# Toward individual displays, the role of color vision

David Alleysson, David Méary

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David Alleysson, David Méary. Toward individual displays, the role of color vision. HCI INTERNA-TIONAL 2024 26th International Conference on Human-Computer Interaction, Constantine Stephani-dis, Jun 2024, Washington DC USA, United States. pp.3-16, 10.1007/978-3-031-76821-7_1. hal-04693165

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# Toward individual displays, the role of color vision

David Alleysson and David Meary

Laboratoire de Psychologie et NeuroCognition, CNRS/UGA UMR 5105 Grenoble,
France

**Abstract.** In order to reduce fatigue and pain caused by prolonged use of a computer screen, visual displays can be adapted to the individual. According to the ecological theory of vision, human bodies have adapted to natural circumstances for vision. Tiredness is the consequence of the unnatural viewing that the visual system is optimized for. Through the prism of human color vision, we examined this question. What does the knowledge of color vision tell us about the inadequacy of displays to reproduce natural viewing conditions? To argue, we will discuss some recent results on human color vision geometry obtained by the authors a few months ago. In a preceding experiment we show that the human visual system operates a decomposition of the display color space into a stack of hyperboloid sheets of different photometrical levels. We will discuss the consequences of this hyperbolic geometry in color vision for the human computer interface.

**Keywords:** Individual displays · Color Vision · Photometry.

## 1 Introduction

Nowadays, most people spend several hours watching a computer screen for work or for entertainment. Prolonged use of a computer display is not without consequence on health and tiredness. Increased prevalence of myopia in the young generation is suspected to correlates with the use of computer displays [1,2,3,4], specifically for children and young adults. But a direct evidence for computer use as a cause for myopia has not been found, and rather this is the lack of outdoor activities that is favor to explain the increase of prevalence for children and young adult [2,4]. However, tiredness and a loose of cognitive abilities is implied by cognitive load induced by watching a display for long specially for learning [5,6].

The main researches on individual displays are done from the point of view of spatial vision. The flatness of the display may induce a lot of optometrical problems because the eye is not longer train for accommodation and vergence over objects placed at different depth [7,8,9]. Color vision is not involved when one talk about ergonomy of the visual displays. One supposes that color in display reproduction is good enough to not perturbed the physiology of the visual system. For us, however, this question remains crucial because the way color reproduction quality is estimated do not take into account the variability of
color metric induced by the local and global context. Objectives of the present
study is to describe the different step in modeling the way a human observer
capture light from the display and interpret it as a colored image.

There are three domains to consider. At first, it is the physical capabilities of light produced by a display that depends on the technology used but are often based on additive color mixture of three primary colors. Then there is the physiology of the retina and visual pathways to the brain. This is for sure a complex system that allows the regularization of light, its encoding and its interpretation. The third one is the appearance or sensation of colorfulness. It is construct by the physiology but not necessary directly derived from physiology activity. There are clear evidences that our conscious perception of colors is constructed from the neural activities but not in a trivial way (Think for example of the blind spot [10] or the lack of S-cone photoreceptors [11] in the center of the fovea - those behaviors are not visible in our conscious vision). Therefore, there is a real difference in light acquisition by the visual system and color perception by our consciousness.

In a previous paper [12], we presented a novel method for measuring the
color space of an observer using stimuli produced by a computer screen. The
advantage of the method is to perform the measurement in-situ, directly on
the display system the user will used. Another advantage is the easiness of the
experiment that allow to perform on a large population. What we found in that
study is (1) that the photometry function in human depends on the resting
adaptation making a hyperbolic color space for the steady state space of color
vision (2) the geometry shows a large variability between observers. This is these
results that we discuss in depth here.

2 Geometry of color spaces

## 2.1 Space of light's spectrum

Usual spectra are emission spectra or the spectral power density of the light source. Alternatively spectra are the reflection of a light source over an object called reflectance spectra. Those two kinds of spectra are called physical because they are positive functions of wavelength, the value of the function being the rate of photon emetted by the light source or reflected by the object. In general here we adopt the position of Schrödinger concerning spectra [13,14,15]. At any moment of time one can consider a light C as defined by the spectral function C(ξ) in the visible domain. The energy element is given by C(ξ)dξ neglecting the other term dω, dσ, dt. Whatever are is the solid angle of view dω, the surface area dσ and the measurement time dt they are consider normalized to 1.

To formalize the notion of spectra let's defined some mathematical objects.
We consider that a spectra is a function of wavelength rather then frequency.
We consider that the visible range of wavelength λ is between 380nm to 780nm
that is reparameterized into a variable ξ = (λ/380) − 1/2 ∈ [−1/2, 1/2] (See
Figure 1(a)). A light that produce a color C is defined by its spectrum C(ξ) as
follow:

$$ C: [-1/2, 1/2] \to \mathbb{R}^+ \\ \xi \mapsto C(\xi) \tag{1} $$

The variable $\xi$ is either a continuous variable or a discrete variable if the spectrum is measured with a spectrophotometer for instance. To account for both these alternatives we define the spectra of a color C as a ket vector as follow:

$$ C(\xi) = |C\rangle = [C(-1/2) \dots C(1/2)]^t \tag{2} $$

This vector is either infinite if the variable $\xi$ is continuous or finite is the variable is discrete. The rule that belong on such mathematical object is described in Dirac's book [16].

A spectrum can be decomposed over a basis. In the case of $\xi$ is a continuous variable the basis in made on Dirac delta distribution $\delta(\xi)$ and the spectrum vector writes $|C\rangle = [C(\xi)] = [\int_{\xi_i} C(\xi_i)\delta(\xi - \xi_i)d\xi_i]$. If the variable $\xi$ is discrete the integral is replaced by a sum, and the Dirac delta distribution by an indexing vector having a 1 at the position $\xi_i$ and 0 elsewhere. The decomposition of the spectrum into the $\delta$ basis has a physical interpretation, it corresponds to what is called a monochromatic light. Every light can be decomposed into a sum (finite or infinite) of monochromatic lights (Figure 1(a)).

The space of light spectra can favorably be constructed with the vector notation representing a spectrum. If two lights are added together, say $C_1$ and $C_2$, the resulting spectrum is the sum of their corresponding spectra, $|C_1\rangle + |C_2\rangle = |C_1 + C_2\rangle$. A physical increase of the power of the light source by a positive factor say $k \in \mathbb{R}^+$, corresponds to a multiplication of a spectrum by the factor $k|C\rangle = |kC\rangle$. The space of light's spectra is also endowed with an identity element, $|0\rangle$ and associativity, commutativity and distributivity are easily verifiable. But this does not confer to the space of light's spectra the nature of a vector space because the subtraction (the inverse of addition) is not defined. It does not exist a light spectrum that could be added to another in order to obtained the zero spectra. Or at least no one knows how to construct such a light couple. But has it is described in [21] the space of positive spectra distribution generate the whole vector space.

## 2.2 Space of absorption's spectrum

The main hypothesis for color vision is that there exists neurons that capture light and able to transfer the count of absorbed photons into electrical and chemical power along the visual pathways [17,18]. Moreover, these neurons can be characterized through their absorption spectra in the visible domain. Absorption spectra are of a different kind because they can take negative values and have not necessarily a physical meaning but rather a physiological meaning.

At the early stage of vision, neurons compare there activity and the retina first layers can be considered as a comparator. As a consequence the signal carried by neurons in the retina is bipolar having positive or negative values. Bipolarity is illustrated by the so-called ON and OFF pathway inside the retina [19].
<figure><img src="image_2.png" /><figcaption>**Fig. 1. Metamerism space.** Metamerism produce by the set of virtual spectra given for the visual system. (a) The continuous spectrum or the spectral power distribution of a light can be decomposed on a succession of vectors for each value of the hue variable ξ corresponding to the wavelength λ. (b) The daylight illuminant correspond to the radiance of the sun by a clear day and is the closest to the black body at 6500K. (c) Simulation of the measurement of the daylight illuminant with the LMS cone fundamentals. Function aD65 is the approximation of the function D65 when represented by the LMS fundamentals (d) Representation of the geometry described by the mathematics of figure (c). See text for explanation.</figcaption></figure>
Because synapses cannot change there properties with the incoming light (they are either excitatory or inhibitory but cannot change along time of with the incoming signal) the retina system had doubling the pathways. The ON pathway carried the positive part of the signal wherever the OFF pathway carried the negative part of the signal. The flexibility imposed by the retina coding may confer to the space of encoded light the nature of a vector space.

A precise mechanism of encoding of positive spectrum of light into bipolar spectrum carried by neuron in the retina is not possible because it involves the spatial, temporal description of the scene seen. But for our purpose here one has to consider that physical light are confined into a positive part of a vector space. The whole vector space may be covered by virtual spectra that correspond to efficiency functions in the visual system.

## 2.3 Color space

Adding with several rules the notion of color space can be derived from the vector space described above. Recall it is the space of spectra function or distribution along the visible spectrum that represents the spectral power distribution to the light under study.

The first rule is to add to the vector space of light's spectra the vector space of the observer. In this system a color is a triplet of values being the orthogonal projection from the space of light's spectra to the three-dimensional space of metamer. This expressed as:

$$c: \mathcal{L} \times \mathcal{L} \to \mathbb{R}^3 \qquad (3)$$
$$ (|\psi\rangle, |C\rangle) \mapsto c = \langle\psi|C\rangle = \int_{\xi} \psi(\xi)C(\xi)d\xi $$

A color, that is to say, the triplet of value that characterized the color in the space, whatever it is a perceptual space or a bio-physical space, it is a three-dimensional object that corresponds to the coordinates of the light C onto the visual system $\psi$ of the observer. Its coordinates depend on both the physical property of the light and the adaptation state of the observer represented by the functions $\psi$ [22].

## 3 Maxwell triangle

The first color space in that sense is the so-called Maxwell space that index color by their proportion in an additive color mixing [24]. Maxwell even proposed what is called the Maxwell's triangle (Figure 2) that is a two-dimensional color space, or a chromaticity diagram, that can serve as an map of the different perceived colors induced by the three primaries. He concluded that three primary colors are necessary and sufficient to provide the whole shade of perceived colors. So from a point of view of color synthesis by mixing, the perceived color space is a vector space of dimension three. A two dimensional chromaticity diagram enable
<figure><img src="image_3.png" /><figcaption>**Fig. 2. Maxwell's triangle.** (a) Maxwell disk. This simple apparatus can be used to simply assess trichromacy in color vision. The three primaries (external disk) are arranged in a proportion that is used to match a color in the center disk (in black/white here). (b) Maxwell triangle that index of color by its coordinates as a proportion in the mixing. (c) The relationship with spectra is that the primary spectra is represented with a unit vector from the origin. Whatever is the real length of the vectors corresponding to a spectra, the projection of the mixing onto the plane $R + G + B = 1$ made the projective point to follow the law of proportions.</figcaption></figure>
the identification of a color by its proportion on the mixed color. Because three
primaries are sufficient and necessary for producing the whole shade of colors,
this defined the perceived color space as a three-dimensional vector space. To
each color produce by the mixing of three primaries (or the maxwell disk one)
we write:

$$
[C] \doteq r[R] + g[G] + b[B], \qquad (4)
$$

with *r*, *g* and *b* are the proportion of the three primary colors [*R*], [*G*] and
[*B*]. Thus (*r*, *g*, *b*) ∈ [0, 1]³ are the coordinates of the color in the color triangle
because in the triangle they are constraint by the rule *r* + *g* + *b* = 1 thus they are
relative proportion. It is not necessary to account for the intensity of [*C*] because
we will represented it in the projective plane. In the three-dimensional space this
representation works if the vector for red, green and blue form an orthogonal
basis. It is not enough that the basis is unitary or orthogonal it should be both
to rely vectors from the three-dimensional vector space to proportions in the
triangle. So if :

$$
[R] = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad [G] = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad [B] = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \qquad (5)
$$

then

$$
[C] = r[R] + g[G] + b[B] \tag{6}
$$

$$
= ((r + g + b)/3) [R + G + B] \tag{7}
$$

$$
-(r/3)[R - (G + B)/2]
$$

$$
-(g/3)[G - (R + B)/2]
$$

$$
-(b/3)[B - (R + G)/2]
$$

$$
= [R G B] \begin{bmatrix} r \\ g \\ b \end{bmatrix}
$$

$$
= [R G B] \left( \ell + \begin{bmatrix} r - \ell \\ g - \ell \\ b - \ell \end{bmatrix} \right) \qquad (8)
$$

If the value *r*, *g* and *b* are such that they are relative proportions of the primary
such that *r* + *b* + *b* = 1, then the decomposition into luminance [*L*] = [*R* + *G* + *B*]
and chrominance [*R* − *L*], [*G* − *L*], [*B* − *L*] provides directly the coordinates of
the point in the triangle.

In the case where r+g+b≠1, there is a transformation that normalized the
values to sum to one. Writing [W] = [R]+[G]+[B] is equivalent to w = r+g+b =
3 with r = 1, g = 1 and b = 1. Let's now called r' = r/w, g' = g/w, b' = b/w.
The projection onto the plane orthogonal to [W] defined [W'] = [W]/3. So this
transformation enable to transform any proportion of the primaries into one that
sum to unity. In turn those coordinates are the coordinates in the triangle.
In some sense, the Maxwell triangle has the property that its coordinates express its position on the triangle as well as the normalized intensity of light that composed the stimulus. This property is given by the central projection itself. The measurement is done by $\ell = v^t x$ with $v = [111]^t$, then the locus of constant measure equal one is the plane $v^t x = r + g + b = 1$.

# 4 Projective hyperbolic color space

There is a way to write this in term of spectra directly. Simply replace a color vector [C] by its spectral vector $|C\rangle$. The definition of the observer space is given through the absorption spectrum matrix called $\langle\varphi|$ represented as a vector of bra spectrum, through the following definition:

$$c = \langle\varphi|C\rangle = \begin{bmatrix} \int_{\xi} \varphi_1(\xi) C(\xi) d\xi \\ \int_{\xi} \varphi_2(\xi) C(\xi) d\xi \\ \int_{\xi} \varphi_3(\xi) C(\xi) d\xi \end{bmatrix}, \qquad (9)$$

where $\varphi_1$, $\varphi_2$ and $\varphi_3$ are the absorption spectra of the three visual color mechanisms that samples the visual light's spectrum. Because they are absorption spectra they remain positive (or null). But has explained above, they will be treated as bipolar signal at the next step of visual processing. Because of the positive constrained they can not be orthogonal. Imagine a set of three distributions that is mutually orthogonal and positive. They must be separated otherwise there exist some wavelength that integrates by the two mechanisms and this positive correlation cannot be compensated by a negative correlation because the function are positives.

The only manner to allowed for orthogonality of color mechanisms is to relax the positive constraint, considering the visual system to function with a bipolar signal (ON and OFF pathways for instance). Generalized spectra are spectral function having positive and negative values. Among them some are constraints in a positive domain, the real spectra of light source and object reflectance.

## 4.1 Metamer orthogonal projection

The first projection to consider is the projection of spectral power distribution onto a vector on the three-dimensional space of metamerism. Let's define an operator $\alpha$ that is a linear application from the space of spectrum into a three-dimensional vector space belonging to $\mathbb{R}^3$.

$$\alpha: \mathcal{L} \to \mathbb{R}^3 \\ |C\rangle \mapsto c = \alpha(|C\rangle) = \langle\varphi|C\rangle \qquad (10)$$

There is however two cases to consider for this operator. Either $\varphi$ is an orthogonal set of function or not. In the first case, we have $\alpha(|\varphi\rangle) = \langle\varphi|\varphi\rangle = \mathcal{I}$ where $\mathcal{I}$ is the identity matrix of $\mathbb{R}^3$. But if $\varphi$ is not made on orthogonal vectors
there is a need to render them orthogonal. The change of space can be given by the following rule. Consider a transformation $\langle\chi| = \langle\varphi|T$. It can be shown that $T = (\langle\varphi|\varphi\rangle)^{-\frac{1}{2}}$ is a symmetric solution $T^t = T$. In that case $\langle\chi|\chi\rangle = \langle\varphi|TT|\varphi\rangle = \langle\varphi|(\langle\varphi|\varphi\rangle)^{-1}|\varphi\rangle = (\langle\varphi|\varphi\rangle)^{-1}\langle\varphi|\varphi\rangle = \mathcal{I}$. Thus $\langle\chi|$ is an orthogonal set of function.

This calculation is illustrated on Figure 1. In Figure 1(b) the spectral power distribution of the daylight illuminant D65 is shown along the wavelength variable. Figure 1(c) shows the LMS absorption spectra taken from [25] and now adopted as a standard by the CIE. They are not orthogonal functions based on the scalar product defined above. Let $\langle\varphi_{LMS}|$ be the matrix composed by the three absorption spectra. The intercorrelation $\Gamma$ between cone signals is given by:

$$ \Gamma = \frac{\langle\varphi_{LMS}|\varphi_{LMS}\rangle}{N} \qquad (11) \\ = \begin{bmatrix} 0.2132 & 0.1717 & 0.0107 \\ 0.1717 & 0.1647 & 0.0171 \\ 0.0107 & 0.0171 & 0.1027 \end{bmatrix}, $$

$N = 401$ being the number of sample of the discrete absorption spectrum given for a range of 380nm to 780nm, 1nm step. This normalisation by the number of sample point is arbitrary but arrange the curves for having almost same maximum. This gives a transformation $T = \Gamma^{-\frac{1}{2}}$ which is a symmetric matrix.

$$ T = \Gamma^{-\frac{1}{2}} \qquad (12) \\ = \begin{bmatrix} 4.4192 & -3.1860 & 0.1393 \\ -3.1860 & 5.3528 & -0.3286 \\ 0.1393 & -0.3286 & 3.1520 \end{bmatrix} $$

The functions $\langle\chi| = \langle\varphi|T$ are represented on Figure 1(c) as $\langle\chi_1|$, $\langle\chi_2|$ and $\langle\chi_3|$. The figure also show the spectra of daylight illuminant D65 as well its approximation aD65 by the visual system based on the LMS absorption spectra and the residual $\delta D65 = D65 - aD65$. Figure 1d) shows the corresponding color space drawn as a three dimensional space by applying the operator $\sigma$ on the spectral functions. Because absorption spectra $\langle\varphi|$ are render orthonormal when transformed into $\langle\chi|$, one can represent the basis formed by $\langle\chi|$ as the orthonormal basis of $\mathbb{R}^3$.

The visual system did an approximation if using only three different absorption spectra. This is illustrated on the Figure 1(c)(d) where D65 is the spectrum of daylight illuminant D65, aD65 its approximation, and $\delta D65$ the residual. In the color space D65 and aD65 match, they are represented by the same point, whereas $\delta D65$ project onto 0 even if it is not a null function. This spectrum belong to the kernel of the application $\sigma$.

From the representation of Figure 1 it is quite clear that the correlation between L and M axis gives to the space of physical stimulus as regarded by the visual system as a flattened rectangle parallelepiped. But, this is also because we defined the scalar product as a uniform integration along the wavelength
variable. This can be discussed and a way to integrate different strength in the wavelength contribution is to integrate a weighting function in the integral as follows:

$$ \langle \varphi | C \rangle_{\ell} = \int_{\xi} \ell(\xi) \varphi(\xi) C(\xi) d\xi \qquad (13) $$

where $\ell(\xi)$ can be chosen as the illuminant [22,23]. Notice that in this case it is equivalent to define a new coordinate system $\langle \varphi' | = \langle \ell \varphi |$. This is a source of ambiguity when considering the estimation of reflectance of object, because the problem of mixing between reflectance and illuminance is multiplicative, its estimation is not trivial. But as we will see on the next paragraph a central projection governed by a Minkowski's quadratic form can help solving the problem by providing a multiplicative decomposition between intensity and color.

## 4.2 Central projection

A way to extend the properties of the maxwell triangle to the spectral domain is to change the way a measure is defined. It is straightforward to consider a quadratic measure such as $r^2 + g^2 + b^2 = 1$ or in the more general manner:

$$ x = [r, b, g] \in \mathbb{R}^3, x^t G x = 1 \qquad (14) $$

The metric $G$ can be used for defining a distance $dist(x, y) = \|x - y\| = \sqrt{(x - y)^t G (x - y)}$ if the metric is definite positive. But more interestingly it can provide a pseudo-distance for the case the metric is a Minkowski metric. Moreover, the projection provided by using that metric has the same flavor than the Maxwell triangle. It is oriented and we show above how important is the orientation in the case of the Maxwell disk. From physics to appearance the conversion must account for the placement of the illuminant in the three dimensional space of stimuli.

Using this decomposition the color vector is equal to [26]:

$$ c = k(\sqrt{2}(\sinh s)u(\xi) + (\cosh s)\mathbf{1}) \qquad (15) $$

where $k$ is the strength of the color, its intensity, $u(\xi) = [\cos 2\pi\xi, \sin 2\pi\xi, 0]^t$ is the hue position, parametrized by the hue angle $\xi$, $\mathbf{1} = [0, 0, 1]^t$ is the white point, the position of the illuminant in the vector space. Variable $s$ is the saturability calculated as $s = \arctanh \sigma/\Sigma$ where $\sigma$ is the saturation and $\Sigma = \sqrt{2}$ the maximal saturation. It could be shown that at color defined as such follow the metrical property of the Minkowski form:

$$ c^t \mathcal{J} c = k^2, \quad \mathcal{J} = \frac{1}{2} \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{bmatrix} \qquad (16) $$

which is the equation of a hyperboloid of radius $k$ inscribed into a cone defined by its envelop $x^t \mathcal{J} x = 0$. But those two later equations are not expressed in the
basis $\langle\varphi|$ nor the basis $\langle\chi|$. It is expressed in the basis called $\langle\psi|$ that writes:

$$ \langle\psi| = P\langle\chi| = PT^{-1}\langle\varphi|, \quad P = \frac{1}{\sqrt{3}} \begin{bmatrix} -\frac{1}{\sqrt{2}} & \sqrt{2} & -\frac{1}{\sqrt{2}} \\ -\frac{\sqrt{6}}{2} & 0 & \frac{\sqrt{6}}{2} \\ 1 & 1 & 1 \end{bmatrix} \quad (17) $$

From a color vector $c$, the parameters $k, s$ and $\xi$ can be expressed as $k = \sqrt{c^t J c}, s = \arctanh \sqrt{c_1^2 + c_2^2/(\sqrt{2}c_3)}, \xi = \arctan c_2/c_1$.

Notice that with the parameter $\Sigma = \sqrt{2}$ we have a canonical transformation between the linear central projection of the Maxwell disk and the central hyperbolic projection. Canonical in the sense that both $\langle\chi|$ and $\langle\psi|$ are orthogonal and normed basis which are canonical basis of $\mathbb{R}^3$ making the transformation between them as a rotation $P$. Moreover, canonical because the expression of linear projective geometry in $\langle\chi|$ is made canonically (i.e. the denominator of the projector is $r+g+b$) as well as the expression of hyperbolic projective geometry in $\langle\psi|$ (matrix $J$ is a diagonal matrix).

The two representations (linear and hyperbolic) are illustrated in Figure 3. Linear projective geometry with the maxwell triangle defined two planes. $P_1$ is the projection plane and $P_\infty$ is the plane to infinity. Any point $p$ corresponding to a color stimulus is included into the cone formed by its envelop $\delta C$ and the plane $P_1$. This is because any reflected light is the product between the illuminant $OW$ and a function restricted to the domain $[01]$ and is limited in saturation by the envelop of the cone. Instead of defining linear geometry one can defined a hyperbolic geometry, given by the cone and the unitary hyperboloid $H_1$.

<figure><img src="image_4.png" /><figcaption>**Fig. 3. Yilmaz Color Solid.** Comparison between linear projective geometry and hyperbolic projective geometry. Linear projective geometry defined two planes $P_1$ and $P_\infty$. A point $p$ corresponding to a color stimulus is projected into the chromaticity diagram or projective plane $P_1$ onto the point $m$. Alternatively it is projected onto the point $M$ in the unitary hyperboloid $H_1$ of the hyperbolic model.</figcaption></figure>
The change of model from the linear projective geometry and hyperbolic projective geometry is not a big gap because the position of the symmetry axis, the white, in the hyperbolic projective geometry is similar to that of linear geometry. Only the distance between colored stimulus is changed over the two geometries. In projective terms, this is the surface of projection which provide an isoluminant surface that change from a plane in linear geometry to a hyperboloid in hyperbolic geometry.

# 5 Conclusion and discussion

For considering color vision aspect in human computer interaction one have to consider three steps instead than two. The two classical steps are biophysics and neurosciences that fill the gap between physics and physiology for the former and the gap between physiology and conscience for the later. Here we show the need for a decomposition into three steps (Figure 4). The physical composition of lights generates a vector space on which the geometry is linear, because spectra add together in a mixing. The physiological space which corresponds to the measurement by the visual system rather interprets as hyperbolic geometry to account for the circularity of the spectrum locus. It is likely that the perceptual space is circular but even more, perfectly circular, to allowed for a constancy of perception whatever is the orientation of the white.

This is the view entertained by Yilmaz and its predecessor working on the geometry of color. But this state of fact can be considered the steady-state of the color vision space. The steady state can only be obtained when the observer see a unique color. When two colors are involved in the stimulus, the adaptation state of the observer can navigate in between them [27]. In the experiment we did [12] for measuring the color space of an individual watching a computer screen, we used a fixed color point *p* that cover the whole pixels of the display. Around that point we perform two sessions of minimum motion in a central visual area (approx. 4°), minimizing the strength of the modulation. Because adaptation state is maintain as fixed as possible, the direction of the luminous vector (estimated as the direction for which their is no motion with the minimum motion experiment, i.e. the photometric direction) can be measured precisely. Those vectors form a vector field that is sightly convergent showing a hyperbolic metric for the whole steady color space. This hyperbolic space is the multi-adapted steady space that represent the global extend of the color space of the observer when steadily adapted to each and every point of the color space. But from one point to the other, the adaptation may have changed and for sure have changed. Again this is the only way to have a fixed adaptation state of the observer and a corresponding multi-adapted global steady color space of the observer.

What happens when the observer leaves the steady state is the future question for a color vision science that may predict or explain how spectra are transformed into colors by the visual system. What is evident from the analysis here is that there is a kind of transformation of geometry from linear to hyperbolic.
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** The wishing conversion between spaces (a) Physical space is designed from the additive synthesis of colors that stipulate a vector addition rule. Notice that this is realistic from the rule of spectra operation because spectra add in a mixing. (b) Physiological space and (c) Perceptual space. Those spaces shared some common components but the relationship between them is still unknown. From physics point of view the decomposition with disciplines is about biophysics of vision and neurosciences of vision. But from a mathematical psychophysics, the decomposition is rather between physical space of light, physiological space of light and perceptual space of light. Those different spaces can be linked together through transformation on projective space of dimension three.</figcaption></figure>
Adaptation can be seen as a homomorphism between hyperbolic spaces having different luminous axis and/or different basis and/or different origins [23]. Using projective geometry of dimension three those transformations should be set up as homographies that change from linear to hyperbolic geometries and from hyperbolic to hyperbolic geometry for accounting adaptation.

There is still a long route for enabling the correspondence between spectra and colors. Displays have been designed on technological constraint and good compromise have been found to render the image perceptually similar to the real scene. But physiological space is different than perceptual space and the best color rendering is not a guarantee of the best safety in using. There is probably a room to modify images content in order to follow the geometry of physiology without perturbing much the appearance of the image.

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