1. Introduction
Yilmaz’s paper [
1] belongs to a surprisingly rich list of contributions to color theory by theoretical physicists. One of the founding fathers of quantum mechanics, E. Schrödinger is among the most famous, with his benchmark axiomatic work on color perception [
2]. In more recent years, also S. Weinberg [
3] and A. Ashtekar [
4], to quote but two, wrote papers about color. The common denominator of all these contributions is the geometry of the perceived colors space
. The geometric models of gravitation and fundamental interactions play a central role in theoretical physics, thus it is not surprising that these scientists were interested in the analysis of the geometry of
.
As we will detail in this paper, H. Yilmaz had the brilliant idea to point out the analogy between perceived colors theory and the special theory of relativity by observing that the saturation of spectral colors is not only an upper bound, but also a perceptually invariant feature with respect to the (broadband) illuminant used to light up a visual scene. The analogy between this property of color saturation and the constancy of the speed of light in vacuum, when measured by inertial observers, in special relativity is clear.
Yilmaz determined, on the base of three results that he claimed coming from experiments, a law for the perceptual effect on color perception induced by a change of illuminant. This law turns out to be the direct analogous of Lorentz transformations. From this, he argued that a color space endowed with a 3D Minkowski metric could be a valid alternative to the classical CIE colorimetric spaces, which are equipped with a Euclidean metric to measure perceived color distances.
The purpose of this paper is double: on one side, we formalize some aspects of Yilmaz’s model that we deem not mathematically clear and, on the other side, we complete its work by showing that it is possible to coherently endow with the Minkowski metric only if we assume the hue-chroma plane to be endowed with the Euclidean metric.
The structure of the paper is as follows: in
Section 2 we start by recalling Yilmaz’s experiments and the results on which his model is based, then, in
Section 3, we recast his description in a clearer mathematical framework and we point out the critical issues of the original model. In
Section 4 the analogy with special relativity, Lorentz transformations and the Minkowski metric is discussed in detail. In
Section 5 the most ambiguous result claimed by Yilmaz, the one referred as experiment 3, is shown to be linked to the relativistic aberration effect. Finally, in
Section 6, we discuss our results together with some ideas for future research perspectives.
3. Recasting Yilmaz’s Model in a Mathematical Framework
In section IV ‘Transformation formulae’ of his paper [
5], Yilmaz’s looked for
a transformation from the coordinates of a color described by an observer adapted to a broadband illuminant I to those of an observer adapted to different broadband illuminant . He deduced, from the three experiments previously discussed, what he claimed to be a linear approximation of this transformation. Such a transformation leaves of course the black point
O fixed.
Unfortunately, the mathematical exposition of Yilmaz lacks of rigor. Our contribution is this section is to introduce a precise notation and a suitable language that will allow us to translate into a rigorous and coherent mathematical framework the information provided by Yilmaz.
We start with the definition of visual stimuli.
If is the compact subset of containing the visible wavelengths, typically nm, then a visible light can be identified with a finite energy non-negative function defined on , i.e., an element of the space ;
We call visual stimulus in Yilmaz’s experiment the spectrum of visible light reflected by either a piece of white paper, or a Munsell chip illuminated by a visible light representing an illuminant entering the eye of an observer. More precisely, the piece of white paper will be illuminated by both broadband and narrow-band illuminants, while the Munsell chips will only be illuminated by broadband illuminants;
will denote a visual stimulus provided by the visible light reflected by an object enlighted by the illuminant I or , respectively. The object surface can be either the piece of white paper, and in that case we will write , or a Munsell chip;
will indicate the visual stimulus provided by the piece of white paper illuminated by the narrow-band illuminants with spike in the red or yellow region, respectively.
Yilmaz assumed that an observer adapted to a broadband illuminant, I or , analyzing colors by the comparison with a set of Munsell chips enlighted by the same illuminant, defines a vector basis of , or , respectively. Thus, the use of and will be always implicitly correlated with a triple given by an illuminant, an observer adapted to it and a set of Munsell chips used for color comparison. If even a single one of these element is lacking, then the color identification process in Yilmaz’s model is not possible.
This observation explains why we will consider the effective available space for perceived colors in Yilmaz’s setting as the volume contained in the
open cone shown in
Figure 4, denoted by
and described by the equation:
We assume to be the same for all observers adapted to any illuminant. Since it is impossible for an observer to replicate with a Munsell chip the maximal saturation of a narrow band visual stimulus, we consider the color sensation produced by such a visible light as a point belonging to the surface of the cone previously defined.
The symbols and will be used as a subscript for the visual stimuli to indicate the illuminant to which the observer is adapted: for I and for . Note that the basis subscript is extremely important because it underlines the central role of the observer, i.e., the basis with respect to which the coordinates are written. Without an observer a “perceived color" is just a “stimulus”, in the same way as a point of a vector space is just an abstract (coordinate-free) concept without a basis which describes it in terms of coordinates.
Yilmaz considers the change of basis from to to be the linear approximation (actually, as we will prove later, this is not an approximation, since the transformation is indeed linear) of the illuminant transformation from I to and denotes the associated matrix as (we recall that indicates the so-called general linear group of degree n over , which is the set of all the invertible matrices with entries in ). is naturally required to be invertible because we can reverse the transformation by switching from one illuminant adaptation to the other, i.e., .
In order to determine the coefficients of
, Yilmaz considered the following equations:
These equations are the translation of the three Yilmaz experiments in our notation.
If we denote with
or
the coordinates of a color perceived by an observer adapted to
I or
, respectively, then
At page 14 of [
5], Yilmaz analyzes, among all possible illuminant changes, the situation in which the couple
I and
produces a color coordinate transformation only along the
-axis, i.e., the
direction, leaving the
-axis totally unaffected, i.e.,
. For this to happen, the second equation of (4) tells us that we must have
and
.
The preservation of the -axis for the inverse transformation, represented by , can be verified with a straightforward computation to imply that .
So, the matrix
has the following form
The remaining coefficients, i.e.,
,
, will be determined by translating into formulae the three Yilmaz’s experiments.
3.1. Coefficients from the First Experiment: The White Point Transformation
Yilmaz’s first experiment contains information about the coordinate change associated to the stimuli
W and
, i.e.,
and
, as depicted in
Figure 5.
Our aim is to write the coordinates of the four points , , and to determine constraints among the coefficients .
An observer adapted to I or , respectively, perceives the piece of white paper placed under the same illuminant as the same white. In terms of coordinates, this means that .
Since the white is achromatic, it must belong to the -axis, so its and coordinates are null. The third coordinate remains free and we can arbitrarily normalize its value to 1 for simplicity, hence .
Let us now look for the coordinates of
and
. As indicated by the notation,
represents the color sensation of an observer adapted to
when he/she looks at the piece of white paper illuminated by
I and compares it with the Munsell chips illuminated by
. The description of
is analogous, with
I and
switched. As reported in
Section 2.2,
is perceived as greenish, i.e., with hue
and saturation
, while
is perceived as reddish, i.e., with
and saturation
.
The -coordinates and are not reported by Yilmaz, thus we are led to introduce two unknown parameters such that and . Note that and are just auxiliary parameters that merely appear in theses intermediate computations and not in the final form of the matrix coefficients of .
The psycho-visual color matching experiments performed by Burnham et al., in the paper [
6] imply that the two parameters
and
are actually different from 1. The test results reported in [
6] led to the determination of matrices that permit, once the XYZ coordinates of a light patch (i.e., a source of light directly emitting a spectrum) perceived by an observer adapted to
I are known, to predict the XYZ coordinates of a different light patch having the same appearance for an observer adapted to
. In particular, experimental data showed that a patch perceived with the same appearance of white by an observer adapted to the standard CIE illuminants
C and
A, i.e.,
, has different colorimetric specifications, thus
. Hence, if we normalize
to 1, the value of
(and vice versa
) must be different than 1.
Since the perceived hue of is greenish, it must lie on the -axis, i.e., , thence . By definition, , but, as reported by Yilmaz, , which gives . Finally, since greenish hues lies in the negative part of the -axis, the correct way to write is as follows: . So, .
Analogously, we obtain , where the positive sign of is due to the fact that, this time, is perceived as reddish.
We can now write explicitly the systems
and
, by obtaining, respectively:
The only relevant information to retain from the previous equations, in order to determine
, is given by the formulae
,
, which allow us to write
as follows:
To determine the remaining parameters we will use the results of the second and the third experiment.
3.2. Coefficients from the Second Experiment: The Red Point Transformation
Our aim here is to determine the coordinates of
and
. Let us denote with
and
the maximally saturated Munsell chips with a hue matching that of
and
, respectively. The perceived saturation of
and
is strictly inferior than
, see the depiction in
Figure 6.
In our mathematical framework, a perceived color is a sensation that can be described in terms of coordinates which come from a match with a set of Munsell chips. The coordinates of and will surely depend on , which cannot be quantified in the Yilmaz’s setting, thus and do not belong to , but to its boundary .
The reason why we consider and on the boundary of and not inside is that we can imagine and as resulting from a limit procedure in which a sequence of Munsell chips with increasing saturation approaches their saturation.
The -coordinate of both and is surely 0 because they lie on the axis. Moreover, their and -coordinates will be and , for , and and , with , . The unknown parameters and are introduced exactly for the same reason as and , i.e., we do not know their lightness. As a consequence, and .
The equation
can be written explicitly as follows:
which implies
The explicit form of will be obtained thanks to the data gathered from the third experiment.
3.3. Coefficients from the Third Experiment: The Yellow Point Transformation
When interpreting the third experiment, we will use the same approach as for the second one. We will denote with
the maximally saturated Munsell chip with a hue matching that of
. Differently than the second experiment, here, when an observer changes the adaptation state from
I to
, the perceived hue of the narrow band stimulus changes from yellow to a greenish yellow, see
Figure 7. For this reason, we denote with
the maximally saturated Munsell chip that best approximates
.
By using the same arguments of the previous subsections, we write the coordinates of
as follows:
,
. Since the hue of
increased by an angle
which satisfies (
2), the coordinates of
are
,
, where the presence of
and
comes from the expression of the hue change in Cartesian coordinates.
The equation
can be written explicitly as follows:
By direct computation and thanks to Yilmaz’s data
(this point will be crucial for the following
Section 3.4), we obtain
So, at the end, the explicit expression of
is:
with
as in Equation (
12).
The variation of the yellow hue effect is said to be “
similar to the aberration effect in special relativity”, by Yilmaz in [
1] at page 132. We will discuss this aspect in
Section 5.
In
Section 4, we will point out the analogy between
and the matrix that represents Lorentz’s transformations in Einstein’s theory of special relativity.
3.4. Critical Issues in Yilmaz’s Model
In the previous subsections our aim was to recast Yilmaz’s model in a rigorous framework, with respect to both its colorimetric interpretation and its mathematical development, remaining as close as possible to what Yilmaz reported.
In this subsection, instead, we point out some critical issues about Yilmaz’s model that are essential to underline before carrying on our analysis about similarities and differences between Yilmaz’s results and special relativity.
The first issue to discuss is the incongruous use of Munsell chips in its experiments. The piece of white paper is used merely as a sort of ideal non-selective Lambertian reflector for the illuminant, thus, comparing the color sensation induced by it with a set of Munsell chips is not coherent. A clear, fundamental, manifestation of this lack of coherence is the fact that no Munsell chip can be found to match the saturation of a narrow band illuminant reflected by the piece of white paper.
While the set of Munsell chips was an obvious choice in 1962, the year of publishing of Yilmaz’s paper, nowadays we can replace it without effort with an emitting display that will also allow us performing comparisons with narrow-band lights. With such a modern experimental apparatus, the color sensations and will be effectively measurable and thus be considered as perceived colors. Moreover, this emitting display will also have the advantage of not be affected by the presence of the illuminant.
Another questionable issue in Yilmaz’s paper is the fact that he does not exclude the dependency of the maximal saturation on the hue . The saturation of a color sensation is defined as a percentage: 100% representing the absence of a washed-out sensation, as it happens for a narrow-band light, and 0% corresponding to the totally washed-out sensation of achromatic stimuli. These measurements are independent on the hue , so it does not make sense to assume that the maximal saturation depends on .
From now on, we will remove the subscript
from
and consider it as a constant. With this choice, Equations (
12) and (
13) become, respectively:
and
Finally, it still remains unclear if the results claimed by Yilmaz have been obtained after actual observations or if they are the results of a gedankenexperiment, i.e., a thought experiment. In the first case, Yilmaz does not report any experimental data and they do not seem to be found anywhere else, this, of course, raises more than a doubt about their validity.
In the second case, it is clear that Yilmaz pushed the gedankenexperiment technique way too far: a thought experiment is used to check what known results of a given theory would predict in an experimental configuration that is not possible to test with the current available technology. No known colorimetric result can be used to predict the outcomes of the three experiments, in particular, we notice that the hue shift in the third experiment represented by Equation (
2) seems to be somehow forced to have that analytical expression to adjust the equations that permit to determine the matrix
with the desired expression of its coefficients.
In spite of the critical issues just underlined, Yilmaz’s paper has the great merit of highlighting the theoretical importance of the assumption that the maximal saturation of the color perceived from spectral lights is invariant w.r.t. changes of illuminants.
4. Yilmaz’s Model and the Standard Formulation of Special Relativity
In this section, we are going to show that it is possible to obtain the same results as Yilmaz’s ones by replacing his controversial experimental results with assumptions on the geometric structure of the perceived color space
. It is important to stress that our aim here is purely theoretical: we are interested in showing to what extent Yilmaz’s model is related to the classical formulation of special relativity and what constraints
must satisfy in order to maximize the analogies between the two theories. These constraints, as we will see, are far from being mild and this is the reason why we consider alternative formulations of special relativity, such as, e.g., those that originated from Mermin’s paper [
7], which are more appropriate to formulate a relativistic theory of color perception. We discuss this issue in the discussion section.
4.1. Elements of Special Relativity
Here we will briefly recap only the basic concepts of special relativity that are needed to show analogies and differences with Yilmaz’s model. The discussion that follows will be faithful to the standard special relativity formulation, see, e.g., [
8,
9].
Special relativity is known to be an extension of Galilean relativity, which is based on the following two postulates:
space is homogeneous and isotropic and time is homogeneous (in this context, isotropy means invariance under rotations, while homogeneity means invariance with respect to translations);
laws of physics have the same form in all inertial (i.e., not accelerated) reference frames, i.e., no inertial reference frame is privileged.
In special relativity, Einstein considered, along with the motion of objects with mass, also the peculiar behavior of electromagnetic signals by adding the following postulate:
- 3.
the speed of light in vacuum has a constant value when measured in all inertial reference frames.
In special relativity, we call
event e a point in
with coordinates written as
, where
and
,
, are, respectively, the time instant and the spatial position of the event as measured by an inertial observer with respect to her/his inertial reference frame
. Using
instead of
t is customary in special relativity: physically, this amounts at replacing the time
t with the corresponding space
traveled by a ray of light during
t. Let us consider, in particular, the following two events: the first,
, consists in a light signal emanating at the time
from the spatial position
; the second,
, consists in the same light signal arriving at the time
in the spatial position
. Since the signal propagates with constant speed
c, the square distance that is traveled is
. If we equip
with the Euclidean metric, this same square distance is equal to
, so the coordinates of the events
and
in the fixed inertial frame
are related by the equation:
being the Euclidean norm in
. Of course, Equation (
16) remains valid for all spacetime differences, also infinitesimal ones, thus we can write the differential version of Equation (
16) as
. In special relativity, the quantity
is called
spacetime interval. From Equation (
16) it follows that the spacetime interval between two events connected by a signal traveling at the speed of light is null. Since the speed of light is an upper limit for velocity, this amounts at promoting it as a reference and at normalizing to 0 the spacetime distance between any two events, no matter how far in space or time, connected by a light-speed signal.
Postulates 1 and 3 imply that the spacetime interval
between two events described in the inertial reference frame
and the spacetime interval
between the same couple of events described in any other inertial reference frame
is exactly the same:
, see, e.g., [
9], page 7 or [
8], page 117, for a rigorous proof.
We will make use of the standard Einstein’s convention which implicitly assumes a sum over repeated indices above and below in an algebraic expression, the sum being of course performed over the range of index variability, e.g., if , then This notation is consistent as long as we agree to write the indices below for the basis vectors and above for the components w.r.t. them. If we write the infinitesimal difference between any two events as the vector , then the spacetime interval can be written as the (non positive-definite) quadratic form , where is the matrix diag . The metric space is called Minkowski spacetime and is the matrix associated to the Minkowski quadratic form. The associated pseudo-norm, i.e., is called Minkowski norm of .
A world-line in is any connected path composed by events between an initial and a final one. Straight lines in correspond to world-lines of inertial movements.
The last information that we must recall is how to relate the coordinates of two inertial frames. First of all, it is simple to deduce from postulate 1 that the coordinate transformation
,
from
to
of an event must be
linear (under the reasonable hypothesis to be differentiable). In fact, by postulate 1, there are no special instants and positions in
, so, the distance between two events remains the same when these are translated by a fixed vector
. This is true independently on the coordinate system used to write the events in two arbitrary inertial reference frames
and
. Let
and
be the coordinates of the two events in
and
and
the coordinates of the same events in
. Since
, we must have
. If we derive the two sides of the last equation with respect to
,
, we obtain
, for all
, since
y does not depend on
x. Thanks to the fact that
b is arbitrary,
represents any vector in
, so the function
is constant, which implies that
for all
,
, i.e.,
The invariance of the spacetime interval imposes a strong constraint on the form of the matrix
: to see this, let us write the difference vector
in the inertial reference frame
by using Equation (
18):
. Thus, on one side,
and, on the other side,
so, the equality
implies:
The set of all these matrices forms a group, called the
Lorentz group classically denoted by the symbols
.
Thus, postulates 1 and 3 imply that the coordinates used to describe the same event in two generic inertial reference frames are related by either non-homogeneous linear transformations of the type
,
,
, called
Poincaré transformations, or, in the special case when
, by linear transformations
called
Lorentz transformations.
4.2. Similarities and Differences between Yilmaz’s Model and Special Relativity
Table 1 provides the list of analogies between Yilmaz’s model and the standard mathematical framework of special relativity just recalled.
Notice that, if homogeneity and isotropy of
are assumed, then the linear nature of Yilmaz transformations is not only approximated, but exact, as previously pointed out in
Section 4.1.
Among the similarities listed above, substantial differences between special relativity and Yilmaz’s model of color perception must be remarked.
The similarities above lead naturally to the question if it is possible to endow with a Minkowski-like metric and, if so, under what set of assumptions about color perception. We investigate this issue in the following subsection.
4.3. The Issue of a Minkowski-Like Metric on
Homogeneity and isotropy of spacetime, the absence of a preferred inertial observer and the constancy of light speed in vacuum for inertial frames naturally leads to endow with the Minkowski metric. Similarly, in Yilmaz’s model, the perceived colors space is homogeneous and isotropic (no combination of hue, saturation and intensity is ‘special’ as long as we remain in ), no observer adapted to a broadband illuminant can be considered privileged, and the maximal saturation is invariant under changes of broadband illuminants.
It is thus tempting to check if it is possible to repeat, in the case of Yilmaz’s color perception model, the same argument used in special relativity to single out the Minkowski metric, which, at page 26 of [
1], is claimed to induce a geometry on the color space which leads to “
a good approximation of color vision phenomena”. As we will see, this is possible only under the hypothesis that the hue-chroma plane is endowed with the Euclidean metric.
Let us start this investigation by considering two spectral colors
and
belonging to
with the same hue
, but with different values of
. Let us fix the axis
in the direction defined by
, so that the coordinates of
and
are:
and
. Since
and
are spectral colors, by definition they have maximal saturation
, so that we can write the system
by subtracting the first equation from the second we obtain
, whose infinitesimal version is
.
This result tells us that, if we endow
the intersection between and the plane defined by any fixed value of ϕ with the metric
then we consider as having 0 perceptual distance any two points in
with the same hue and maximal saturation, in spite of having different chroma and lightness, when described by two observers adapted to different illuminants. Thus, the distance (
23) comes from promoting spectral lights, having the maximal attainable saturation, to a reference and normalizing their distance to 0.
Without specific hypotheses, there are infinite ways to extend the metric
, defined for all
fixed , on the whole
. In fact, every metric of the type
where
are are arbitrary functions of
,
, is an extension of (
23) one because, when (
24) is restricted to the plane defined by
constant, the differential of
vanishes and all the terms containing
disappear.
In an orthogonal coordinate system (w.r.t., the Euclidean inner product), the metric (
24) becomes diagonal, so that we can write
We are going to discuss the assumptions that permit to single out the specific analytical form of
f. The first assumption seems the most reasonable one: if we make the hypothesis that no hue can be considered as special, then
f does not depend on
and (
25) becomes:
where
is the metric
restricted to the chromaticity plane by fixing
.
Let us now introduce
the strongest hypothesis of our extension procedure: we assume that the chromaticity plane is endowed with the Euclidean metric. This is the standard choice that is made in classical colorimetry (actually, in classical CIE colorimetry, the whole color space is endowed with a Euclidean color metric), see, e.g., [
12] and the references therein, however, mathematically speaking,
this choice is completely arbitrary.
is coherent with a Euclidean metric in polar coordinates on the chromaticity plane, i.e.,
, if and only if
f has the following separate dependence on
and
:
, where
is an arbitrary positive-valued function of
, where positivity must be requested to preserve the signature of the Euclidean metric. The term ‘coherent’ here is used in the sense that, for every fixed value of
, the metric restricted to the plane
is the Euclidean metric in polar coordinates up to a constant. The constant is
and, since it is positive, we can rescale the variable
in order to obtain exactly the Euclidean metric on the plane. To resume, the hypothesis that the chromaticity plane is equipped with a Euclidean metric implies that (
26) must have the following expression:
The only degree of freedom that remains left is the function
, however, we are going to show that, if one also assumes the homogeneity hypothesis of
under scaling of
and
, then
must be constant. In fact, if we transform the color coordinates with a uniform scaling, i.e., we perform
,
, then, homogeneity implies that the metric must change as follows:
. This implies that
for all
, i.e., that
,
constant, for all
.
If we rescale the hue
as follows
, then (
27) becomes
which is the expression of the Minkowski metric on
in polar coordinates.
To resume, the classical hypothesis on the geometry of the space of perceived colors , i.e., homogeneity w.r.t and , isotropy w.r.t and the choice to measure distances on the hue-chroma plane by means of a Euclidean metric, single out the Minkowski metric on as the only one compatible with these classical colorimetric hypothesis.
If Yilmaz’s assumptions about the perceptual effect of (broadband) illuminant changes being described by Lorentz transformations and about the constancy of
are accepted, then the same arguments used in the standard formalism of special relativity, and quoted in
Section 4.1, can be used to check that the Minkowski metric on
is invariant under (broadband) illuminant changes.
Finally, we remark that Yilmaz’s transformations, represented by the matrix
of Equation (
15), are isometries for the Minkowski metric. In fact, if
has coordinates
w.r.t to a basis
, then its Minkowski norm is:
while
has Minkowski norm
i.e.,
for all
.
5. Relativistic Aberration and Yilmaz’s Third Experiment
In this paragraph, we want to discuss Yilmaz’s most ambiguous assumption, represented by the result of the third experiment, in a relativistic framework and show that it is the translation, in the colorimetric context, of the relativistic aberration effect. This phenomenon expresses how the angle of incidence of a ray of light changes with the inertial frame of reference and it is a direct application of Lorentz transformations.
Let and be two inertial reference frames, with moving with respect to with constant speed v along the x-direction. Without loss of generality we can consider a photon moving towards the origin of the frame and whose spatial trajectory is a straight line contained in the plane . Of course, in both and , the speed of the photon will be c.
We suppose that its trajectory forms the angle
(resp.
) in
(resp.
), with the
x-direction shared by both
and
. Our aim is to show the functional relationship between
and
. In
the photon’s world-line is given by
, to obtain it with respect to
we need to apply the so-called Lorentz boost as follows:
with
.
In particular and .
Hence
By a straightforward calculation we obtain
thus
where only the positive determination of the square root is compatible with the fact that, if
, then we must have
. Moreover
indeed
, so
and
.
We have now all the information to discuss Yilmaz’s third experiment: taking into account the analogies that we have commented in
Section 4 together with Equation (
35), we have that
For the spectral yellow, we have that
, so Equation (
Section 5) becomes
, but since
, we get
which corresponds to the Equation (
2) reported by Yilmaz, concerning the hue variation of the spectral yellow.
6. Discussion
Yilmaz’s contribution [
1], based on the analogies between color perception and special relativity, stood out from the classical CIE color space theories for at least two reasons: the first is that he was able to model the perceptual effect of illuminant changes via Lorentz transformations and the second is that he put the accent on the possibility to replace the standard Euclidean CIE metrics with a 3D version of the Minkowski metric.
However, as we have shown in
Section 4.3, the possibility to endow
with the Minkowski metric coherently with the rest of Yilmaz’s assumptions requires to impose a Euclidean metric on the hue-chroma plane obtained by fixing a constant value for the lightness
. This underlines a systematic and important difference between Yilmaz’s model and the CIE classical color space models. In fact, while these latter are globally Euclidean spaces, the fact that Yilmaz introduced the Minkowski metric in the space
allows us to investigate naturally hyperbolic structures associated with the Minkowski metric, i.e., hyperbolic leaves that foliate
that coincide with surfaces of constant Minkowski norm. This natural association is not possible in the Euclidean space endowed with the canonical Euclidean norm.
We are currently investigating an alternative formulation of Yilmaz’s model in which hyperbolicity is present also on the chromatic plane: in fact, if we consider it to be the hue-saturation plane instead of the hue-chroma one, then it is possible to endow it with a hyperbolic metric. The reason behind this fact is that, while the hue-chroma plane is simply a slice of the complete color space, the hue-saturation plane is obtained via a hyperbolic projection.
The determination of the formal relationship between these two alternative formulations is still an open problem that clearly underlines the important issue of a coherent understanding and definition of the colorimetric attributes.
This involves also the correct interpretation of the coordinate
: there are several visual phenomena, e.g., the Helmholtz-Kohlrausch effect, the Bezold-Brücke hue shift and the Hunt Effect [
10], which show that treating
as independent of the chromatic coordinates is not coherent with human perception. A rigorous mathematical model of the relationship between
and the chromatic coordinates of a perceived color is a subtle open problem that we deem important to solve.
Yilmaz himself, in the last part of his paper, quoted the Bezold-Brücke hue shift as “a departure from the Minkowski metric”, this observation led him conjecture that, in order to take into account this effect, should be a negatively curved color space endowed with a more complicated metric.
Remarkably, these Yilmaz’s speculations had a strong impact on H.L. Resnikoff who, in the paper [
13] published twelve years later, acknowledged Yilmaz for his intuition and proved rigorously that a hyperbolic color space, i.e., a homogeneous space with constant negative curvature, is perfectly compatible with the phenomenology of color perception, see also [
14] for a modern discussion.
Finally, we would like, in future investigations, to understand how the mathematical properties of the perceived color space relate with the physics of color, which is still a hard open problem to solve.