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IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

[DOI: 10.2197/ipsjtcva.7.41]

Research Paper

Computer Simulation of Color Confusion for Dichromats

in Video Device Gamut under Proportionality Law

Hiroshi Fukuda1,a) Shintaro Hara2,b) Ken Asakawa1,c) Hitoshi Ishikawa1,d)

Makoto Noshiro1,e) Mituaki Katu ya 3,f)

Received: September 9, 2014, Accepted: Februar y 24, 2015, Released: May 7, 2015

Abstract:

Dichromats are color-blind persons missing one of the three cone systems. We consider a computer simu-

lation of color confusion for dichromats for any colors on any video device, which transforms color in each pixel into

a representative color among the set of its confusion colors. As a guiding principle of the simulation we adopt the

proportionality law between the pre-transformed and post-transformed colors, which ensures that the same colors are

not transformed to two or more diﬀerent colors apart from intensity. We show that such a simulation algorithm with the

proportionality law is unique for the video displays whose projected gamut onto the plane perpendicular to the color

confusion axis in the LMS space is hexagon. Almost all video display including sRGB satisfy this condition and we

demonstrate this unique simulation in sRGB video display. As a corollary we show that it is impossible to build an

appropriate algorithm if we demand the additivity law, which is mathematically stronger than the proportionality law

and enable the additive mixture among post-transformed colors as well as for dichromats.

Keywords:

dichromatic simulation, proportionality law, sRGB, video device gamut

1. Introduction

About 2.5% of male population are dichromats. Dichromats

are color-blind persons missing one of the three cone systems in

their eyes, and they cannot distinguish some colors that normal

trichromats, persons having normal color vision, can distinguish.

Such colors are called confusion colors.

Almost all dichromats are inherited and cannot change their

color vision in their life. Thus, the color universal design to avoid

danger and inconvenience caused by confusion colors are pre-

ferred. The most common tools used by designers are the com-

puter simulation of images which transforms images into those

that dichromats see.

Standard computer simulation algorithms of color appearance

for dichromats were proposed by Brettel et al. in 1997 [1], [2].

Nowadays this algorithm is implemented in many computer ap-

plications, such as Vischeck [3] for image processing software

and Chromatic Vision Simulator[4], [5] for smart phones, and is

chosen as a reference data among several more elaborate dichro-

matic simulation algorithms [6]. We denote this algorithm as

A97.

1Graduate School of Medical Sciences, Kitasato University, Sagamihara,

Kanagawa 252–0373, Japan

2Graduate School of Medicine, The University of Tokyo, Bunkyo, Tokyo

113–0033, Japan

3University of Shizuoka, Shizuoka 222–8526, Japan

a) fukuda@kitasato-u.ac.jp

b) hara@bme.gr.jp

c) asaken@kitasato-u.ac.jp

d) hitoshi@kitasato-u.ac.jp

e) noshiromakoto@gmail.com

f) mitsuaki.katsuya.jurigi@gmail.com

It is well known that A97 cannot simulate all colors that video

devices can support as the authors already indicated in their pa-

per [1]. This is the reason why they proposed another algorithm

in 1999 [7] which can simulate all colors on the device. We de-

note this algorithm as A99. The modiﬁcation in A99, however,

has defects as the simulated colors are not the confusion colors.

In any dichromatic simulation the simulated color at least needs

to be a confusion color. However, this defect may be practically

unworthy of attention because this simulated color is very close

to the confusion color in the standard sRGB video display [8].

Furthermore A99 is a display dependent algorithm and it works

only for the sRGB video display. Thus A99 is not a satisfactory

modiﬁcation of A97.

In this work, rather than pursuing the simulation of color per-

ception, we consider the simulation of color confusion for dichro-

mats for all colors that video devices can support under some

guiding principles. From a point of view of the computer algo-

rithms, both simulations are about choosing a representative color

s(Q) among the set of confusion colors for a given color Q.Inthe

simulation of color perception, the representative or the function

of Q,s(Q) is determined based on the reports on unilateral inher-

ited color vision deﬁciencies [9], [10], [11] or the human color

vision mechanism [6]. Instead, in our simulation of color con-

fusion, we will determine s(Q) by the demand that for any Qin

the display color gamut G,s(Q) exists in Gas well under some

reasonable guiding principles. Recent works [6], [12] on dichro-

matic simulations are not focused on display color gamut like

A99 or the present work but on better algorithms of color percep-

tion from the viewpoint of human color vision mechanism.

In Section 2, after explaining A97 and A99 with the same stan-

c

2015 Information Processing Society of Japan 41

IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

dard device sRGB and in the same cone fundamentals, we make

the relation between A97 and A99 clear, and show how numeri-

cally serious the problems are. In Section 3, we turn to a guiding

principle in the simulation of color confusion, namely a propor-

tionality law and propose a new algorithm for general devices in-

cluding sRGB in Section 4. In Section 5, experimental results of

responses from dichromats for A97, A99 and our new algorithm

are shown. In Section 6, we discuss another guiding principle,

namely additivity law. Section 7 is the summary and discussion.

2. Relation between the Brettel et al. 1997 and

Vi´

enot et al. 1999 Algorithms

In this paper, we use CIE 1931 XYZ color speciﬁcation system

and use LMS system derived from Smith and Pokorny[7], [8] for

normal trichromats (normal people),

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

L

M

S⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=U⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

X

Y

Z⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,(1)

where

U=⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0.15514 0.54312 −0.03286

−0.15514 0.45684 0.03286

000.01608 ⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.(2)

L,M,andSspecify colors in terms of the relative excitations

of longwave sensitive, middlewave sensitive, and shortwave sen-

sitive cones, respectively. It is assumed that the three kinds of

dichromats, protanopes, deuteranopes and tritanopes, cannot per-

ceive any change in L,M,andS, respectively. In other words,

the confusion colors of a color stimulus Q=(L,M,S)T, where T

denotes the matrix transpose, are on the line

la(Q)={Q+ˆat|t=real}(3)

parallel to a=L,Mand Saxis passing through Qin this LMS

space, respectively. ˆain Eq. (3) represents the unit vector for

a=L,Mand Saxis.

In order to deﬁne a representative color s(Q) among the confu-

sion colors la(Q), the surface

Σ={⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

L

M

S⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

|u(L,M,S)=0}(4)

is introduced, where u(L,M,S) is a real function in the form

u(L,M,S)=⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

L−uL(M,S) for protanopes,

M−uM(S,L) for deuteranopes,

S−uS(L,M) for tritanopes.

(5)

Thus the color s(Q) is the crossing point between the surface Σ

and the line la(Q)

s(Q)=Σ∩la(Q),(6)

where a=L,M,orSfor protanope, deuteranope, and tritanope,

respectively. Figure 1 shows this geometrical scheme among Q,

lL(Q), Σand s(Q) for protanopes. In Fig. 1, the bended surface is

Σwhich is used in A97 explained in the next section.

Fig. 1 Geometrical scheme to determine s(Q)fromQas crossing point be-

tween lL(Q)andΣfor protanopes in LMS space. The bended surface

is ΣwhichisusedinA97.

We call s(Q)assimulation function and Σas simulation sur-

face, respectively, and deﬁne simulation of color confusion.

Simulation of color confusion. Input: a picture I composed of

w×h pixels whose color at (i,j)position is Qij. Output: a pic-

ture s(I)composed of w×h pixels whose color at (i,j)position is

s(Qij).

Note that there is no more restriction on simulation function

s(Q) and accordingly on simulation surface Σ. Thus, in the sim-

ulation of color confusion, simulation surface Σcan be either

piecewise planar as shown in Fig. 1 or non-planar (curved) sur-

face used in [6].

2.1 A97: Brettel et al. 1997 Algorithms

In accordance with the reports on unilateral inherited color vi-

sion deﬁciencies [9], [10], [11], which stated that all colors were

seen as colors with dominant wavelength either λ1or λ2,A97

deﬁned the simulation surface Σas

Σ(λ1,λ

2)=Σ(λ1)∪Σ(λ2),(7)

Σ(λi)={αE+βC(λi)|α≥0,β≥0},i=1,2,(8)

where E=U(1,1,1)Tis equal-energy stimulus and

C(λ)=U⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

¯x(λ)

¯y(λ)

¯z(λ)⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(9)

is monochromatic stimulus. Here ¯x(λ), ¯y(λ) and ¯z(λ) are the CIE

1931 standard color matching functions. Σ(λi) in Eq. (8) is a pla-

nar region bounded by two half-line from origin parallel to E

and C(λi). For protanopes and deuteranopes λ1=475 nm and

λ2=575 nm are adopted, and for tritanopes, λ1=485 nm and

λ2=660 nm. The simulation surface Σshown in Fig. 1 is actu-

ally Σ(475,575).

In these cases, the simulation surface is called stimulus surface

because it is a common color stimulus perceived by both dichro-

mats and normal trichromatic observers. In general,

Simulation of color perception. The simulation of color confu-

c

2015 Information Processing Society of Japan 42

IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

Fig. 2 Intersection of the stimulus surface Σ(475,575) of A97 and the sRGB-parallelepiped. If we look

at the left ﬁgure from the Maxis, we obtain the right ﬁgure. Dashed line lMin the left ﬁgure,

a line parallel to M-axis, is passing through the sRGB-parallelepiped but does not intersect with

Σ(475,575) in the sRGB-parallelepiped. A point near S-axis in the right ﬁgure is the lM.

sion s(I)of a picture I is called the simulation of color perception

s∗(I)if stimulus surface is used as the simulation surface.

If a unilateral subject looks at the simulation of color percep-

tion s∗(I) with his normal trichromatic eye and at the original

picture Iwith his dichromatic eye, he cannot ﬁnd any diﬀerences

between two pictures. However, if he looks at the simulation of

color confusion s(I) instead of s∗(I)hewillﬁnddiﬀerences. Of

course, dichromats whose eyes are both dichromatic cannot dis-

tinguish three pictures I,s(I)ands∗(I).

Now we introduce sRGB video devices as the most popular

type of displays. The sRGB color stimuli are represented by 8-bit

DAC values (r,g,b) for r,g,b=0,1,2,...,255 and these stim-

uli are included in the parallelepiped deﬁned by Red=(255,0,0),

Green=(0,255,0) and Blue=(0,0,255) primaries in sRGB. We

call this parallelepiped the sRGB-parallelepiped.

In Fig. 2 (left), the intersection of the stimulus surface

Σ(475,575) shown in Fig. 1 and the sRGB-parallelepiped is

shown. If we look at the Fig. 2 (left) from the Maxis, we obtain

the Fig. 2 (right). We can see from the Fig. 2 (right) for some stim-

ulus Qin the sRGB-parallelepiped, the line lM(Q) does not cross

the stimulus surface Σ(475,575) in the sRGB-parallelepiped, that

is, the intersection of the stimulus surface Σ(475,575) and the

sRGB-parallelepiped. The dashed line labeled lMin Fig. 2 (left)

is an example of such lM(Q). The lMhas points in the sRGB-

parallelepiped, Q, but does not intersect with Σ(475,575) in the

sRGB-parallelepiped. A point near S-axis in Fig. 2 (right) is the

lM. This means that the stimulus surface Σ(475,575) cannot be

adopted to simulate all 2563=16777216 colors for deuteranopes.

This problem also exists for protanope and tritanope simulations.

In Table 1, we present the number of sRGB colors (r,g,b)

which cannot be simulated by A97. We consider those numbers

as rather large to neglect. Note that irrespective of the devices

some colors cannot be simulated by A97 as shown in Fig. 6 of

Ref. [6]. To our knowledge, it is not known whether the dichro-

matic eye and normal trichromatic eye of unilateral dichromats

can match these colors or not.

Tab l e 1 Number of sRGB colors which cannot be simulated by A97. In the

column labeled Nthe numbers of colors which cannot be simulated

by A97 are tabulated, and in “Ratio” the ratios between Nand the

total number of sRGB colors (2563).

Type of simulation NRatio

Protanopes 4,669,975 27.8%

Deuteranopes 2,621,467 15.6%

Tritanopes 2,797,874 16.7%

Tab l e 2 Number of sRGB colors which cannot be simulated by A99 with-

out color domain transformation (11). In the column labeled Nthe

numbers of colors which cannot be simulated by A99 without do-

main transformation (11) are tabulated, and in “Ratio” the ratios

between Nand the total number of sRGB colors (2563).

Type of simulation NRatio

Protanopes 190,447 1.1%

Deuteranopes 634,406 3.8%

2.2 A99: Vi´

enot et al. 1999 Algorithms

For sRGB video devices, in A99, E,C(475), and C(575)

which determine Σ(475,575) in A97 are approximated by col-

ors of 8-bit sRGB value Uf(255,255,255), Uf(0,0,255) and

Uf(255,255,0), respectively, where f(r,g,b) is the column vec-

tor function which transforms (r,g,b)to(X,Y,Z)Taccording to

the formula in Ref. [8]. Since f(255,255,255) =f(0,0,255) +

f(255,255,0), two planar parts Σ(475) and Σ(575) in Σ(475,575)

become parallel in A99 and stimulus surface for protanopes and

deuteranopes is the plane

Σ(99) ={αUf(0,0,255) +βUf(255,255,0)|αand βare real}.

(10)

A tritanope simulation and video devices other than sRGB are not

supported in A99.

Again, not all lines lL(Q)orlM(Q) for Q=Uf(r,g,b) cross the

stimuli surface Σ(99) in the sRGB-parallelepiped. Thus there also

exists the same problem in the Σ(99) as in A97. In Ta b l e 2 ,we

show the number of sRGB colors which cannot be simulated by

Σ(99).

Since these numbers are small, A99 introduced the following

color domain transformation

f∗(r,g,b)=c1f(r,g,b)+c2f(255,255,255) (11)

c

2015 Information Processing Society of Japan 43

IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

where c1=1.0092 and c2=−0.0046 for protanopes, and

c1=0.9420 and c2=0.0264 for deuteranopes. These coeﬃ-

cients are slightly diﬀerent from those in Ref. [7], since in Ref. [7]

adiﬀerent LMS space is adopted. For the modiﬁed stimulus

Q∗=Uf∗(r,g,b) all lines lL(Q∗)orlM(Q∗) cross the stimulus

surface Σ(99) in the sRGB-parallelepiped. Namely in A99 after

color domain transformation (11) all colors of sRGB can be sim-

ulated.

We should notice that although the transformation (11) is al-

most identical, the simulated color s(Q∗) is not the confusion

color of Qbut the confusion color of Q∗. This means dichromats

may perceive the diﬀerence between Qand Q∗; therefore, strictly

speaking, A99 with this transformation is not a dichromatic sim-

ulation.

Note that the transformation of a unit or scale in L,M,orS

does not aﬀect the above mentioned problem either for A97 or

Σ(99), that is, Table 1 and Table 2 are unchanged.

3. Proportionality Law

The stimulus surface Σ(λ1,λ

2) of A97 has a special shape,

called cone. If a point is on the stimulus surface Σ(λ1,λ

2), the line

segment connecting the point and the origin is always included in

the same surface Σ(λ1,λ

2). This means in the simulation sur-

face with cone shape, if a color stimulus Qis simulated by s(Q)

then color stimulus αQproportional to Qis simulated by αs(Q).

Thus A97 implies that the proportionality law, s(αQ)=αs(Q),

between the dichromatic eye and normal trichromatic eye holds

although the authors of A97 did not refer to this law explicitly.

We simply call this law the proportionality law.

Proportionality law. If color stimulus Qis simulated by s(Q),

then proportional color stimulus αQis simulated by αs(Q), that

is, s(αQ)=αs(Q)holds.

Inversely we can state that if the simulations do not contradict

with this proportionality law, their stimuli surfaces must be a cone

with its apex at the origin.

For normal trichromats, the vector αQproportional to Qrep-

resents the same color with diﬀerent intensity, and vectors with

diﬀerent direction do not represent the same color. Thus, if the

proportionality law holds, the same color αQwith diﬀerent in-

tensity are simulated by one color αs(Q), and vice versa. Without

the proportionality law, the same color αQwith diﬀerent intensity

will be simulated by several diﬀerent colors.

Since this seems to be a fundamental requirement in dichro-

matic simulation in early-stage, we will discuss a simulation

function s(Q) which can support all colors on a device while sat-

isfying this proportionality law.

4. Simulation Function s(Q) in a Device

Gamut with the Proportionality Law

We discuss the general device described by an ICC proﬁle with

device color primaries E1,E2and E3in a LMS space as deﬁned

in Section 2. For sRGB these are

E1=Red =Uf(255,0,0),

E2=Green =Uf(0,255,0),

E3=Blue =Uf(0,0,255).(12)

We will show that for a given set of E1,E2and E3,thesim-

ulation function s(Q) which can support all colors on the device

gamut while satisfying the proportionality law is uniquely deter-

mined except for two rare cases. In general, if we look at three

primary vectors E1,E2,E3from one of axes a=L,Mor S,we

will see three non-zero vectors, for instance those labeled “Red,”

“Green” and “Blue” in Fig. 2 (right) for sRGB, which are pro-

jected vectors of E1,E2,E3into the plane perpendicular to the

axis a=M. We will consider this case in the following and ex-

ceptional cases where we cannot see three projected vectors will

be considered in Appendix A.1 for completeness’ sake.

We deﬁne projection operator

P=⎛

⎜

⎜

⎜

⎜

⎝

010

001

⎞

⎟

⎟

⎟

⎟

⎠,or ⎛

⎜

⎜

⎜

⎜

⎝

100

001

⎞

⎟

⎟

⎟

⎟

⎠,or ⎛

⎜

⎜

⎜

⎜

⎝

100

010

⎞

⎟

⎟

⎟

⎟

⎠(13)

for protanopes, deuteranopes and tritanopes, respectively. Then,

we consider three projected vectors in two dimensions, PEk,k=

1,2,3. We rename vectors Ekusing descending order of the ﬁrst

component of the normalized projected vectors ˆ

Vk, deﬁned by

ˆ

Vk=PEk/|PEk|for PEk0and ˆ

Vk=0for PEk=0.

Theorem 1. When ˆ

V10,ˆ

V20,ˆ

V30and ˆ

V1ˆ

V2

ˆ

V3ˆ

V1, the simulation function s(Q)with the following proper-

ties (P1) and (P2) are unique. (P1) For all Qin the parallelepiped

E1E2E3s(Q)is also in the parallelepiped E1E2E3.(P2)s(Q)

satisﬁes the proportionality law, s(αQ)=αs(Q). The simulation

surface which deﬁne the s(Q)is

Σ(g)=Σ

1∪Σ2∪Σ3∪Σ4,

Σ1={pE1+q(E1+E2)|p,q≥0,p+q≤1},

Σ2={p(E1+E2)+q(E1+E2+E3)|p,q≥0,p+q≤1},

Σ3={p(E1+E2+E3)+q(E2+E3)|p,q≥0,p+q≤1},

Σ4={p(E2+E3)+qE3|p,q≥0,p+q≤1}.(14)

Proof: The parallelepiped E1E2E3spanned by E1,E2and E3is

projected by Pto a hexagon and the edges of the hexagon are

the projections of the following six edges of the parallelepiped

E1E2E3

e1={tE1|0≤t≤1},

e2={tE1+(1 −t)(E1+E2)|0≤t≤1},

e3={t(E1+E2)+(1 −t)(E1+E2+E3)|0≤t≤1},

e4={t(E1+E2+E3)+(1 −t)(E2+E3)|0≤t≤1},

e5={t(E2+E2)+(1 −t)E3|0≤t≤1},

e6={tE3|0≤t≤1}.(15)

Figure 2 (right) is an example of sRGB for deuteranopes. The

projected vectors PE1,PE2and PE3are the vectors denoted

by “Blue,” “Green” and “Red,” respectively, and the ei,i=

1,2,...,6 are the edges in the hexagonal envelope (outer most

c

2015 Information Processing Society of Japan 44

IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

hexagon) of the sRGB-parallelepiped.

Since Qon eiin the parallelepiped E1E2E3has no common

point with the la(Q) except for Qitself, all edges ei,i=1,2,...,6,

must be included in the simulation surface Σwhich deﬁnes s(Q)

in order for s(Q) to be in the parallelepiped E1E2E3as well. On

the other hand, the proportionality law requires that the line seg-

ment connecting the origin and any point in eialso be included in

the simulation surface Σ. This determines the simulation surface

Σuniquely as Eq. (14). See Fig. 3. Q.E.D.

We illustrate the simulation surface (14) in Fig. 4 by using

the data of sRGB video device in LMS space deﬁned in Sec-

tion 2. The left ﬁgure is the surface Σ(g)for protanopes and

tritanopes, and the right one for deuteranopes. They are diﬀer-

ent since for deuteranopes the primary vectors E1,E2and E3are

the sRGB primaries “Blue,” “Green” and “Red,” respectively, as

shown in Fig. 2 (right), while for protanopes and tritanopes those

are “Blue,” “Red” and “Green,” respectively (“Green” and “Red”

are interchanged).

We can see that the stimulus surface of A97 for protanopes

and deuteranopes shown in Fig. 2 (left) are very diﬀerent from

the simulation surface Σ(g)shown in Fig. 4 (left) for protanopes

and in Fig. 4 (right) for deuteranopes. If we look at these simu-

lation surfaces from the Laxis and Maxis, respectively, we will

not ﬁnd any gaps such as we see in Fig. 2 (right) for A97. Ac-

cordingly ratios shown in Table 1 for A97 become 0%’s for Σ(g)

as we intended.

The condition ˆ

V10,ˆ

V20,ˆ

V30,ˆ

V1ˆ

V2ˆ

V3ˆ

V1

Fig. 3 Origin and six edges ei,i=1,2,...,6 and four planar pieces Σi,

i=1,2,...,4 of the simulation surface Σ(g).

Fig. 4 Surface Σ(g)for protanopes and tritanope (left), and for deuteranope (right) in sRGB. In the left

ﬁgure, the primary vectors E1,E2and E3are the vectors denoted by “Blue,” “Red” and “Green,”

respectively, while in the right ﬁgure, those are “Blue,” “Green” and “Red,” respectively (“Green”

and “Red” are interchanged).

in theorem 1 is identical with the geometrical statement that the

projection of the device gamut onto the plane perpendicular to the

color confusion axis in LMS space is hexagon. The majority of

devices including sRGB are this type and for these devices Σ(g)in

Eq. (14) is the only possible surface deﬁning s(Q) for all colors

on the device while satisfying the proportionality law. We call

our algorithm by theorem 1 as APL (algorithm based on propor-

tionality law).

5. Comparison of the Three Algorithms

We tested if the algorithms A97, A99, and our APL worked

in the sense that the dichromatic observer could not ﬁnd any dif-

ferences between the original pictures and the transformed ones.

Figure 5 presents the original picture which is similar to the pic-

ture used in Ref. [1] and consists of 25 color cells selected ran-

domly from sRGB 2563colors. The sRGB 8-bit (r,g,b) values of

these 25 colors are listed in Table 3.

As a typical sRGB video display we used Mitsubishi LCD

(Liquid Crystal Display), Dyamondcrysta RDT231WLM. The

accuracy of this display is Δ=0.064 several hours after the power

is on, where

Fig. 5 The original picture (25 colored cells).

Tab l e 3 sRGB 8-bit (r,g,b) values corresponding to the color cells in Fig. 5.

(222,244,69) (191,56,78) (33,27,174) (222,47,47) (95,96,5)

(14,97,103) (38,223,240) (227,100,70) (205,248,189) (200,149,238)

(133,72,133) (37,175,207) (252,57,6) (32,64,133) (46,171,174)

(211,131,223) (250,92,93) (154,95,155) (12,232,135) (54,119,69)

(4,7,55) (55,179,139) (209,114,99) (227,205,73) (116,28,79)

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Fig. 6 Protanopic (left) and Deuteranopic (right) simulations of the origi-

nal picture by A97. The colors which cannot be simulated by A97

are indicated by black. There are ﬁve black cells in protanopic and

deuteranopic simulations.

Fig. 7 Protanopic (left) and Deuteranopic (right) simulations of the original

picture by A99.

Fig. 8 Pictures by APL for Protanopic (left) and Deuteranopic (right).

Δ= 1

n

i

(f(ri,g

i,bi)−(Xi,Yi,Zi)T)2

for these n=25 colors. (ri,g

i,bi) is the sRGB 8-bit value and

(Xi,Yi,Zi)Tis the corresponding measured value by a color an-

alyzer (Display Color Analyzer CA-210, Konica Minolta, Inc)

normalized as Y=1 at (255,255,255) sRGB 8-bit value.

Figure 6 illustrates the simulation result by A97, where ﬁve

cells with black in protanopic and deuteranopic simulations show

the colors which cannot be simulated by A97. These ratios of the

number of the cells with black, 5/25 =20% for both simulations,

do not statistically contradict with the ratios 27.8% and 15.6%

tabulated in Table 1, respectively.

On the other hand, for A99, we used color domain transforma-

tion (11) and thus all 25 colors are simulated as shown in Fig. 7

though all of them are not confusion colors.

For our APL all 25 colors are displayed as shown in Fig. 8

since APL can support all colors for any video devices. As we

explained in Section 2, Fig. 6 is the most reliable dichromatic

simulation and Fig. 7 is its approximation, and thus they are sim-

ilar. However, Fig. 8 is quite diﬀerent from Fig. 6 (or Fig. 7) be-

cause Fig. 8 is not the result of the simulation of color percep-

tion but of color confusion using the simulation surfaces shown

in Fig. 4 which have completely diﬀerent shapes from the well-

known stimulus surface of A97 shown in Fig.2 (left).

We studied whether three dichromatic observers, one

protanope and two deuteranopes, could distinguish color cells in

original pictures from those by A97, A99 and APL or not.

The tasks of the experiment are as follows:

( 1) In a darkroom, we show two pictures in the display, the orig-

inal picture Fig. 5 on the left hand side, and the simulated

picture which is one of pictures shown in Fig. 6–Fig. 8 or tri-

tanopic results by A97 and APL on the right hand side.

( 2 ) We point one of 25 cells in the original picture by mouse

pointer in raster scan order, and ask a test subject whether

the color in the cell is similar to the corresponding cell on

the right hand side.

( 3 ) We change the picture on the right hand side after 25 ques-

tions and continue to ask the next 25 questions.

( 4) The picture on the right hand side is changed 20 times in the

following order:

(1-A97-P/D), (1-APL-P/D),

A97-P, A97-D, A97-T, APL-P, APL-D, APL-T, A99-P,

A99-D,

(2-A97-P/D), (2-APL-P/D),

A97-P, A97-D, A97-T, APL-P, APL-D, APL-T, A99-P,

A99-D,

The label in the form X-Y-Z or Y-Z indicates the picture. No

X means the picture in Fig. 5 and diﬀerent X’s the other orig-

inal pictures. Y is the method, and Z the type of dichromats.

P/D corresponds to the type of the test subject P or D. Two

series of 8 succeeding terms beginning from the third and

13th term are the same.

( 5 ) The 4 results in the parentheses are discarded, and we have

checked that the two series of 8 results coincide.

After this experiment, we gained the results that they found no

diﬀerence between original and the transformed pictures adjusted

to their dichromatic types, and when protanopic (deuteranopic)

patients looked at the pictures for the deuteranopic (protanopic)

results they found some diﬀerences. Although our test subjects

are only three without tritanopes, we conclude that all three meth-

ods, A97, A99 and APL work as simulation of color confusion

because they are based on a well-established model for color con-

fusion. For A99, though it uses approximate confusion colors, all

test subjects could not detect this approximation in this experi-

ment.

We note that the APL is the algorithm for any display devices

in which sRGB display is not necessarily assumed. In this section

we have shown the APL woks on sRGB as a sample of display

devices. Therefore we assure that APL can work on any other

display devices.

6. Additivity Law

The stimulus surface Σ(475,575) of A97 for protanopes and

deuteranopes shown in Fig. 2 (left) is almost like a plane, that is,

two parts Σ(475) and Σ(575) are almost parallel. If they are ex-

actly parallel the following additivity law also holds for dichro-

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matic simulation like the proportionality law.

Additivity law. If color stimuli Q1and Q2are simulated by s(Q1)

and s(Q2), respectively, then an additivity color stimulus Q1+Q2

is simulated by s(Q1)+s(Q2), that is, s(Q1+Q2)=s(Q1)+s(Q2)

holds.

For both normal trichromats and dichromats, an addition of two

vectors, Q1+Q2is the additive mixture of the two colors Q1and

Q2. Therefore, for simulated colors it is preferable that an addi-

tion s(Q1)+s(Q2) is also a simulated color. Since the additivity

law ensures this property, we are interested in the additivity law.

The inverse statement that if the additivity law holds the sim-

ulation surface must be a plane which includes the origin, is eas-

ily veriﬁed because the proportionality law always holds when

the additivity law holds. Mathematically if a continuous func-

tion s(Q)ofvectorQwith real component is additive, i.e.,

s(Q1+Q2)=s(Q1)+s(Q2), then s(Q) is proportional, i.e.,

s(αQ)=αs(Q). Therefore we will obtain following corollary

1 from the theorem 1 for the majority of the devices. Exceptional

cases where we cannot see three projected vectors will be consid-

ered in corollaries in Appendix A.1 for the sake of completeness.

Corollary 1. In the notation of theorem 1, when ˆ

V10,ˆ

V20,

ˆ

V30and ˆ

V1ˆ

V2ˆ

V3ˆ

V1, the simulation function

s(Q)having both properties (P1) in theorem 1 and (P3) in the

following does not exist. (P3) s(Q)satisﬁes the additivity law,

s(Q1+Q2)=s(Q1)+s(Q2).

Proof: Suppose s(Q) satisﬁes both (P1) and (P3) then s(Q) satis-

ﬁes proportionality law as stated above. Thus by theorem 1, s(Q)

is unique and deﬁned by the surface Σ(g)in Eq. (14).

We investigate Σ1and Σ2in (14) which constitute Σ(g).Σ1is on

Fig. 9 Simulation results for protanopes (P), deuteranopes (D) and tritanopes (T) of a photograph of

Kitasato University by the three methods A97, A99 and APL.

the plane S1spanned by E1and E1+E2,andΣ2is on the plane

S2spanned by E1+E2and E1+E2+E3.SinceE1,E2and

E3are independent, after some vector algebra, we can show that

S1∩S2is a line passing through the origin directed to the vector

E1+E2. This means S1and S2are not parallel and accordingly

Σ1and Σ2are not. Therefore the surface Eq. (14) is not a plane.

This contradicts that s(Q) satisﬁes the additivity law. Q.E.D.

7. Summary and Discussions

We have assumed a standard model of confusion color for

dichromatic observer, which is composed of the LMS space such

as Eqs. (1) and (2), and the set of confusion colors in Eq. (3).

Within this model, we have shown our main result that a dichro-

matic simulation function s(Q) for any color Qin display color

gamut G, is uniquely determined if we demand the proportion-

ality law, s(αQ)=αs(Q), for devices whose projected gamut

Gonto the plane perpendicular to the color confusion axis is

hexagon in LMS space. The s(Q) is given by the intersection

between the unique simulation surface Σ(g)and the line la(Q)in

Eq. (3). For devices with n=3 video device primaries such as

sRGB video devices in Eq. (12), the unique simulation surface

Σ(g)is given by Eq. (14).

In Fig. 9, examples for simulation results on a real image, a

photograph of Kitasato University, created by the three methods

A97, A99 and APL for protanopes (P), deuteranopes (D) and tri-

tanopes (T) are shown. We note the followings.

( 1 ) Black pixel corresponding to non-black pixel in the original

image shows that the pixel cannot be simulated. We call this

pixel skipped pixel below.

( 2 ) Skipped pixels exist only in the results by A97; however,

A97 is the most reliable simulation of color perception.

( 3 ) The results of A99 are similar to those of A97, and those of

A99 have no skipped pixel. A99, however, does not satisfy

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IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

the fundamental requirement that simulated colors have to

be confusion colors within the standard model.

( 4 ) The results of APL are not similar to those by A97, since

APL is not the simulation of color perception but the sim-

ulation of color confusion. However, APL has no skipped

pixel and satisfy the proportionality law.

For devices with n≥4 video device primaries such as “Quat-

tron” developed by Sharp Corporation, the parallelepiped of dis-

play gamut Gand its projected hexagon in Fig. 3 will be zonohe-

dron and convex polygon, respectively. In general, any edge of

the convex polygon which is a projection of the zonohedron by P

in Eq. (13) corresponds to an edge of zonohedron (not more than

two edges). In this “general case” for n≥4, the projected convex

polygon corresponding to Fig. 3 will have 2nvertices

ui=

i

j=1

PEj,un+i=

i

j=1

PEn−j−1,i=1,2,...,n,

and 2nedges

ei=ui+1−ui,(u2n+1=u1),i=1,2,...,2n,

with ejen+jPEj,j=1,2,...,n. We will obtain the 2n−2

planar parts Σifrom this ﬁgure as in Fig. 3 and then the unique

simulation surface

Σ(g)=Σ

1∪Σ2∪···∪Σ2n−2.

As a corollary of our main results, we have shown that it is

impossible to build the algorithm for such devices if we demand

the additivity law instead of proportionality law. It is obvious

from the above discussion that this corollary also holds for de-

vices with n≥4 video device primaries whose projected color

gamut onto the plane perpendicular to the color confusion axis is

convex polygon with 2nedges.

Finally we note again that our result is not a simulation of color

perception for dichromats based on human color vision mecha-

nism but a simulation of color confusion derived from the demand

on the device color gamut in computer vision algorithm.

Acknowledgments The research of HF was supported by

Grant-in-Aid for Scientiﬁc Research 24500212 JSPS.

References

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ıguez-Pardo, C.E. and Sharma, G.: Dichromatic color perception

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Appendix

A.1 Simulation Function s(Q) under Propor-

tionality and Additivity Law for Excep-

tional Cases

We consider simulation function s(Q) under proportionality

and additivity law when we cannot see three projected primary

vectors in Section 4. Using notations in Section 4, since three

primary vectors E1,E2and E3must be independent, at least two

normalized projected vectors among ˆ

V1,ˆ

V2and ˆ

V3are non zero.

If all normalized projected vectors are non zero, it is impossible

that all three normalized projected vectors are equal. Thus there

are following three cases. 1) All normalized projected vectors,

ˆ

V1,ˆ

V2and ˆ

V3, are non zero and all of them are non-equal. 2)

All normalized projected vectors, ˆ

V1,ˆ

V2and ˆ

V3, are non zero and

two of them are equal. 3) One of the normalized projected vec-

tors, ˆ

V1,ˆ

V2and ˆ

V3, is zero. The ﬁrst case is considered by the

theorem 1 and the corollary 1. We show below the simulation

functions s(Q)’s under proportionality law in theorem 2 and 3 for

the second and third case, respectively, and those under additivity

law in corresponding corollaries.

Theorem 2. When all normalized projected vectors, ˆ

V1,ˆ

V2and

ˆ

V3are non zero and two of them are equal, i.e., ˆ

V10,ˆ

V20,

ˆ

V30and ˆ

V1=ˆ

V2ˆ

V3, the simulation function s(Q)with

the properties (P1) and (P2) in the theorem 1 are determined

uniquely by the simulation surface

Σ(2) ={p(E1+E2)+qE3|p>0,q>0}.(A.1)

Proof: The parallelepiped E1E2E3is projected to a parallelogram

and two parallel edges of the parallelogram are the projections of

the following two edges of the parallelepiped E1E2E3

e7={t(E1+E2)+(1 −t)E3|0≤t≤1},

e8={tE3|0≤t≤1}.(A.2)

Since Qon e7and e8in the parallelepiped E1E2E3has no

common point with the la(Q) except for Qitself, two edges e7

and e8must be included in the simulation surface. Thus from

the requirement of proportionality law the simulation surface is

uniquely determined as a plane (A.1) spanned by E1+E2and

E3. Q.E.D.

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IPSJ Transactions on Computer Vision and Applications Vol.7 41–49 (May 2015)

This is a case where one of the axes L,Mor Salong which

dichromats cannot perceive change of color is parallel to the plane

deﬁned by the two primary vectors E1and E2.

Corollary 2. When all normalized projected vectors, ˆ

V1,ˆ

V2and

ˆ

V3are non zero and two of them are equal, i.e., ˆ

V10,ˆ

V20,

ˆ

V30and ˆ

V1=ˆ

V2ˆ

V3, the simulation function s(Q)having the

properties (P1) and (P3) in corollary 1 are determined uniquely

by the simulation surface (A.1).

Proof: Suppose s(Q) satisﬁes both (P1) and (P3) then s(Q) satis-

ﬁes proportionality law as stated in Section 6. Thus by the theo-

rem 2, s(Q) is unique and deﬁned by the surface Σ(p)in Eq. (A.1).

However the surface Σ(p)is a plane and s(Q)alsosatisﬁesaddi-

tivity law. Q.E.D.

Thus for this type of special device there exists unique algo-

rithm which simulates all colors on the device while satisfying

the additivity law.

Theorem 3. When one of the normalized projected vectors is

zero, i.e., ˆ

V3=0, the simulation function s(Q)having the proper-

ties (P1) and (P2) in theorem 1 are determined by the simulation

surface

Σ(3) =Σ

5∪Σ6

Σ5={p(E1+rE2+g(r)E3)|0≤p≤1,0≤r≤1},

Σ6={p(E2+(1 −r)E1+g(r)E3)|

0≤p≤1,1≤r≤2},(A.3)

where g(r)is a single-valued continuous real function deﬁned in

0≤r≤2with 0≤g(r)≤1.

Proof: The parallelepiped E1E2E3is projected to a parallelogram

and all edges of the parallelogram are the projections of the faces

spanned by E1and E3,orE2and E3. Therefore the simulation

surface is not uniquely determined and is expressed as Eq. (A.3).

Q.E.D.

This is the case where one of the axes L,Mor Salong which

dichromats cannot perceive change of color is just the primary

E3.

Corollary 3. When one of the normalized projected vectors is

zero, i.e., ˆ

V3=0, the simulation function s(Q)having the proper-

ties (P1) and (P3) in corollary 1 are determined by the simulation

surface spanned by two vectors E1+g(0)E3and E2+g(2)E3,

where 0≤g(0) ≤1and 0≤g(2) ≤1are two free parameters.

Note that in this case since simulation surface includes two

free parameters g(0) and g(2), the simulation function s(Q) is not

unique.

Hiroshi Fukuda is Associate Professor, Graduate School of

Medical Sciences, Kitasato University. He received his Ph.D. in

Engineering from University of Tsukuba in 1989. His research in-

terests are Computer Vision, Discrete Geometry, and Three Body

Problem. He is a member of IPSJ.

Shintaro Hara is Doctoral student, Department of Biomedical

Engineering, Graduate School of Medicine, The University of

Tokyo. He recieved his Master of Medical Science from Kitasato

University in 2012. His research interests are Computer Vision,

Computational Fluid Dynamics and Artiﬁcial Organ.

Ken asakawa is Instructor, Graduate School of Vision Science,

Kitasato University. He received his Ph.D. in Ophthalmology and

Vision Science from Kitasato University in 2010. His research in-

terests are Functional evaluation and Morphological observation

of Retinal Photoreceptor Cells.

Hitoshi Ishikawa is Professor, Graduate School of Medical Sci-

ences, Kitasato University. He received his Ph.D. in Medicine

from Kitasato University in 1994. His research interests are Or-

thoptics and Visual Science.

Makoto Noshiro received his Ph.D. in engineering from The

University of Tokyo in 1981. He is currently a professor emeritus

in Kitasato University. His research interests are signal process-

ing in and identiﬁcation of biomedical systems.

Mituaki Katuya is Professor Emeritus, School of Administra-

tion and Informatics, University of Shizuoka. He received his

Ph.D. in Science from Hokkaido University in 1973. His research

interests are Color Vision and Particle Physics.

(Communicated by Yasutaka Furukawa)

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