomalous trichromats. J. Opt. Soc.
Am. A, 9(9):1465–1476, 1992.
[8] M. D. Fairchild. Color Appearance Models. Addison Wesley, 1997.
[9] D. Farnsworth. The Farnsworth-Munsell 100-hue test for the examination
of color discrimination. Munsell Color Company, NY, 1957.
[10] C. H. Graham and Y. Hsia. Studies of color blindness: A unilaterally
dichromatic subject. Proc. Natl. Acad. Sci. USA, 45(1):96–99, 1959.
[11] C. G. Healey. Choosing effective colours for data visualization. In Proc.
of the 7th IEEE Conference on Visualization, pages 263–270, 1996.
[12] S. Ishihara. Tests for colour-blindness. Kanehara Shuppan Co., 1979.
[13] D. B. Judd. Color perceptions of deuteranopic and protanopic observers.
J. Opt. Soc. Am., 39(3):252, 1949.
[14] D. B. Judd. Response functions for types of vision according to the M ¨
uller
theory. J. Res. Natl. Bur. Std., 42(1):1–16, January 1949.
[15] D. B. Judd. Fundamental studies of color vision from 1860 to 1960. Proc.
Natl. Acad. Sci. USA, 55(6):1313–1330, 1966.
[16] S. Kondo. A computer simulation of anomalous color vision. In Y. Ohta,
editor, Color Vision Deficiencies, pages 145–159. Symp. Int. Res. G. on
CVD, Kugler & Ghedini, 1990.
[17] G. R. Kuhn, M. M. Oliveira, and L. A. F. Fernandes. An efficient
naturalness-preserving image-recoloring method for dichromats. IEEE
TVCG, 14(6):1747–1754, 2008.
[18] H. Levkowitz and G. T. Herman. Color scales for image data. IEEE
CG&A, 12(1):72–80, 1992.
[19] D. McIntyre. Colour Blindness: Causes and Effects. Dalton Publ., 2002.
[20] G. W. Meyer and D. P. Greenberg. Color-defective vision and computer
graphics displays. IEEE Comput. Graph. Appl., 8(5):28–40, 1988.
[21] M. Neitz and J. Neitz. Molecular genetics of color vision and color vision
defects. Arch. Ophthalmol., 118(3):691–700, 2000.
[22] J. Pokorny and V. C. Smith. Evaluation of single-pigment shift model of
anomalous trichromacy. J. Opt. Soc. Am., 67(9):1196–1209, 1997.
[23] K. Rasche, R. Geist, and J. Westall. Re-coloring images for gamuts of
lower dimension. Comput. Graph. Forum, 24(3):423–432, 2005.
[24] P. Rheingans. Task-based color scale design. In SPIE-Int. Soc. Opt. Eng.,
volume 3905, page 3343, 2000.
[25] C. Rigden. The eye of the beholder - designing for colour-blind users. Br
Telecomm Eng, 17, 1999.
[26] W. A. H. Rushton. A cone pigment in the protanope. Journal of Physiol-
ogy, 168(2):345–359, September 1963.
[27] L. T. Sharpe, A. Stockman, H. J¨
agle, and J. Nathans. Color Vision: From
Genes to Perception, chapter Opsin genes, cone photopigments, color vi-
sion, and color blindness, pages 3–51. Cambridge University Press, 1999.
[28] V. Smith and J. Pokorny. Spectral sensitivity of the foveal cone photopig-
ments between 400 and 500 nm. Vision Res., 15(2):161–171, 1975.
[29] J. J. Vos and P. L. Walraven. On the derivation of the foveal receptor
primaries. Vision Res., 11(8):799–818, 1971.
[30] C. Ware. Color sequences for univariate maps: Theory, experiments and
principles. IEEE C&GA, 8(5):41–49, 1988.
[31] M. F. Wesner, J. Pokorny, S. K. Shevell, and V. C. Smith. Foveal
cone detection statistics in color-normals and dichromats. Vision Res.,
31(6):1021–1037, 1991.
[32] G. Wyszecki and W. S. Stiles. Color Science: concepts and methods,
quantitative data and formulae. John Wiley and Sons, 2nd edition, 2000.
[33] S. Yang, Y. M. Ro, E. K. Wong, and J.-H. Lee. Quantification and
standardized description of color vision deficiency caused by anomalous
trichromats - Part I: Simulation and measurement. EURASIP Journal on
Image and Video Processing, 2008(1), 2008.