Analysis of the colorimetric properties
of goniochromatic colors using
the MacAdam limits under
different light sources
Esther Perales,1,* Elísabet Chorro,1 Werner R. Cramer,2
and Francisco M. Martínez-Verdú1
1
Department of Optics, Pharmacology and Anatomy, University of Alicante,
Carretera de San Vicente del Raspeig s/n 03690, Alicante, Spain
2
Consultant for interference pigments, Hafenweg 22, D-48155 Muenster, Germany
*Corresponding author: esther.perales@ua.es
Received 11 November 2010; revised 23 March 2011; accepted 28 May 2011;
posted 4 August 2011 (Doc. ID 138062); published 15 September 2011
Technological innovation in all areas has led to the appearance in recent years of new metallic and pearlescent materials, yet no exhaustive studies have been conducted to assess their colorimetric capabilities.
The chromatic variability of these special-effect pigments may largely be due to the three-dimensional
effect of their curved shapes and orientations when they are directionally or diffusely illuminated. Our
study examines goniochromatic colors using the optimal colors (MacAdam limits) associated with normal
colors (photometric scale of relative spectral reflectance from 0 to 1) under certain conventional illuminants and other light sources. From a database of 91 metallic and interference samples and using a multigonio-spectrophotometer, we analyzed samples with lightness values of more than 100 and others with
lightness values of less than 100, but with higher chromaticities than optimal colors, which places them
beyond the MacAdam limits. Our study thus demonstrates the existence of chromatic perceptions
beyond the normal solid color associated with these materials and independent of the light source.
The challenge for future research, therefore, is to replicate and render these color appearances in current
and future color reproduction technologies for computer graphics. © 2011 Optical Society of America
OCIS codes: 160.1190, 330.1710, 330.1720, 300.6550.
1. Introduction
In recent years, technological innovation in all areas
has led, among other things, to the appearance
of new materials such as metallic and pearlescent
objects developed from special-effect pigments that
produce goniochromatic effects, i.e. they present
notable color changes under different illuminationviewing conditions These pigments are used in many
industrial activities, such as automotive coatings,
cosmetics, plastics, security inks, building materials,
0003-6935/11/275271-08$15.00/0
© 2011 Optical Society of America
and the visual simulation of virtual environments.
Their popularity is due to the fascinating interplay
of colors and to effects produced by the various
materials used in their layered structures [1–3]. Refractions and reflections of light at and within these
layers cause interferences that yield certain colors [4]
in an attempt to replicate natural colors seen in
lesser animals such as butterflies and insects [5].
Interference pigments can be classed by either the
manufacturing method used or by their structure.
Substances such as titanium dioxide or iron oxide
that have high indices of refraction may, for example,
be deposited on a transparent substrate such as
mica, as in the case of Iriodin, silicon dioxide, as
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in the case of Colorstream, or aluminum oxide, as
in the case of Xirallic. Such pigments are produced
using wet-chemical processes, whereas those with an
aluminum layer as an internal reflector (Variochrom,
ChromaFlair, and SpectraFlair) are manufactured
in a high vacuum. Liquid crystals are also classed
as interference pigments. Aluminum pigments are
based on pure aluminum powder, which is broken
down in a dry or wet milling process. Depending on
the starting material, different types of patterns
are obtained such as “corn flakes”, which have an
uneven platelet structure, or discs that are known as
“silver dollars”. Unlike interference pigments based
on transparent substrate, these materials are not
transparent.
The optical behavior of these materials is determined by the spectral bidirectional reflectance distribution function (BRDF), defined as the spectral
ratio between the radiance of the sample in a given
direction and the irradiance over that sample. Some
instruments are designed to measure the spatial
distribution of the reflectance factor in different geometrical configurations. The geometries needed for
evaluating and observing interference and metallic
pigments may be derived from their properties. Measurements are made at specified angular intervals,
referred to as the angle of specular reflection, which
yields a series of measured spectral data. Analyses of
data acquired for three or four aspecular angles (the
difference between the angle of specular reflection
and the angle of observation) are frequently regarded
as sufficient, although they cover and use a mere
fraction of the colorimetric data available. As the
color shift toward shorter wavelengths is a property
peculiar to the interference pigment involved, it
should be used in visually and instrumentally characterizing interference pigments. This means that
different angles of illumination are required and
measurements close to the angle of specular reflection, at an aspecular angle of 15°, should be made for
each one. Additional measurements for a constant
45° angle of illumination and various aspecular angles are needed to acquire overall color impressions.
In this way, measurements made at various angles
of illumination for the same difference angle (15°)
with respect to the specular direction produce an
interference line that is peculiar to the particular interference pigment involved. Measurements made at
a constant angle of illumination (e.g. 45°) for various
angles of observation and difference angles produce
an aspecular line (see Fig. 1). Together, those two
lines resemble an anchor, where neither is necessarily a straight line. Although aspecular lines usually
all have a uniform shape, they may well extend over
several quadrants of the a b-coordinate system in
the case of diffraction pigments.
As can be seen in Fig. 2, a goniochromatic sample
changes hue and chroma along the interference line,
but not as much along the aspecular line. On the contrary, the lightness variation is usually greater along
the aspecular line than along the interference line.
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Fig. 1. (Color online) Illumination and observation angles of the
measures of the Datacolor MultiFX10 gonio-spectrophotometer
in accordance with the DIN 6175-2 and ASTM E2194 standards.
Top: aspecular line; bottom: interference line.
For instance, in both examples relating to Fig. 2,
the effect of pigment formulation can be seen. In
the first case, the color formulation is pure, with only
special-effect pigment, so the lightness values are
higher than 100 for certain measurement geometries. In contrast, the color formulation for the second
Fig. 2. (Color online) Color shifts encoded in CIELAB color space
of two goniochromatic samples belong to two conventional objects
showing the interference and aspecular lines obtained from a
multi-gonio-spectrophotometer with 10 measurement geometries
(Datacolor MultiFX10). Top: Colorstream pigment (pure formulation) belong to a piece of a handbag. Bottom: Flame Blue pigment
(mixed formulation) belonging to a car body.
case is mixed, sharing the same special-effect pigment as the first case, but incorporating certain
absorbing pigments, so the lightness values are
lower than 100, but have higher chroma values than
those of the first case.
Although in daily situations we usually perceive
complex scenes with diffuse rather than directional
illumination, many conventional objects (automotive
coatings, plastics, cosmetics, etc.) are manufactured
with semiflat or curved shapes, even using many
different orientations as regards to observer position.
For instance, these optical and visual behaviors may
also be slightly different by simply changing the
orientation of the sample and applying the same
measurement geometry, particularly for pearlescent
objects [6]. Therefore, to reach a full and exhaustive
understanding of the optical and perceptual behavior
of these special-effect pigments and their colored materials, the complexity of crossing object shape and
orientation, object material, and illumination geometry needs to be reduced.
On the other hand, according to traditional
color science, the colors distinguishable by the human visual system define a three-dimensional (3D)
structure called a color solid. The colors that define
the limits are known as optimal colors and were studied by MacAdam [7,8]. These colors are associated
with the normal reflectance/transmittance spectrum
(photometric scale from 0 to 100%), with maximum
colorfulness for a given luminance factor (Y, lower
than 100%). The main characteristic of optimal colors is the shape of their reflectance curve. Its possible
values are zero and one, with only two possible transitions between these two values throughout the
visible spectrum. Two types of optimal colors can
therefore be distinguished: type 1, with a mountainlike spectral profile and type 2, with a valleylike
spectral profile. However, a recent theorem has
been proved from which the number of transitions
depends on the shape of the cone fundamentals [9].
However, the completed spectral contour resulting
from the color matching functions adopted by the
CIE as the standard colorimetric observer is convex. This indicates that, for this observer the twotransition assumption holds true. Optimal colors do
not really exist; that is, they are not found in nature,
or cannot be obtained by means of colorant formulation as the reflectance curve is extremely abrupt.
Figure 3 shows the differences between optimal reflectances and reflectances from a real color sample.
However, optimal colors delimit the conventional
color solid of human perception, and evaluate the
colorimetric quality of colorants [10–15]: when colorants approach the MacAdam limits, a greater range
of reproducible colors can be obtained (color gamut).
For instance, in 1980 M. R. Pointer [16,17] used these
colorimetric data to compare several industrial color
gamuts.
MacAdam worked in the CIE-xy chromaticity
diagram. Since the CIE-XYZ color space is not visually uniform, it is better to calculate the color solid
in a more perceptual uniform color space. CIELAB is
a uniform color space that is widely used in industry
as it allows the color solid to be visualized more
realistically. The color solid can be obtained for any
illuminant or light source.
As goniochromism is the effect of abrupt color
depending on the illumination/observation angle, it
is interesting to consider whether these color variations are inside or outside the Rösch–MacAdam color
solid and whether this depends on the pigment type,
measurement geometry, or light source used. Providing an answer to these two questions is the main
objective of this work, i.e. to analyze the colorimetric
characteristics of goniochromatic samples using the
theoretical color solid associated with a standard
observer and with different light sources associated
with normal colors. This will make it possible to
Fig. 3. (Color online) Differences between the spectral reflectance curves associated with optimal colors and a real color.
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determine the colorimetric behavior of goniochromatic samples in relation to the light source and
measurement geometry.
2. Materials and Methods
Our goniochromatic database comprises 91 metallic
and interference color samples collected from the
technical color charts of various manufacturers. Each
spectral reflectance relating to a standard matte
and nonfluorescent white was read by the Datacolor
MultiFX10 multi-gonio-spectrophotometer. This instrument has ten illumination/observation geometries in accordance with the ASTM 2194 and DIN
6175-2 standards [18,19].
Based on the relative spectral reflectance curves
associated with each measurement geometry, the
CIELAB values were calculated by taking the CIE
1931-XYZ standard observer [20] and different light
sources (D65, A, F11, HP1 and wLED) into account.
The correlated color temperatures (T c ) and CIE general color rendering indexes (Ra ) used were 6500 K,
2856 K, 4000 K, 1959 K, and 7800 K, and 100, 100,
83, 8 and 81, respectively. These light sources represent daylight and incandescent illumination, typical
fluorescent lamps, typical high-pressure metal halide lamps, and illumination based on white LEDs
that are in great demand on the market. How the
correlated color temperature and color quality (color
rendering) of the used light sources interacted with
the goniochromatic effects in our database was also
studied.
The theoretical color solid obtained from the optimal colors was taken into account when evaluating
the colorimetric characteristics of the goniochromatic
samples. Therefore, the color solid associated with
the CIE 1931 standard observer and different illuminants, was calculated [21,22], plotting the 91 goniochromatic samples together with the corresponding
color solids for each illuminant and measurement
geometry. However, the same color solid was considered for each measurement geometry because the
Rösch–MacAdam theorem is valid and applicable
regardless of the type of color measurement. Furthermore, in these 3D plots (L versus a versus b),
the goniochromatic samples were shown with the
MacAdam limits in constant lightness profiles
to allow for a more detailed study and to analyze
the chromatic perceptions obtained with these
materials.
Therefore, even though the gonio database of 91
samples provides a large amount of color data, with
ten measurement geometries and five light sources,
determining whether CIELAB values are inside or
outside the (normal) MacAdam limits is a straightforward matter.
3. Results
Firstly, color solids are shown together with the
goniochromatic samples in the CIELAB color space
under the illuminant D65 for all measurement geometries. As shown in Fig. 4, there are samples with
lightness values greater than 100, around L ¼ 135,
for the 45°=150° and 75°=120° measurement geometries, which are associated with the interference line.
With the exception of the 25°=170° measurement
geometry (see Fig. 5), which is a very flat geometry
of the interference line (15°), the other geometries
of this line show gonio samples beyond the MacAdam
limits due to the existence of additive color mixing
near the specular direction up to 20°–25°. These
selected colors are therefore very different to the
conventional colors produced by subtractive color
mixing, and are obtained in the same samples for
the aspecular angles. The results relating to the
25°=170° geometry (shown in Fig. 5) are not plausible
due to optical laws of interference, which we consider
Fig. 4. (Color online) Rösch–MacAdam color solid associated with the CIE 1931 standard observer and the D65 illuminant together with
the 91 goniochromatic samples measured by the Datacolor FX10 multi-gonio-spectrophotometer with different measurement geometries.
a) Aspecular line. b) Interference line.
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Fig. 5. (Color online) Rösch–MacAdam color solid associated
with the CIE 1931 standard observer and the D65 illuminant
together with the 91 goniochromatic samples measured by the
Datacolor FX10 multi-gonio-spectrophotometer for the 25°=170°
and 25°=140° measurement geometries.
to be more or less artifacts caused by the proximity
of angle 180° and/or by instrumental error in the
angular tolerances for this measurement geometry.
In contrast, and reinforcing our argument, the other
measurement geometry for the same illumination
direction of 25°=140° clearly shows many gonio
samples beyond MacAdam limits, as shown in Fig. 5.
As a result, the color encodings relating to the aspecular line seem to be inside the corresponding color
solid, whereas the color encodings relating to the
interference line are sometimes outside the corresponding color solid.
It is important to bear in mind that in the a
versus b chromatic diagram, most of the samples
are less saturated than what the human visual system can distinguish, although there are some samples that are close to MacAdam limits (45°=150°,
75°=120°, etc.). However, to check this fact, the 91
samples must be compared with the MacAdam limits
at constant lightness profiles. Figure 6 shows four
constant lightness planes (L ¼ 30, 60, 90, 99) for
some of the measurement geometries, with some of
the samples beyond the MacAdam limits. It seems
logical to assume that it is possible to find samples
outside the lightness scale, as some samples have reflectance factors of more than 100% in some spectral
bands and therefore have a lightness value of more
than 100. However, it is interesting to see samples
with lightness values of less than L < 100 that
are beyond MacAdam limits. To study this behavior
in more detail, Figs. 7 and 8 show the spectral reflectance of 17 samples that are beyond MacAdam limits, with L higher and lower than 100 for some
measurement geometries of the interference line.
As can be seen below, the existence of relative spectral reflectance values of more than 100% are necessary, but not sufficient, to lead to CIELAB values
beyond MacAdam limits. The subsequent chromatic
perception will depend on many parameters relating to the spectral profile, such as photometric
range, contrast between the maximum and minimum values, the existence of a large number of
peaks, etc. Here, for our examples, the photometric
range of spectral reflectances is clearly more than
100%, with some values even exceeding 200%. Furthermore, for some of the gonio samples located outside MacAdam limits with L values of less than 100,
their reflectance spectrum is plotted in Fig. 8. It is
interesting to observe how the photometric contrast
or dynamic range can be the key feature to be
encoded beyond conventional chromatic limits.
The other aim of this work is to evaluate this
behavior using other illuminants (A, F11, HP1 and
Fig. 6. (Color online) Comparison of the goniochromatic samples (red circles) with MacAdam limits (solid line) at constant lightness
profiles for the measure geometries 25°=140° and 45°=120°.
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Fig. 7. Relative spectral reflectances along the interference line
associated with 14 gonio samples with CIELAB values positioned
outside (with L values >100) MacAdam limits.
wLED). By way of example, Fig. 9 shows the color
solids relating to the goniochromatic data associated
with the 45°=120° measurement geometry, with
gonio samples lying outside of MacAdam limits independently to the type and color quality of the light
source. Although the spectral content of the illuminant or light source greatly affects the shape and
volume of its color solid—influenced by its correlated
color temperature and color rendering index, these
parameters or those relating to light technology
are not the main reasons for their position beyond
MacAdam limits. The main reason for this significant conclusion is closely related to the physical
and chemical nature of these special-effect pigments
and their microarrangement inside the colored substrates (materials). However, the spectral content of
these light sources (and consequently the correlated
color temperature and color quality) does influence
the chromatic variability of these gonio samples
since the number of samples beyond MacAdam
limits are not always the same, as can be observed in
Fig. 10. Only three gonio samples are placed outside
the color solids for the five light sources whose spectral reflectance is shown in Fig. 8, for the different
measurement geometries associated with the interference line.
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Fig. 8. Relative spectral reflectances along the interference line
associated with three gonio samples with CIELAB values positioned outside (with L values <100, but Cab > Cab;MacAdam )
MacAdam limits.
A theoretical study can then be carried out
relating to spectral reflectance values, dynamic
ranges, peaks, etc, to analyze how these parameters
result in chromatic perception beyond the (normal)
MacAdam limits. Therefore, taking this hypothesis
into account, theoretical spectral reflectances located
beyond the normal color solid could be predicted. The
next challenge would be reproduction using current
special-effect pigments which could be used to explore special-effect pigment formulations with or
without additional absorbing pigments.
4. Discussion
In terms of conventional color science analysis, this is
a highly descriptive work at the qualitative level,
using conventional spectral and chromatic data
and diagrams. However, at implicit levels the study
is more complex as many of the calculations and
color measurements combine 91 samples and 10
geometries, which are all related to the color solids
associated with illuminants and real light sources.
These few mathematical steps therefore make our
argument a solid one for the main purpose of
this work, to determine whether any chromatic
perceptions lie outside the conventional chromatic
MacAdam limits. As a result of this study, the answer
Fig. 9. (Color online) Rösch–MacAdam color solid associated with
the CIE 1931 standard observer and the different light sources
together with the 91 goniochromatic samples measured with
Datacolor FX10 multi-gonio-spectrophotometer for the 45°=120°
measurement geometry.
to this question is clearly affirmative, although the
study could be extended with a larger gonio database of special-effect pigments and a range of daily
objects with these special color appearances. The
main interest of these experiments is undoubtedly
the physical and chemical nature of the special-effect
pigments and their special microarrangement inside
the colored materials. However, in our opinion, future works should focus on the influence of the color
formulation of these special-effect pigments. In our
study, we were unable to differentiate between pure
and mixed color formulations that affect the outer
position, and this would be an interesting direction
for new research to take.
Secondly, replicating these color appearances
found in real gonio samples is a major challenge for
color technologies applied to computer graphics (3D
cinema, virtual reality, display technologies, etc). As
these special chromatic perceptions on semiflat and
curved shapes of many objects are a daily occurrence,
even with diffuse illumination, the question is
whether these colors can be reproduced at least at
a relative colorimetric level, as it seems very difficult
to reproduce them at a spectral and absolute level.
Thirdly, having found that relative spectral reflectance values of more than 100% can result in chromatic perception beyond conventional limits, the
next challenge would be to find an algorithm such
as a pass/fail test with which to note which spectral
reflectances with spectral values of more than 100%
will be within or beyond MacAdam limits, taking different spectral profiles and mathematical analyses
(Fourier, PCA, etc) into account. This also relates to
the challenge of whether current or future specialeffect pigment technologies will be able to reproduce
these spectral reflectances in formulation and what
the effect would be of working with pure and mixed
formulations of effect and absorbing pigments.
Finally, at an instrumental level, it would be interesting to improve coverage of the geometry range
with other geometries nearer the specular angle, or
even to study other interference lines, with angular
values ranging from 5° to 25° and not only the conventional geometries of 15°. More information
could then be obtained on how additive and subtractive color mixtures interact with effect pigments.
However, all these results are associated with the
incidence plane and it would be of great interest
to expand this work with off-plane measurement
geometries by using the X-Rite MA98 multi-goniospectrophotometer, for instance.
5. Conclusions
Fig. 10. Number of samples placed beyond the MacAdam limits
for the different light sources and different (interference line) measurement geometries with Cab > Cab;MacAdam .
The concept of optimal color, as described in
the Rösch–MacAdam theory, needs to be revised.
Special-effect pigments are now used in many objects
to create metallic and pearlescent appearances,
in automotive and industrial paint formulations, cosmetics, plastics, and so on; their popularity is due to
the fascinating interplay of colors and effects. However, until now there has been an insufficient amount
of research into the colorimetric capacities of these
materials that takes the chromatic variability of
these special-effect pigments into account. This
may largely be due to the 3D effect of their curved
20 September 2011 / Vol. 50, No. 27 / APPLIED OPTICS
5277
shapes and orientations in daily objects when directionally or diffusely illuminated.
This work has shown that with a large database of goniochromatic samples, a multi-goniospectrophotometer and the conventional chromatic
limits for normal spectral reflectances (ranging from
0 to 100%), known as MacAdam limits, samples exist
with lightness values of less than 100, but with chromaticities that are higher than for optimal colors and
are thus beyond MacAdam limits. This conclusion is
valid even for different illuminants and real light
sources, in particular for measurement geometries
belonging to the interference line. This result does
not therefore depend on the spectral content and
color quality of the illuminant and light sources, but
rather on the physical and chemical nature of these
special-effect pigments and their microarrangement
inside the colored materials.
Consequently, this study demonstrates the existence of chromatic perceptions outside the normal
color solid (MacAdam limits) associated with these
materials (special-effect pigments), regardless of the
type and quality of light source. The next challenge
will be to replicate and render these color appearances in current and future color reproduction technologies for computer graphics. A complete theory of
the spectral reflectances associated with specialeffect pigments to obtain chromatic perceptions
outside the classical chromatic limits is also a future
challenge. If current and future special-effect pigments could reproduce these optimal colors, it would
be a great help in formulating colors.
This work is therefore useful for future research
into the manufacture of colored materials with chromatic perceptions outside the classical chromatic
limits. For instance, the methodology may be useful
in showing potential clients for special-effect pigments the predictions of color appearance for their
color designs and samples beyond the conventional
color perceptibility limits.
This research; was supported by means of the
grant number DPI2008-06455-C02-02 of the Spanish
Ministry of Science and Innovation.
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