Influence of colour matching functions on threshold and large

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					J. Opt. 28 (1997) 26–36. Printed in the UK                                                          PII: S0150-536X(97)81573-0

      Influence of colour-matching
      functions on threshold and large
      colour differences
J A Mart´nez, A J Poza, M Melgosa and E Hita
Departamento de Optica, Facultad de Ciencias, Universidad de Granada, 18071
Granada, Spain

Received 16 January 1996, accepted 2 November 1996

Abstract. In the present work, we study the influence of colour-matching functions
on the determination of colour-threshold differences (part I) and of greater size
(part II). For this, we have used the colour-matching functions of 16 different
observers, the five CIE centres and 200 Munsell samples. A previously operated
and fully automated Wright colorimeter was used for the determination of
colour-matching functions and colour-discrimination thresholds. The results for
part I indicate that the discrimination ellipses obtained with different
colour-matching functions (with some exceptions) had highly similar orientations,
semiaxis relationships and areas, whereas the centres appeared to shift with
respect to each other, primarily in the blue area. The results for part II reveal
important discrepancies between the different observers in evaluating any colour,
and close agreement in evaluating colour differences. On the basis of the overall
study, we deduce that the colour-matching functions of the observers have
significant influence on the specification of individual colours, but only slight effects
on the determination of small or large colour differences.

Keywords: Colour-matching functions, discrimination threshold, large colour

Influence des fonctions colorimetriques au seuil
de discrimination et pour les grandes
differences de couleur
   ´    ´            ´                    ´
Resume. Nous etudions dans le present travail l’influence des fonctions
        ´                  ´                            ´
colorimetriques sur la determination du seuil de difference de couleur (partie I) et
                   ´                                                   ´
des grandes differences (partie II). Pour ce but, nous avons utilise les fonctions de
                        ´                               ´
16 observateurs differents cinq centres CIE et 200 echantillons de Munsell. Un
        `                                  ´ ´     ´                           ´
colorimetre automatique de Wright a ete utilise pour notres mesures. Les resultats
de la premiere partie font apparaˆtre que les ellipses de discrimination obtenues
            `                         i
          ´                             ´            `                       `
avec differentes fonctions colorimetriques ont (a quelques exceptions pres) des
                         ´                                       ´
orientations, ellipticites et aires similaires, mais sont excentrees les unes par
rapport aux autres, principalement dans le bleu. Les resultats de la seconde partie
  ´ `                          ´                                     ´
revelent d’importantes differences entre les observateurs pour l’evaluation de
                                              ´                       ´
chaque couleur, mais non pour les differences de couleur. Les resultats de
                         ´                  ı
l’ensemble de cette etude font apparaˆtre l’influence importante des fonctions
        ´                    ´                                         ´
colorimetriques sur la specification des couleurs et faible sur la determination des
differences petites ou grandes.

        ´                   ´                                              ´
Mots cles: Fonctions colorimetriques, seuil de discrimination, grandes differences
de couleur.

General framework                                                  (thresholds and larger) which should be appropriately de-
                                                                   termined for numerous industrial applications in colorime-
The present study, divided into two parts, analyses the in-        try. This approach loosely follows the recommendations
fluence of colour-matching functions on colour differences          made by the CIE for coordinating work on evaluating colour

0150-536X/97/010026+11$19.50      c 1997 IOP Publishing Ltd
                                              Influence of colour-matching functions on threshold and large colour differences

differences [1]. Among other things, the CIE recommends
the study of ten experimental parameters related to chro-
matic discrimination and to avoid broadly scattered results,
the study of five specific zones of the chromatic diagram,
called CIE centres.
     One of these parameters is ‘variability between
observers’, directly related, at least partly, to the colour-
matching functions, the main focus of our study. In this
sense, K Witt prepared an outline towards the end of
1994 [2] to lend continuity to the CIE recommendations
[1]. In that work, Witt underlines the value of studying
the variability of the colour-matching functions in the
judgement of colour differences among different normal
observers and also mentions the interest of working with
observers having defective colour vision. In this context,
the present work has also emerged to provide coherence to
prior results obtained by our research group [3–6].
     Part I of our study describes the experimental
determination by our real observers of the chromaticity
thresholds corresponding to five CIE centres [1]. For this,
we used the same Wright colorimeter previously employed
to determine colour-matching functions [6]. These centres
have been studied in other works determining differential
colour thresholds with presentation both in object mode
[7–9] and in aperture mode [10–12]. In fact, the basic
aim of the present work is to ascertain the influence of the
colour-matching functions on these thresholds, using our          Figure 1. Scheme of the Wright visual colorimeter (see
own colour-matching functions [6] as well as other classical
ones, such as those of the CIE 1931 Standard Observer [13],
of the CIE Standard Deviate Observer (SDO) [14] and those         the photometric cube P5 . An electromechanical shutter
proposed by Vos [15].                                             placed at K2 controls the exposure time of the visual
     In Part II of our study, we analyse the influence of the      field. Three grey filters (neutral glass filters or attenuators)
colour-matching functions on the discrimination of colour         with a continuous optical-density variation can be displaced
differences of sizes larger than the threshold. For this, we      independently in front of the reflecting prisms T1 , T2 , T3 .
used a broad group of glossy samples from the Munsell             A set of grey filters with optical densities between 0.1
Atlas [16], as well as the colour-matching functions of 16        and 1.0, also available, was used in front of the prisms
observers: our three observers [6], the CIE 1931 Standard         T5 , T6 , T7 , which generate a fixed stimulus called the
Observer [13], the CIE Standard Deviate Observer (SDO)            ‘reference stimulus’. Thus, the quantities of the two sets
[14], that proposed by Vos [15], and the ten observers of         of three primaries to be mixed can be modified to provide
the pilot research of Stiles–Burch for small fields [17, 18].      several pairs of stimuli for comparison.
                                                                       For the colour measurements of the stimuli generated,
PART I. THRESHOLD COLOUR DIFFERENCES                              we used a spectroradiometer Photo Research SpectraScan
                                                                  PR-704. This instrument was placed on a tripod and
1. Experimental device                                            focused at the anterior surface of the photometric cube P5 .
                                                                  The estimated accuracy of these measurements is ±0.003
The experimental device used to measure the thresholds            for the chromaticity coordinates and ±5% for the luminance
was the same as for the experimental determination of             readings.
the colour-matching functions of our three observers (MM,
JAM and CF) [6]—that is, a Wright visual colorimeter.             2. Experimental measurements
    Figure 1 shows a schematic view of our Wright
visual colorimeter, a detailed description of which, and          The experimental parameters used for determining the
its calibration, can be found in previous publications            colour-matching functions [6] remained identical in
[19]. In short, the luminous source (a halogen stabilized         our experiment of determining chromaticity thresholds.
lamp) is placed at M and the two systems of reflecting             Nevertheless, it would be helpful to explain some of the
prisms (T5 , T6 , T7 and T1 , T2 , T3 ) each generate three       aspects concerning the choice and characteristics of the
nearly monochromatic lights (primaries) that are additively       reference stimuli used (CIE centres), as well as the method
mixed. Using the reflecting prisms, P3 and P4 , placed at          followed.
different heights, we direct two different stimuli towards            Due to the high number of measurements needed
the observer (located after diaphragm R4 ). The two stimuli       to determine each threshold, we restricted ourselves to
are juxtaposed by the reflecting prism P6 , together with          five CIE centres [1]. The luminance level of the stimuli

J A Mart´nez et al

stayed within the photopic range, being of the same
order of magnitude as that used in the colour-matching
determinations [6]. In any case, previous works [20, 21]
indicate that the luminance level does not significantly
affect the results of chromatic discrimination within the
photopic level.
    The experimental method chosen was that of constant
stimuli, in our opinion the most suitable of all those
used in the literature (e.g. methods of limits, matching,
comparison with reference pairs, categorical assignment,
arrangement, etc). Of all the methods we have available,
this was the easiest to apply in visual colorimetry and,
given that the observer does not act on the experimental         Figure 2. Scheme illustrating the different directions of
device, avoids tactile effects or learning, which could          movement to obtain the differential colour threshold.
affect the measurements, as well as the effects of the
chromatic adaptation, etc. Works carried out previously          have studied discrimination at constant luminance [23, 24],
in our laboratory [11, 12, 22] demonstrate that this method      we believe our results to be more useful, since the
provides optimal objectivity and reproducibility of the          discrimination involved conditions closer to the visual
measurements.                                                    functioning in everyday life.
    Calibrating our device, we fixed the reference stimulus           Each stimulus, to compare with the reference, was
(one of five CIE centres) by the additive mixture of the          presented to the observer ten times at random and thus
luminous signals reflected by the reflecting prisms T5 ,           characterized by a weight factor equal to the number of
T6 and T7 , using the sets of grey filters with optical-          times that the observer judged it equal to the reference.
density 0.1 to 1.0. These prisms were placed in the              The coordinates of the stimuli observed produce a cloud of
positions corresponding to the same wavelengths used in          points in the colour space, each point characterized by its
the determinations of the colour-matching functions [6]—         corresponding weight. A statistical fit for this cloud was
that was, 650 nm, 530 nm and 460 nm, with half-widths            carried out following the method proposed by Wyszecki
of 10 nm, 6 nm and 5 nm, respectively. In this way,              [25] and used by us in previous works [11, 12, 22]. This fit
we obtained a reference stimulus very close to the CIE           gave an ellipsoid representative of the corresponding colour
centre desired. Given that we also knew the calibrations         threshold, the principal section of which (result of cutting
corresponding to the luminous signals through the grey           the ellipsoid by a constant-luminance plane passing through
filters which slip in front of the reflecting prisms T1 , T2       the centre) is an ellipse defining the chromaticity threshold.
and T3 , we were able to situate the filters to obtain a              Of all the parameters of this ellipse, the semiaxes,
stimulus equal to that of the reference. These three prisms      area and orientation are of special interest, given that
were placed at the same wavelengths as mentioned above,          their analysis provides revealing data concerning the
although the half-widths were slightly different (12 nm,         discrimination. Thus, for example, greater areas would
7.2 nm and 6.0 nm, respectively), making the resulting           reflect worse discrimination, and smaller areas better.
match quasi-isomeric.                                            Similarly, one predominant size of a semiaxis of the ellipse,
    After performing this procedure based on the                 as well as the orientation of this semiaxis, would indicate
colorimetric calibration of our device, each observer, acting    preferential directions in the discrimination; the relationship
by means of the control system, could move the grey filters,      between semiaxes would also serve to characterize the
thereby verifying that the stimulus selected matched to the      more-or-less uniform behaviour of the colour-representation
reference one. Thus, each observer made a preliminary            system chosen.
match which invariably proved quite close to that obtained           Finally, we should emphasize that, as an essential
by the calibrations.                                             aspect of our experiments, any effect from the observer
    With the match fixed, we automatically recorded the           learning in the process of measurement was excluded. The
observer responses (Yes/No) on the equality of the reference     observer had no access to the colorimetric controls and
stimulus and the variable (the latter obtained by different      did not know the sequences in the presentation of the
positions of the grey filters). For each grey filter, two          stimuli, being limited to positive or negative responses to
positions were chosen—one on top and another below the           the equality of paired stimuli.
central match—and for each of these positions, sweeping
movements were made with the two remaining grey filters.          3. Results and discussion
In this way, stimuli were systematically presented near the
reference in many different directions, following a versatile    Table 1 presents the chromaticity coordinates of each
systematic method. Figure 2 schematically illustrates            centre, as well as the orientation, semiaxis relationship and
the various directions studied using this procedure, in a        area of the corresponding discrimination ellipses obtained
hypothetical colour-representation system. It is important to    experimentally by the observer JAM. Table 2 provides the
stress that, in the stimuli generated, there were simultaneous   same data for the ellipses obtained by observer CF. As these
modifications in the luminance and chromaticity with              tables show, the chromaticity thresholds were calculated
respect to the reference stimulus. Although some authors         using four different groups of colour-matching functions:

                                             Influence of colour-matching functions on threshold and large colour differences

Table 1. Centre, orientation, semiaxis relationship and area
                                                                         Table 2. Same as table 1, for observer CF.
in the x y plane, of the discrimination ellipses obtained by
the observer JAM applying the colour-matching functions of
the observers CIE 1931, CIE SDO, Vos and the own
                                                                                          Observer: CF
observer JAM, referred to the space of unreal primaries
XY Z.                                                                          CIE-31       SDO       Vos        CF
                                                                                                CIE Green
                     Observer: JAM
                                                                 x               0.2589      0.2580     0.2683     0.2876
              CIE-31   SDO         Vos            JAM            y               0.3582      0.3489     0.3952     0.4063
                                                                 θ(deg)         81.9        81.7       86.6       86.1
                              CIE Green
                                                                 b /a            0.50        0.48       0.54       0.57
x               0.2593      0.2584      0.2687     0.2539        Area (104 )     8.36        8.23       8.53       7.64
y               0.3583      0.3490      0.3952     0.2946
θ(deg)         83.2        82.7        89.9       77.9                                          CIE Yellow
b /a            0.57        0.55        0.62       0.37
                                                                 x               0.3652      0.3636     0.3727     0.3865
Area (104 )     8.09        7.98        8.25       5.39
                                                                 y               0.3975      0.3925     0.4184     0.4237
                                                                 θ(deg)         66.7        66.7       70.9       71.4
                              CIE Yellow
                                                                 b /a            0.46        0.44       0.54       0.57
x               0.3645      0.3629      0.3721     0.3423        Area (104 )     6.56        6.72       5.81       4.93
y               0.3969      0.3918      0.4180     0.3750
θ(deg)         62.8        63.1        65.8       65.9                                           CIE Red
b /a            0.46        0.44        0.56       0.26
                                                                 x               0.4704      0.4685     0.4782      0.4863
Area (104 )     5.05        5.17        4.48       5.07
                                                                 y               0.3121      0.3089     0.3256      0.3345
                                                                 θ(deg)         30.7        33.5        3.5         1.5
                               CIE Red
                                                                 b /a            0.77        0.74       0.91        0.91
x               0.4708      0.4689      0.4786     0.4399        Area (104 )     7.57        7.78       6.53        5.65
y               0.3117      0.3086      0.3252     0.3228
θ(deg)         29.7        32.3        12.4       46.3                                           CIE Grey
b /a            0.72        0.70        0.86       0.44
                                                                 x               0.3192      0.3174     0.3312     0.3476
Area (104 )     7.20        7.40        6.22       9.26
                                                                 y               0.3320      0.3245     0.3618     0.3729
                                                                 θ(deg)        101.7        95.5      120.8      122.9
                               CIE Grey
                                                                 b /a            0.85        0.85       0.77       0.75
x               0.3193     0.3174       0.3313     0.2977        Area (104 )    14.22       14.15      13.97      12.50
y               0.3320     0.3245       0.3618     0.2882
θ(deg)        118.3      103.2        132.3       65.6                                           CIE Blue
b /a            0.93       0.94         0.79       0.63
                                                                 x               0.2339      0.2339     0.2454     0.2628
Area (104 )    13.09      13.02        12.84      10.24
                                                                 y               0.2346      0.2254     0.2714     0.2884
                                                                 θ(deg)         73.5        73.9       74.3       72.9
                               CIE Blue
                                                                 b /a            0.22        0.22       0.25       0.24
x               0.2362      0.2360      0.2478     0.2327        Area (104 )     1.85        1.76       2.18       2.16
y               0.2361      0.2269      0.2728     0.1798
θ(deg)         73.2        73.6        74.2       75.1
b /a            0.28        0.27        0.30       0.19
Area (104 )     1.44        1.37        1.68       0.76          Observer. This similarity between the results of different
                                                                 observers was expected on the basis of prior research
                                                                 [11, 12, 22], bearing in mind that the matching was quasi-
                                                                 isomeric and that the colorimetric measurements made
the classical observers in the first three columns—CIE 1931       with the spectroradiometer to obtain an initial match used
Standard Observer [13], CIE Standard Deviate Observer            the colour-matching functions of the CIE 1931 Standard
[14] and that proposed by Vos [15]—and in the fourth those       Observer.
of the observer performing the experiment [6]. All these             The parameters of the ellipses of our two observers
colour-matching functions are referred to the same unreal        JAM and CF were also similar when calculated with the
primaries X Y Z [6] (equivalent to the CIE XY Z).                colour-matching functions of the CIE Standard Deviate
     The results in table 1 are also shown in figures 3 to        Observer (SDO) and those proposed by Vos, though greater
7 below, where we have represented the discrimination            differences are appreciable, especially in the coordinates
ellipses corresponding to JAM (semiaxes enlarged five             of the centre of the ellipses. Nevertheless, this is not
times) with each of the four colour-matching functions           true when these parameters are calculated with the colour-
mentioned. Similar results are obtained for observer CF.         matching functions of the observers who directly carry
     According to tables 1 and 2, the chromaticity ellipses      out the experiments: significant differences are found
in the five CIE centres [1] for the observers JAM and CF          between our two observers and also between them and the
are closely similar in their centres, orientation, semiaxis      other observers. These results agree with previous ones
relationships and areas, calculating these parameters with       [6] concerning colour-matching functions, thus confirming
the colour-matching functions of the CIE 1931 Standard           certain discrepancies with the CIE 1931 Standard Observer.

J A Mart´nez et al

                                                                    Figure 4. Same as figure 3, for the CIE yellow centre.
Figure 3. Discrimination ellipses corresponding to the
observer JAM, (semiaxes enlarged five times) for the CIE
green centre, obtained with the colour-matching functions
of the observers CIE 1931, CIE SDO, Vos and JAM.

    According to tables 1 and 2, with regard to the
orientation, the greatest differences between the observers
JAM and CF appear in the achromatic and red centres,
though this is barely significant, taking into account that the
relationships of semiaxes of these ellipses are near unity,
so that the orientation of these tend to be indeterminate.
With respect to the semiaxis relationships and areas, the
most marked differences between our two observers occur
in the red centre, in accord with the results in a previous
study [12].
    Overall, the variability found between the ellipses
computed at each of the CIE centres (figures 3–7) appears
to be negligible, from the point of view of their orientations,
semiaxes, relationships and areas. The most notable
discrepancy in results appeared in the location of the centres
of the ellipses, especially in the blue area, interpretable
as the consequence of the greater variability in the                  Figure 5. Same as figure 3, for the CIE red centre.
colour-matching functions at the short wavelengths of the
spectrum, in agreement with the analysis made in a previous
work [6].                                                         uses the colour-matching functions of its Standard
                                                                  Observers, as shown by the well-known calculation of
PART II. LARGE COLOUR DIFFERENCES                                 the tristimulus values, the basic elements in this system.
                                                                  Therefore, any modification of the Standard Observers
1. Introduction                                                   proposed by the CIE—that is, the proposal of new colour-
                                                                  matching functions which better characterize the average
In this second part of our work, as indicated in the              behaviour of the human visual system—would imply an
introductory layout of our study, we have expanded the            immediate change in the basic specification of colour
group of colour-matching functions used in part I; this is        given by these tristimulus values. Unquestionably, these
helpful in order to generalize our conclusions, and suitable      changes should be approached with caution and, though it is
because now we will not be working from the experimental          desirable at present to have colour specifications and colour
measurements made by our observers.                               differences in better agreement with visual perception [26],
    As often assumed, the perceived colour of a sample            it is equally certain that any modification would also raise
depends basically on the spectral distribution of the             diverse problems related to the colorimetric instrumentation
luminous source used, on the reflectance/transmittance of          and the industrial applications of colorimetry; therefore, it
the sample and on the colour-matching functions of the            is essential that these modifications represent a significant
observer performing the experiment. The CIE colorimetry           statistical improvement [27].

                                             Influence of colour-matching functions on threshold and large colour differences

                                                                 difference formula’. The colour differences calculated with
                                                                 various formulae share no explicit analytic relationships,
                                                                 a situation which complicates the task of comparing and
                                                                 describing the results. To avoid this drawback, in 1976 the
                                                                 CIE recommended the use of the colour-difference formulae
                                                                 CIE 1976 (L∗ , u∗ , v∗ ) and CIE 1976 (L∗ , a∗ , b∗ ), more
                                                                 commonly known as CIELUV and CIELAB, respectively.
                                                                     In our current study, we used the Munsell atlas
                                                                 and the CIELAB system, calculating the coordinates
                                                                 of several samples and the colour differences between
                                                                 certain pairs of contiguous samples. These computations
                                                                 were performed using different groups of colour-matching
                                                                 functions, specifically those of the 16 observers mentioned
                                                                 above. We used only CIELAB in our work for three
                                                                 main reasons: (1) this system is one of the two currently
                                                                 recommended by the CIE; (2) it is frequently used
                                                                 especially in relation to object colours (e.g. Munsell
                                                                 samples); and (3) it is the most widespread system today
                                                                 in industrial settings [29, 30].
   Figure 6. Same as figure 3, for the CIE grey centre.
                                                                 2. Experimental results

                                                                 In the present work, we used samples with a glossy finish
                                                                 from the Munsell atlas [16]. Page by page, we selected a
                                                                 central sample (specifically that with value 5 and chroma 6,
                                                                 designated as 5/6). Thus, from each page, it was possible
                                                                 to choose another four samples so that two differed from
                                                                 the previous one only in value (samples 3/6 and 7/6) and
                                                                 the other two only in chroma (samples 5/4 and 5/8). The
                                                                 samples chosen with respect to the central one (5/6) had
                                                                 a visibly different scale in chroma or in value. We chose
                                                                 a total of 200 samples, measuring their reflectances with a
                                                                 Hunterlab PR-107 spectrophotometer from 400 to 700 nm
                                                                 in intervals of 10 nm. From these measurements, we
                                                                 calculated the tristimulus values of the samples, using the
                                                                 groups of colour-matching functions described above (a
                                                                 total of 16 different observers) and assuming D65 as the
                                                                     The 40 pages of the Munsell atlas give us a complete
                                                                 sampling of the tones. From these, for the sake of brevity,
   Figure 7. Same as figure 3, for the CIE blue centre.           we have chosen the five main tones (5R, 5Y, 5G, 5B, 5P)
                                                                 and the five intermediate tones (5YR, 5GY, 5BG, 5PB,
                                                                 5RP) characteristic of this atlas.
     The just-perceptible or threshold differences, although         On this basis, we made the following two studies.
usually having greater scientific and industrial interest,            A. We considered a single sample viewed by each of
are not the only ones considered in colorimetry. Thus,           the 16 observers, and thus with different associated a∗ , b∗ ,
the colour order systems, or colour atlases, present             L∗ values. In the calculation of these coordinates with
collections of samples with differences far beyond the           CIELAB, we considered for each observer a reference white
threshold, these having also been submitted to rigorous          (the result of applying its own colour-matching functions,
colorimetric analyses. Within the numerous colour order          using the illuminant D65 and the equi-energetic white).
systems that have been proposed [28], the Munsell system             We chose Munsell samples which were close to the
[16] is perhaps the best known and the most studied              five CIE centres [1], calculating their CIELAB coordinates
scientifically, because of its careful formulation.        In     with the colour-matching functions of different observers
this system, the samples are structured in scales (value,        (16 in total). For example, figure 8 presents the a∗ , b∗
hue, chroma), following the three classical attributes in        coordinates for each observer with the sample 7.5PB3/6
chromatic perception and seeking to maintain constant            close to the blue CIE centre. In this figure, the points
perceptible differences between contiguous samples; that         corresponding to the three most classical observers—
is, the perceptive distribution is uniform.                      CIE 1931 Standard Observer [13], CIE Standard Deviate
     Colour differences are usually calculated from the          Observer (SDO) [14] and that proposed by Vos [15]—
tristimulus values of the samples applying a ‘colour-            can be distinguished from the rest: the ten Stiles–Burch

J A Mart´nez et al

Table 3. Results of the mean and standard deviation of the CIELAB colour differences of the 120 observer pairs (first two
columns) against the colour differences obtained by the observer pairs CIE 1931–SDO, CIE 1931–Vos and SDO–Vos, (last
three columns) for the samples with value 5 and chroma 6 in each of the mean and intermediate tones of the Munsell atlas.

     Munsell          Mean colour-differences     Standard     Colour-difference    Colour-difference    Colour-difference
     samples   Hue    CIELAB                      deviation    CIE31–SDO            CIE31–Vos            SDO–Vos
     5R5/6     5R     4.589 603                   3.975 238    0.298 230            0.337 810            0.457 180
     5YR5/6    5YR    5.972 735                   5.204 120    0.927 610            0.360 410            1.111 340
     5Y5/6     5Y     5.342 593                   4.989 737    1.240 060            0.333 210            1.502 770
     5GY5/6    5GY    3.241 156                   2.962 575    1.367 160            0.373 100            1.731 570
     5G5/6     5G     2.301 208                   1.817 915    0.261 250            0.432 030            0.689 860
     5BG5/6    5BG    4.308 172                   3.784 015    0.884 260            0.955 960            1.714 070
     5B5/6     5B     5.758 665                   5.076 335    1.242 750            0.783 260            1.855 440
     5PB5/6    5PB    4.513 532                   4.113 862    1.141 250            0.720 940            1.754 640
     5P5/6     5P     1.606 553                   1.243 188    0.679 310            0.260 910            0.929 270
     5RP5/6    5RP    2.485 122                   2.109 084    0.172 860            0.315 160            0.472 060

                                                                  the results of this mean and standard deviation for the
                                                                  samples with value 5 and chroma 6 in each intermediate and
                                                                  main tone. The last three columns of this table show the
                                                                  CIELAB colour differences corresponding to the three pairs
                                                                  of observers formed from the three most classical observers
                                                                  mentioned above (CIE 1931, CIE SDO and proposed by
                                                                      B. A second study consisted of considering pairs of
                                                                  Munsell samples for each intermediate and main tone,
                                                                  which differed from the central sample of value 5 and
                                                                  chroma 6 only in chroma (table 4) or only in value (table 5).
                                                                  Each of these tables shows, in the first columns, the mean
                                                                  and standard deviation of the CIELAB colour differences
                                                                  appreciated by the 16 observers for each pair of samples.
                                                                  The last three columns contain the colour differences
                                                                  corresponding to the three classical observers for each pair
                                                                  of samples considered. Some of the results presented in
                                                                  these tables are also illustrated in figures 9 and 10.

                                                                  3. Analysis of results and conclusions

                                                                  In figure 8, it is striking that the a∗ , b∗ coordinates obtained
                                                                  on applying the colour-matching functions proposed by Vos
                                                                  are situated exactly over the trend marked by the group of
Figure 8. The a∗ , b∗ coordinates for each of the 16              16 observers, reaffirming the goodness of the correction
observers with the sample 7.5PB3/6 (near the CIE blue             by Vos [15], especially for the short wavelengths. Both
centre).                                                          for the sample 7.5PB3/6 (figure 8), as well as for the
                                                                  others near the CIE centres, we find a cloud of points
                                                                  and therefore scattered results similar to the situation with
observers [18] and our three observers [6]. In this figure,        the centres of the thresholds calculated with different
the trend of the cloud of points is shown with a simple           colour-matching functions, including those of the observers
first-order fit.                                                    themselves (part I of the present study). In this sense, it
    To quantify differences between observers for the same        is also noteworthy that the two most extreme points of
sample, we paired the 16 observers, obtaining a total of 120      each of these clouds correspond to the a∗ , b∗ coordinates
different pairs of observers. For each of the 200 samples         obtained on applying the colour-matching functions of the
studied, we computed the CIELAB colour difference                 observers MM and JAM, whereas the a∗ , b∗ coordinates
corresponding to these pairs of observers, yielding 120           corresponding to the observer CF are closer to those
colour differences per sample.                                    obtained by the ten Stiles–Burch observers than to those
    The mean and standard deviation were calculated for           belonging to the classical observers and MM or JAM. All
these 120 colour differences and for each of the five              these results agree with our previous analyses of colour-
samples considered at each of the ten pages mentioned             matching functions [6].
above (main and intermediate tones). For example, and                 An analysis of table 3 underscores the fact that the mean
again for brevity, the first two columns of table 3 show           is greatest for the samples with tones 5B, 5R and 5YR, this

                                              Influence of colour-matching functions on threshold and large colour differences

Table 4. Results of the mean and standard deviation of the CIELAB colour differences of the 16 observers (first two
columns) against the colour differences obtained by the observers CIE 1931, SDO and Vos, (last three columns) for the
Munsell sample pairs, having only a chroma difference with respect to the central sample 5/6.

Munsell                    Mean colour-differences     Standard    Colour-difference    Colour-difference    Colour-difference
samples            Hue     CIELAB                      deviation   CIE 31               SDO                  Vos
5R5/6–5R5/8        5R       7.748 840                  0.206 979    8.653 665            8.764 104            8.549 885
5R5/6–5R5/4        5R       7.918 202                  0.234 165    9.079 384            9.105 491            8.989 153
5YR5/6–5YR5/8      5YR      6.274 786                  0.071 259    6.624 914            6.621 768            6.579 828
5YR5/6–5YR5/4      5YR      9.326 783                  0.117 311    9.600 604            9.562 478            9.562 478
5Y5/6–5Y5/8        5Y       5.595 857                  0.176 572    5.270 463            5.353 172            5.213 056
5Y5/6–5Y5/4        5Y       9.949 855                  0.269 193    9.389 921            9.562 128            9.242 566
5GY5/6–5GY5/8      5GY      8.733 867                  0.180 918    8.338 598            8.629 911            8.233 207
5GY5/6–5GY5/4      5GY     11.035 15                   0.186 834   10.655 52            11.017 22            10.512 73
5G5/6–5G5/8        5G       9.168 104                  0.180 301   10.436 56            10.340 12            10.352 11
5G5/6–5G5/4        5G       9.638 952                  0.209 417   10.830 62            10.740 62            10.820 56
5BG5/6–5BG5/8      5BG      7.813 713                  0.418 203   10.079 99             9.415 022           10.266 91
5BG5/6–5BG5/4      5BG      6.738 322                  0.323 660    8.399 960            8.016 918            8.524 884
5B5/6–5B5/8        5B       6.369 764                  0.215 756    7.721 834            7.252 079            7.780 118
5B5/6–5B5/4        5B       5.882 087                  0.132 045    6.710 571            6.464 880            6.656 217
5PB5/6–5PB5/8      5PB      7.487 970                  0.228 424    6.935 845            7.078 294            6.728 188
5PB5/6–5PB5/4      5PB      6.836 700                  0.187 766    6.511 370            6.583 733            6.309 228
5P5/6–5P5/8        5P       8.212 570                  0.087 426    7.952 441            8.097 976            7.889 331
5P5/6–5P5/4        5P       9.015 975                  0.108 207    8.600 378            8.677 116            8.700 453
5RP5/6–5RP5/8      5RP      7.595 629                  0.130 495    7.922 584            8.156 176            7.778 331
5RP5/6–5RP5/4      5RP      8.485 134                  0.218 740    9.499 564            9.538 707            9.509 995

                            Table 5. Same as table 4, but for pairs of samples differing in value.

Munsell                    Mean colour-differences     Standard    Colour-difference    Colour-difference    Colour-difference
samples            Hue     CIELAB                      deviation   CIE 31               SDO                  Vos
5R5/6–5R7/6        5R      18.588 230                  0.008 378   18.571 750           18.583 260           18.571 330
5R5/6–5R3/6        5R      18.591 610                  0.011 417   18.610 980           18.589 360           18.605 870
5YR5/6–5YR7/6      5YR     18.558 910                  0.005 788   18.560 900           18.567 420           18.557 250
5YR5/6–5YR3/6      5YR     20.298 280                  0.043 922   20.337 700           20.339 860           20.275 050
5Y5/6–5Y7/6        5Y      19.706 640                  0.021 689   19.642 240           19.611 980           19.650 240
5Y5/6–5Y3/6        5Y      10.634 730                  0.097 672   10.460 860           10.541 200           10.392 740
5GY5/6–5GY7/6      5GY     19.189 350                  0.061 346   18.984 420           18.898 240           19.052 670
5GY5/6–5GY3/6      5GY     21.803 420                  0.144 299   21.512 690           21.745 970           21.363 070
5G5/6–5G7/6        5G      19.297 910                  0.007 923   19.260 410           19.261 930           19.263 740
5G5/6–5G3/6        5G      19.582 130                  0.023 270   19.668 900           19.706 360           19.657 100
5BG5/6–5BG7/6      5BG     19.056 670                  0.008 881   19.070 960           19.083 420           19.068 460
5BG5/6–5BG3/6      5BG     19.224 370                  0.015 460   19.275 440           19.343 500           19.263 810
5B5/6–5B7/6        5B      19.774 880                  0.008 267   19.759 920           19.758 090           19.782 960
5B5/6–5B3/6        5B      18.864 250                  0.026 054   18.957 060           19.035 000           18.907 140
5PB5/6–5PB7/6      5PB     18.844 790                  0.007 470   18.833 950           18.833 550           18.845 180
5PB5/6–5PB3/6      5PB     18.157 260                  0.010 292   18.152 410           18.162 200           18.137 840
5P5/6–5P7/6        5P      18.988 740                  0.015 006   18.947 040           18.931 420           18.983 890
5P5/6–5P3/6        5P      17.947 030                  0.002 163   17.941 310           17.937 750           17.936 280
5RP5/6–5RP7/6      5RP     19.718 830                  0.006 857   19.734 250           19.733 930           19.745 080
5RP5/6–5RP3/6      5RP     18.466 640                  0.008 656   18.426 530           18.417 620           18.421 790

occurring in general for all the samples studied. The lowest       other.
values of this mean are given also, in a general way, for the          An overall view of this analysis is presented in figure 11
samples with tone 5G and particularly for those of tone 5P.        where, in addition to the above findings, we encounter the
The standard deviation values generally agree with those of        strongest discrepancies of the classical observer pairs with
the mean.                                                          respect to the others, particularly evident in the sample
     In table 3, it is notable that the CIELAB colour              of tone 5YR, though appreciable in all the samples. In
differences calculated with the pairs of classical observers       addition, this figure shows that the pair formed by the
(last three columns) render values which are close to each         CIE 1931 Standard Observer and the observer proposed by
other, but very small compared to the mean—a general               Vos gives the least CIELAB colour-difference values and
occurrence for all the samples. This again confirms a               the pair formed by SDO–Vos is the one that most closely
discrepancy in these classical observers with respect to the       approaches, in a relative way, the combined mean of the

J A Mart´nez et al

Figure 9. Mean of the CIELAB colour differences of the 16      Figure 10. Same as figure 9, but for pairs of samples
observers, against the colour differences obtained by the      differing in value.
observers CIE 1931, SDO and Vos, for the Munsell sample
pairs, in which, compared with the central sample 5/6,
there is a difference only in chroma.                              It should be stressed that for samples varying only
                                                               in chroma (table 4), the mean of the colour differences
                                                               appreciated by the 16 observers contrasts with that
120 observer pairs. This also occurs in general for all the    appreciated by the classical observers by around 1 CIELAB
samples studied.                                               unit at most (except for the sample 5BG). The standard
    The foregoing analyses are consistent with the one         deviations register quite small values, as opposed to the
performed on the colour-matching functions [6] and on          situation with mean and standard deviations for colour
thresholds (part I), in the sense of revealing discrepancies   differences for the 120 observer pairs (table 3). When the
between the entire group of observers, or more specifically     comparison involves varying the sample value (table 5), the
between the classical observers and those of Stiles–Burch      disagreement between the overall mean of the 16 observers
or the three of our study (MM, JAM, CF), principally in        and the classical observers is around 0.1 CIELAB units
the blue and red tones.                                        and the standard deviations prove much smaller than when
    In general, the coordinates of any of the samples change   comparisons are made varying only chroma (table 4).
significantly upon altering the colour-matching functions in        In tables 4 and 5, the differences appearing in the
the total group of 16 observers, as indicated by the fact      means of colour differences between the various sample
that the mean of the colour differences has a value of         pairs simply reflect the lack of uniformity in CIELAB
several CIELAB units, with strong differences appearing        [31], in this case when measuring colour differences far
within each group of observers, as shown by the high           beyond the threshold (over 5 CIELAB units in all cases).
value of the standard deviation. This fact contrasts with      Nevertheless, the most important aspect of these two tables
the situation in which the colour differences between pairs    is, in our opinion, that the standard deviations register
of Munsell samples are evaluated using different observers,    decidedly small values, as might also be expected from the
as explained below.                                            comparison of the last three columns, which present quite
    From the analysis carried out in study B, table 4 shows    similar values. In fact, the differences between observers
that the means for the CIELAB colour differences give the      are on the order of tenths of CIELAB units (figures 9 and
highest values for the sample pairs 5GY5/6–5GY5/4 and          10), contrasting, as mentioned above, with the situation in
the lowest for the pairs 5Y5/6–5Y5/8, table 5 reflecting        which mean and standard deviations are obtained with the
similar results.                                               CIELAB colour differences between the 120 observer pairs

                                              Influence of colour-matching functions on threshold and large colour differences

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