EN ISO 18314-4:2024

SLOVENSKI STANDARD
01-marec-2024
Nadomešča:
SIST EN ISO 18314-4:2021
Analizna kolorimetrija - 4. del: Metamerični indeks parov vzorcev pri spremembi
vrste svetila (ISO 18314-4:2024)
Analytical colorimetry - Part 4: Metamerism index for pairs of samples for change of
illuminant (ISO 18314-4:2024)
Analytische Farbmessung - Teil 4: Metamerie-Index von Probenpaaren bei
Lichtartwechsel (ISO 18314-4:2024)
Analyse colorimétrique - Partie 4: Indice de métamérisme de paires d'échantillons pour
changement d'illuminant (ISO 18314-4:2024)
Ta slovenski standard je istoveten z: EN ISO 18314-4:2024
ICS:
17.180.20 Barve in merjenje svetlobe Colours and measurement of
light
87.060.10 Pigmenti in polnila Pigments and extenders
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 18314-4
EUROPEAN STANDARD
NORME EUROPÉENNE
January 2024
EUROPÄISCHE NORM
ICS 87.060.10 Supersedes EN ISO 18314-4:2021
English Version
Analytical colorimetry - Part 4: Metamerism index for
pairs of samples for change of illuminant (ISO 18314-
4:2024)
Analyse colorimétrique - Partie 4: Indice de Analytische Farbmessung - Teil 4: Metamerie-Index
métamérisme de paires d'échantillons pour von Probenpaaren bei Lichtartwechsel (ISO 18314-
changement d'illuminant (ISO 18314-4:2024) 4:2024)
This European Standard was approved by CEN on 20 October 2023.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2024 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 18314-4:2024 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
This document (EN ISO 18314-4:2024) has been prepared by Technical Committee ISO/TC 256
"Pigments, dyestuffs and extenders" in collaboration with Technical Committee CEN/TC 298 “Pigments
and extenders” the secretariat of which is held by DIN.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by July 2024, and conflicting national standards shall be
withdrawn at the latest by July 2024.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 18314-4:2021.
Any feedback and questions on this document should be directed to the users’ national standards
body/national committee. A complete listing of these bodies can be found on the CEN website.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and the
United Kingdom.
Endorsement notice
The text of ISO 18314-4:2024 has been approved by CEN as EN ISO 18314-4:2024 without any
modification.
International
Standard
ISO 18314-4
Second edition
Analytical colorimetry —
2024-01
Part 4:
Metamerism index for pairs of
samples for change of illuminant
Analyse colorimétrique —
Partie 4: Indice de métamérisme de paires d'échantillons pour
changement d'illuminant
Reference number
ISO 18314-4:2024(en) © ISO 2024

ISO 18314-4:2024(en)
© ISO 2024
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
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Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii
ISO 18314-4:2024(en)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 2
5 Reference illuminant . 3
6 Test illuminant . 3
7 CIELAB coordinates L*, a*, b* . 3
8 Metamerism index for change in illuminant . 4
8.1 General calculation methods .4
8.2 Basic calculation of the metamerism index from colour differences .5
8.3 Correction methods .5
8.3.1 Additive correction .5
8.3.2 Multiplicative correction .5
8.3.3 Spectral correction.6
8.4 Test report .10
Annex A (informative) Calculation examples .11
Bibliography .24

iii
ISO 18314-4:2024(en)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out through
ISO technical committees. Each member body interested in a subject for which a technical committee
has been established has the right to be represented on that committee. International organizations,
governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely
with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are described
in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the different types
of ISO document should be noted. This document was drafted in accordance with the editorial rules of the
ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent
rights in respect thereof. As of the date of publication of this document, ISO had not received notice of (a)
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this may not represent the latest information, which may be obtained from the patent database available at
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Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and expressions
related to conformity assessment, as well as information about ISO's adherence to the World Trade
Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 256, Pigments, dyestuff and extenders, in
collaboration with the European Committee for Standardization (CEN) Technical Committee CEN/TC 298,
Pigments and extenders, in accordance with the Agreement on technical cooperation between ISO and CEN
(Vienna Agreement).
This second edition cancels and replaces the first edition (ISO 18314-4:2020), which has been technically
revised.
The main changes are as follows:
— a brief introduction about differentiation between metamerism and paramerism has been added in 8.1;
— Formula (1) has been updated to align with Formulae (2) and (4) to (24);
— the key of Figure A.1 has been updated.
A list of all parts in the ISO 18314 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
ISO 18314-4:2024(en)
Introduction
This document distinguishes three kinds of metamerism of pairs of samples:
a) Illuminant metamerism occurs if both of the object colours of a pair of samples are perceived as being
the same only under a specific illuminant (e.g. under illuminant D65), while they differ under a different
illuminant (e.g. illuminant A).
b) Observer metamerism occurs if the object colours of a pair of samples are perceived as being the same
by one observer, while a different observer perceives a colour difference under the same illuminant and
the same reference conditions.
NOTE 1 The observer metamerism is caused by differences between the distributions of spectral colour
matching functions of different observers.
c) Field-size metamerism occurs if both of the object colours of a pair of samples are perceived as being the
same on the retina for a size of an observation field (e.g. determined by the 2° standard observer), while
they differ for a different observation field on the retina (e.g. 10°).
NOTE 2 The reason for field-size metamerism is based on the existent colour matching functions of an observer
during an observation situation. The colour matching functions change with the size of the observation field on
the retina. Such change of the observation field can also occur if, for example, the pair of samples is examined
from different distances.
v
International Standard ISO 18314-4:2024(en)
Analytical colorimetry —
Part 4:
Metamerism index for pairs of samples for change of
illuminant
1 Scope
This document specifies a formalism for the calculation of the illuminant metamerism of solid surface
colours. It cannot be applied to colours of effect coatings without metrical adaptation.
This document only covers the phenomenon of metamerism for change of illuminant, which has the greatest
meaning in practical application. In the case where chromaticity coordinates of a pair of samples under
reference conditions do not exactly match, this document gives guidance on which correction measures to
take. Regarding the reproduction of colours, the metamerism index is used as a measure of quality in order
to specify tolerances for colour differences between a colour sample and a colour match under different
illumination conditions.
The quantification of the illuminant metamerism of pairs of samples is formally performed by a colour
difference assessment, for which tolerances that are common for the evaluation of residual colour differences
can be used.
NOTE In the colorimetric literature and textbooks, the term geometric metamerism is sometimes used for the
case where two colours appear to be the same under a specific geometry for visual assessment and selected standard
observer and standard illuminant pair, but are perceived as two different colours at changed observation geometry.
The term geometric metamerism is different to metamerism described in this document.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes
requirements of this document. For dated references, only the edition cited applies. For undated references,
the latest edition of the referenced document (including any amendments) applies.
ISO/CIE 11664-1, Colorimetry — Part 1: CIE standard colorimetric observers
ISO/CIE 11664-2, Colorimetry — Part 2: CIE standard illuminants
ISO/CIE 11664-4, Colorimetry — Part 4: CIE 1976 L*a*b* colour space
CIE 015, Colorimetry
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/

ISO 18314-4:2024(en)
3.1
metamerism
property of spectrally different colour stimuli that have the same tristimulus values in a specified
colorimetric system
[SOURCE: CIE S 017:2020, 17-23-006]
3.2
paramerism
characteristic of a pair of samples with spectral colour stimulus functions which have different fundamental
colour stimulus functions as well as different residuals or metameric black values within the visible spectral
range
Note 1 to entry: Parameric objects are characterized by the fact that they reflect colour stimuli of different spectral
power distribution functions under a specified standard illuminant, which cause approximately the same colour
perception under the selected observation conditions.
3.3
colour difference
ΔE
difference between two colour stimuli, defined as a distance between the points representing them in a
specified colour space
3.4
reference illuminant
illuminant with which other illuminants are compared
[SOURCE: CIE S 017:2022, 17-22-108]
3.5
test illuminant
illuminant, for which the colour difference (3.3) between the two samples to be tested is assessed
3.6
metamerism index for change in illuminant
M
t
colour difference (3.3), ΔE , between the two samples under test illuminant (3.5) if Δ=E 0 is observed under
the reference illuminant (3.4)
3.7
correction method
algorithm for theoretically eliminating a colour difference (3.3) of the pair of samples under the reference
illuminant (3.4)
4 Symbols
For the application of this document, the symbols given in Table 1 apply.
Table 1 — Symbols
Symbol Identification
X, Y, Z standard tristimulus values of a measured object colour
standard tristimulus values of the used illuminant
X , Y , Z
n n n
x , y , z colour-matching functions
* * *
basic coordinates of the CIELAB system
L , a , b
ΔL*, Δa*, Δb* differences between basic coordinates of the CIELAB system
M metamerism index for change in illuminant
t
ISO 18314-4:2024(en)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Identification
 
vector of the radiometric function of a sample with associated fundamental colour stimulus (f) and
NN,,N
fr
metameric black (r)
λ wavelength
S relative spectral distribution function of an illuminant

vector of the standard tristimulus values
W
w integration weights for the calculation of the standard tristimulus values
A matrix of the integration weights w for the calculation of the standard tristimulus values
R projection matrix
I identity matrix
Index spl sample
Index std standard
Index t colour under test illuminant
Index corr corrected value
Index multipl multiplicative correction
Index f fundamental colour stimulus
Index r metameric black values (residuals)
Index ref reference illuminant
Index T transposed matrix
5 Reference illuminant
The standard illuminant D65 shall be chosen as reference illuminant in accordance with ISO/CIE 11664-2.
Other reference illuminants required in special cases shall be specified.
6 Test illuminant
The selection of the test illuminant depends on the application. If the test illuminants are not particularly
specified, standard illuminant A in accordance with ISO/CIE 11664-2 and/or illuminants of the fluorescent
lamp type, such as FL11 in accordance with CIE 015, shall be selected. The test illuminant used shall be
indicated as an index to M, e.g. M or M .
A FL11
When calculating the standard tristimulus values X, Y, Z under the selected test illuminants, the basic raster
of wavelengths shall comply with those given in ISO/CIE 11664-2 or CIE 015 for A and D65, and in CIE 015
for FL11 and FL2. In cases of missing measuring values of the standard or sample for these wavelengths,
these values shall be interpolated and/or extrapolated.
7 CIELAB coordinates L*, a*, b*
* * *
The metamerism index, M , is based on the CIELAB coordinates L , a , b of samples 1 and 2 which are
t
* * *
compared. L , a , b shall be calculated in accordance with ISO/CIE 11664-4 from the standard tristimulus
values X, Y, Z. These values are derived from the sample for the CIE 1964 10° standard observer in accordance
* * *
with ISO/CIE 11664-1 for the reference illuminant and the selected test illuminant. If calculating L , a , b
under the test illuminant, the respective standard tristimulus values X , Y , Z of the entirely matt white
n n n
surface shall be used in accordance with CIE 015. For the standard illuminants A and D65 or for the
illuminant recommendation FL11, the standard tristimulus values X , Y , Z of the entirely matt white
n n n
surface apply in accordance with Table 2.
Table 2 specifies standard tristimulus values for the frequently used standard illuminants D65 and A as well
as illuminant FL11 and both of the standard observers according to CIE 015.

ISO 18314-4:2024(en)
Table 2 — Standard tristimulus values
2° standard observer 10° standard observer
Standard
tristimulus Illuminant
values
D65 A FL11 D65 A FL11
X 95,04 109,85 100,96 94,81 111,14 103,86
n
Y 100,00 100,00 100,00 100,00 100,00 100,00
n
Z 108,88 35,58 64,35 107,32 35,20 65,61
n
For fluorescent samples, the illuminant used for measurement shall be adjusted as close as possible to that
illuminant for which the standard tristimulus values are determined.
NOTE In contrast to non-fluorescent samples, the calculation of metamerism indices for fluorescent samples is
erroneous if the samples are measured only under one illuminant.
8 Metamerism index for change in illuminant
8.1 General calculation methods
Metamerism implies no colour difference under the reference illuminant. The colour difference under the
test illuminant is used as metamerism index. This index is described in Formula (1):
22 2
** *
ML=Δ +Δab+Δ (1)
() () ()
tt tt
where
t is the colour under test illuminant;
* * *
ΔL =−LL ;
t splc,,orrt stdt,
* * *
Δa =−aa ;
t splc,,orrt stdt,
* * *
Δb =−bb .
t splc,,orrt stdt,
In case of a small colour difference already present under reference illuminant conditions, the colour
difference at change of illuminant is called paramerism. To eliminate the effect of the difference under
reference illuminant, a mathematically corrected virtual sample is created, having no remaining colour
difference under the reference illuminant.
Three different correction methods for calculating a metamerism index in the case of paramerism have
been proposed in References [6] to [13]. All methods assume that, for practical cases, there can already be
a small difference between the colours of the sample and the standard, even under the reference illuminant
from the very beginning, due to problems of fabrication. In the case of two methods, called the additive and
the multiplicative correction, these inherent colour differences often merge with the difference introduced
by the change of the illuminant. The third method, the spectral correction, works more fundamentally by
the separation of inherent colour differences under the reference illuminant from those introduced by the
change of the illuminant.
NOTE Annex A includes calculation examples.

ISO 18314-4:2024(en)
8.2 Basic calculation of the metamerism index from colour differences
After this correction (see 8.1) leading to the virtual sample, the common formula for a metamerism index
at change in illuminant, expressed in CIELAB coordinates for the test illuminant (t), is given by Formula (2):
22 2
** *
ML()x =Δ +Δab+Δ (2)
() () ()
tcorrcorrcorr
where
t is the colour under test illuminant;
* * *
ΔL =−LL ;
corr splc,,orrt stdt,
* * *
Δa =−aa ;
corr splc,,orrt stdt,
* * *
Δb =−bb ;
corr splc,,orrt stdt,
x nominates the correction method.
Formulae (1) and (2) are provided as examples if using the CIELAB colour space.
Analogous equations apply for other Euclidian colour spaces such as DIN 99o as specified in DIN 6176. In non-
[6]
Euclidian colour spaces such as CIE 94 or CIEDE2000, the specific colour differences are provided with
colour-space dependent weight functions and, in regard to the latter case, are expanded by an additional
rotation term. The CIELAB metric used in this document is an example and should be replaced in practical
applications by one of the more recent metrics mentioned (e.g. CIE 94, CIEDE2000, DIN 99o), which are
significantly more uniform than the CIELAB model.
8.3 Correction methods
8.3.1 Additive correction
When using the additive correction, the differences of colorimetric coordinates between the standard (std)
and the sample (spl) under the reference illuminant (ref) are added to the respective differences between
the standard and the sample under the test illuminant (t). The resulting calculation for the metamerism
index M (add), expressed in differences of CIELAB coordinates, is then given by Formula (3):
t
22 2
** *
ML= ΔΔ+ ab+ Δ (3)
() () ()
t corr corr corr
()add
where
* * **
ΔL = LL−−ΔL ;
corr splt, stdt, ref
* * *
ΔL = LL− .
ref splr, ef stdr, ef
Analogous relationships apply for Δa* and Δb*. It should be noted that slightly different results are to be
expected, if the correction is applied to standard tristimulus values prior to transformation into a uniform
colour space such as CIELAB or DIN 99o.
8.3.2 Multiplicative correction
NOTE The multiplicative correction is specified in CIE 015 as the correction method.
When using the multiplicative correction, the standard tristimulus values of the sample (spl), which are
observed under test conditions (t) are multiplied with the quotient of the standard tristimulus values

ISO 18314-4:2024(en)
of standard (std) and sample (spl), which are obtained under reference conditions (ref). The resulting
calculation is given in Formula (4):
Y
stdr, ef
YY= (4)
spl,corr,tspl,t
Y
splr, ef
in which case, again, analogous combinations for X and Z apply. Subsequently, a transformation into
corr corr
a uniform colour space (e.g. CIELAB) takes place and results in Formula (5):
22 2
** *
ML()multipl =Δ +Δab+Δ (5)
() () ()
tcorrcorrcorr
with
* * *
Δ=LL − L .
corr splc,,orrt stdt,
* *
Analogous relationships apply for the two remaining specific differences Δa and Δb .
corr corr
8.3.3 Spectral correction
The spectral method considers that under the reference illuminant, minor differences between the
tristimulus values of the sample and the standard can already exist, which are not relevant for the
metamerism characteristics. In order to first mathematically compensate them and only determine the
effective component of metamerism at change in illuminant of sample pairs with given spectral reflectance,
it is possible to mathematically split a spectral reflectance into two additive components.
One component describes only the function that is effective for the formation of the colour stimulus under
the reference illuminant and the other component describes a function, which does not lead to a contribution
to the colour stimulus when integrating via the stimulus under the reference illuminant.
This function necessarily includes positive and negative components. The fundamental colour stimulus
function results from the first component of the spectral reflectance under the reference illuminant. This
is effective for the formation of the colour. The respective second part of the colour stimulus function leads
to a metameric black of the decomposition (residue), i.e. an invisible contribution with a resulting colour
stimulus identical to zero.
The compensation of the deviations of the colour stimuli of a test sample from the standard sample, which
are non-effective for metamerism characteristics, is realized by replacing the fundamental colour stimulus
of the sample by that of the standard. The component that is effective for the metamerism characteristic
remains unchanged, i.e. a new colour stimulus function of the sample is generated from the sum of the
replaced fundamental colour stimulus and the unchanged second component. The sum determines the
metamerism at change in illuminant with regard to the standard.
The mathematical description of the method of spectral correction starts with the general definition of the
spectral reflection function of a sample in Formula (6) and the definition of a matrix of spectral weights
[Formula (9)] to calculate the expected tristimulus values. This matrix of weights is composed from the
spectral illuminant and the spectral matching functions of the observer in Formula (8). The product of the
matrix of weights with the spectral reflection function results in the tristimulus values in Formula (11),
which appear under the illuminant considered.
As specified in this document by Formulae (12) to (15), a decomposition into visible and invisible parts of
colour stimuli uses the splitting of the spectral reflectance function under the defined reference illuminant
into a “fundamental reflection function” [Formula (13)], and a “black reflection function” [Formula (14)].
These parts lead to the visible fundamental colour stimulus and the invisible black colour stimulus
functions for the illuminant considered. So, it should always be noted that this method is only valid under
the assumption that these components of the spectral reflection function describe the visual effects in
combination with the spectral distribution of the reference illuminant (D65 in this document). Consequently,
the distribution function of the reference illuminant is inherently included in the decomposition of the
reflectance functions.
ISO 18314-4:2024(en)
In order to highlight this connection, the components are additionally marked in the text by “for the reference
illuminant “, when decomposing the reflectance function of a colour in Formulae (16) to (18). These formulae
describe differences and resulting correction terms for a pair of samples.
[9],[11]
In the model of the spectral decomposition developed by Cohen and Kappauf, the spectral reflectance
of an object colour obtained in the visible spectral range is summarized in the vector in Formula (6):
ρλ
()
 
 
 ρλ()
 2 
N = (6)
 

 
 
ρλ
()
 n 

The components of the vector N are the reflectance values ρλ in=…12,, , of the examined colour,
() ()
i
which are discretely present on n intervals. For the calculation of the respective standard tristimulus values
XY,,Z from the reflectance functions, the product based on the supporting points of the distribution
function of the used illuminant S()λ , the respective standard colour-matching function [see Formula (7)],
the distance of supporting points Δλ and a normalization constant k shall be determined.
αλ()= xy()λλ,, () z()λ (7)
[]
i ii i

The components ρλ() of the vector N are the reflectances of the examined colour. Considering the
i
illuminant S()λ , the standard tristimulus components X, Y, Z are calculated from the sum of the products
i
S()λρ()λα()λλΔ k with the normalization constant k. The term αλ()=[]xy()λλ,, () z()λ describes
ii i i ii i
n
the colour matching functions. The constant k is determined from k=1/ Sy()λλ()Δλ for the
ii
∑
i
Y-component of the illuminant considered.
These products mentioned above are introduced as weights in Formula (8):
wk()λλ= S()α()λλΔ (8)
α ii i
All weights of all supporting points for the standard colour-matching curves are given in matrix form in a
n × 3 matrix in Formula (9):
w ()λ
w ()λ w ()λ 
y 1
x 1 z 1
 
w λ
w λ ()w λ
() ()
y 2
 x 2 z 2 
A= (9)
 

 
 
 
w ()λ
w ()λ w (λ ))
yn
 xn zn 
ISO 18314-4:2024(en)
The transposed vector of the tristimulus values in Formula (10):

T
WX={},,YZ (10)
results in accordance with Formula (11):

T
WA=⋅N (11)

T
from the matrix multiplication of the transposed weighting matrix A with the radiometric function N .
[9],[11]
From the matrix A of the integration weights, Cohen and Kappauf constructed an orthogonal
n × n matrix shown in Formula (12):
−1
TT
RA=⋅AA A . (12)
()
The application of the matrix in Formula (13):

RN⋅=N (13)
f

to the radiometric function N of an object colour isolates its fundamental reflectance function for the

reference illuminant N , which actively forms the process of colour perception.
f

The tristimulus value W assigned to the difference in Formula (14):
r
 
NN=− N (14)
rf
is calculated from Formula (15):

T
WA=⋅N (15)
rr
and results in zero.
This contribution, identified as metameric black value or residuals, does not bear any colour information

and does not become visible during the colour-perception process under the reference illuminant. N
r
contributes positively as well as negatively. Also, regarding the obtained fundamental colour stimuli,

negative vector elements can occur. It is essential that Ni≥∀0 applies for the reflectance function, which is
i
composed of fundamental reflectance function and the metameric black values, in order to be physically
realized. The designations of fundamental or black parts of reflectance function consider their action in
combination with the reference illuminant, which is included in the weights according to Formula (8). The
n × n projection matrix R depends on the standard observer (2°, 10°) and the used standard illuminant.
[10],[11],[12],[16]
Fairman proposed a model of residual colour difference of parameric pairs of samples with
minor colour difference. This model was based on the described decomposition of a spectral reflectance
function into a fundamental reflectance function for the reference illuminant and the residuals as metameric
black values of the reflectance function.

ISO 18314-4:2024(en)
 
The vectors N and N describe the reflectance functions of a parameric pair of samples, the n elements
std spl
of which represent the measured reflectance values of the respective object colour within the visible
spectral range. They can be decomposed into their fundamental reflectance functions for the reference
illuminant and metameric black functions (r) for the reference illuminant by means of the Cohen-Kappauf
decomposition in Formula (16) and Formula (17):
 
NN=+ N (16)
stdf ,,stdr std
 
NN=+ N . (17)
splf ,,splr spl
By addition of the residuals of the sample (indicated by the index r,spl) and the fundamental reflectance
function of the standard for the reference (indicated by the index f,std), a virtual sample with corrected

reflectance function N is generated. This composition does not show any residual colour difference
splc, orr
with regard to the standard under the reference illuminant, but still has the same metameric characteristics
compared to the standard. The resulting Formula (18) is:
 
NR=⋅NI+−RN⋅ (18)
()
splc, orrstd spl
where I is the n × n identity matrix and R is the decomposition or projection matrix developed by Cohen and
Kappauf. For the calculation of a specific metamerism index for the illuminant metamerism, the experimental
 
reflectance function of sample N is replaced by the reflectance function N , which has been modified
spl splc, orr
by means of the spectral correction.
NOTE 1 Annex A includes calculation examples.
In summary, the calculation of the metamerism index for the spectral correction method requires the
following steps.
 
1) Measure the radiometric functions N and N of the reference and the sample, respectively.
std spl
2) Calculate the weighting matrix A for the reference illuminant and the respective weighting matrix A for
t
the test illuminant;
3) Calculate the Cohen−Kappauf matrix R for the reference illuminant as shown in Formula (19):
−1
TT
RA=⋅AA A (19)
()
4) Calculate the radiometric function of the corrected sample, according to Formula (20):
 
NR=⋅NI+−()RN⋅ (20)
splc, orrstd spl
where I is the identity matrix;
5) Calculate the tristimulus values for the reference illuminant in Formula (21) and Formula (22):

T
WA=⋅N (21)
std std

T
WA=⋅N (22)
splc,,orr splcorr
and for the test illuminant in Formula (23) and Formula (24):

T
WA=⋅N (23)
stdt, t std
ISO 18314-4:2024(en)

T
WA=⋅N (24)
splc,,orrt splc, orr
t
  
NOTE 2 WW= under the reference illuminant and the fundamental parts of W and W
stdspl,corr stdt, splc,,orrt
 
under the test illuminant are equalized. Any differences between W and W are introduced only by the
splc,,orrt stdt,

residual part of N that appears black for the reference illuminant and changed for the test illuminant.
splc, orr
6) Calculate the colour differences between W and W using any appropriate system in mind
stdt, splc,,orrt
and calculate the metamerism index. For the CIELAB-colour space, for example, the resulting formulae
are shown as Formulae (25) to (28):
22 2
 ** * 
ML()spectr =Δ +Δab+Δ (25)
() () () 
tcorrcorrcorr
 
* * *
Δ=LL − L (26)
corr splc,,orrt stdt,
* * *
Δ=aa − a (27)
corr splc,,orrt stdt,
* * *
Δ=bb − b . (28)
corr splc,,orrt stdt,
8.4 Test report
In the test report, the used correction method for the residual colour difference under reference conditions
shall be indicated in the formula of the metamerism index as argument. In the identifier M ()x , the index
t
“t” is for the used test illuminant (e.g. t = A for test illuminant A or FL11 for test illuminant FL11, etc.) and the
argument “x” is for the correction method (x = add for the additive correction, x = multipl for the multiplicative
correction, and x = spectr for the spectral Cohen-Kappauf correction). For example, M (spectr) is for a
A
metamerism index for the test illuminant A using the spectral Cohen-Kappauf correction.
Further, colour differences should not exceed the range recommended in the respective standards of the
*
colour difference formulae (for CIELAB ΔE <5 should be fulfilled).
ab
ISO 18314-4:2024(en)
Annex A
(informative)
Calculation examples
The following calculation examples serve to verify the programmed implementation of the correction
methods recommended in this document. As examples, pairs of green samples Green-1, Green-2 and Green-3
have been chosen. Their spectral reflectance factors are given as a function of wavelength in the second and
fifth column in Tables A.2 and A.4, respectively. The wavelength ranges from 400 nm to 700 nm with data
each 10 nm, based on illumination D65, 10° CIE 1964 observer. Green-1 serves as the standard and Green-2
and Green-3 as the samples, respectively. The colorimetric data of all samples had been calculated in the
range of 400 nm to 700 nm using the weighting functions given in ASTM E308-22, Table 5.19. Values at
wavelengths below 400 nm and above 700 nm are summed up and included in the results of the 400 nm and
*
700 nm values. All colour differences are calculated in the CIELAB ΔE . For both test examples of green
ab
pairs under reference conditions (illuminant D65, CIE 1964 10° standard observer), there is a residual colour
*
difference of about Δ=E 3 units.
ab
All relevant data of both examples are summarized in Table A.2 to Table A.5.
In a first step, the projection matrix R is calculated for the selected reference conditions (illuminant D65,
CIE 1964 10° standard observer) in accordance with 8.3. The result of this calculation is given in Figure A.2
and in Table A.1.
 
Applying this matrix R to the reflectance functions of standard ()N and sample ()N leads to the
std spl
 
vectors for the fundamental reflectance functions N and N , the black parts of the reflectance
fs, td fs, pl
 
functions (residuals N and N ) and to the summarized spectrally corrected reflectance function of
rs, td rs, pl

sample N . These are given for the colours Green-1 as standard and Green-2 as sample (see Table A.2)
splc, orr
as well as for the colours Green-1 as standard and Green-3 as sample (see Table A.4). In accordance with this
method, the corrected functions do not show residual colour differences any more with regard to the
standard (sample Green-1) under the reference illuminant D65.
Figure A.1 a) shows the reflectance functions of the standard Green-1 and of the sample Green-2 and, in
comparison, the corrected reflectance function of the sample Green-2. This insignificantly differs from the
original reflectance function, particularly at short wavelengths. Figure A.1 b) shows the corresponding
fundamental reflectance functions of standard Green-1 and of the sample (Green-2) which differ
insignificantly.
Both the reflectance functions illustrate the components that actively create the visible colours under
the reference illuminant D65. Figure A.1 c) shows the residuals of the standard and the sample under the
reference illuminant D65, demonstrating the black parts of the reflectance functions, which do not lead
to visible colours. These functions necessarily also include negative values, since their integrals, assessed
by the spectral power distribution of the reference illuminant and the spectral matching functions of the
observer are identical to zero.
Further calculation provides a change of illuminant D65 to illuminant A. All colorimetric data of the pair of
samples Green-1 and Green-2 for the reference illuminant and illuminant A are summarized in Table A.3.
For illuminant D65, the corrected reflectance function of the sample Green-2, in accordance with the
specification, does not lead to a colour difference compared to Green-1, while the parameric residual colour
difference of the original sample Green-2 compared to the standard Green-1 is given in column D65/Green-2/
spl and equals 3,005 1.
Under illuminant A, the colour calculated from the corrected reflectance function also shows a difference
compared to the colour of the standard Green-1. Accordingly, the metamerism index is 3,541 7 in this case.
For comparison, the data resulting from the additive and the multiplicative correction and the respective

ISO 18314-4:2024(en)
metamerism indices (3,79) are also given in Table A.3. In this case, these data give slightly higher values
...