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# Projective Colorimetry

David Alleysson

► To cite this version:

David Alleysson. Projective Colorimetry. 2021. hal-04772685

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Distributed under a Creative Commons Attribution 4.0 International License
# Projective Colorimetry

DAVID ALLEYSSON

This manuscript was compiled on November 8, 2024

Here color spaces are constructed as three-dimensional projective spaces $E = P^3(\mathbb{R}) = \mathbb{P}(\mathbb{R}^4)$, with a Minkowski's quadratic form as projective invariant. The quadratic form provides a pseudo-metric for color space allowing the definition of distance between colors even visually far apart. The space $E$ is deduced from the real vector space of dimension four $\mathbb{R}^4$. A collinearity equivalence relation reduces the dimension to three and shapes color spaces as a pile of proportional hyperboloid surfaces of constant perceptual lightness. Color spaces constructed here are unified by considering them as different map representations of the same projective space. Adaptation can also be accounted for by a change in the projective frame or in the projective invariant. Vector space $\mathbb{R}^4$ is based on four spectral functions instead of three. The fourth function is interpreted as the projective direction of the projective frame. It can change with the observer's adaptation state or along neural visual pathways. This interpretation provides a novel relationship between physical, physiological, and perceptual color spaces. It can potentially link objective and subjective color measures through transformations in $E$.

http://dx.doi.org/XX.XXXXXX

## INTRODUCTION

Human observers measure colors in photopic conditions using three photoreceptors. They are called L, M and S for their corresponding sensitivities to light in the Long, Middle and Short wavelength range of the visible spectrum [1–3]. These sensitivities have been derived from the color matching experiment compensated for by the ocular media transfer function [4]. The color matching experiment provides three wavelength spectral functions [Fig. 3, 5][Fig. 2, 6]. They correspond to the gain of three monochromatic primaries used to match monochromatic lights in the visible domain. These functions are then transformed into CIERGB by accounting for the spectral efficiency function of the eyes, $V(\lambda)$, and then into CIEXYZ for being a positive coordinate system [7, 8], which constitutes the space for standard colorimetry. Color spaces LMS, CIERGB and CIEXYZ are related by linear transformations and correspond to the basis change of the same vector space representing metamerism space for human color vision.

Advanced colorimetry defines several other color spaces such as CIELAB, CIELUV. They integrate nonlinearities and adaptations found in color appearance phenomena. The choice was made to apply a non-linear adaptive transformation to each channel in CIEXYZ and to measure distance between colors as a Euclidean distance in the transformed coordinates (La*b* for instance). This measure of distance, however, is not suitable for large color differences. More sophisticated spaces have been designed, such as the CIECAM, to better accommodate adaptation.

These different color spaces of human observers can be con-

sidered different levels of visual representation. Color space LMS can be called physical space (or rather bio-physical space if preferred) because it relates to the relative positive absorption rate of light's photons by photoreceptors [9]. Color spaces CIERGB and CIEXYZ can be considered physiological spaces because they include the eye's spectral efficiency function. Perceptual spaces are CIELAB, CIELUV, and CIECAM.

Here, the approach is quite different because non-linearity is derived by a projection onto a quadratic surface directly from the metamerism color vector space without any transformation of coordinates. The metamerism vector space is based on a simple model of spectral functions suggested by Yilmaz [10–12]. Distance is calculated by the norm derived from the quadratic form that produces the projective surface and works for the whole color space even for colors that are far apart or close to the spectrum locus. Adaptation is given either by changing the coordinate system on the projection quadratic surface or by transforming the surface of projection. Using projective geometry of dimension three, $E = \mathbb{P}^3$, physical, physiological and perceptual spaces constructed here can be unified as a single projective space. Those which are related linearly or by a change of coordinates correspond to a change in the projective frame with a fixed invariant. Conversely, those with a transformed projective surface indicate a changed projective invariant.

Projective geometry was widely used in colorimetry [10, 13–15] but usually restricted to two dimensions $P^2$. It is supported by the fact that the color of a light doesn't change as the light level increases or decreases. Thus lights which differ only by

<footnote n="1">¹Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LPNC, 38000 Grenoble, France</footnote>

<footnote>*Corresponding author: David.Alleysson@email.univ-grenoble-alpes.fr</footnote>
their level belong to the same equivalence class for their col-
ors. From color space CIEXYZ a projective plane is constructed,
called the chromaticity diagram CIExy [13, 16, 17], as a central
projection onto the plane X+Y+Z=1. Any point of coordinates
(X, Y, Z), corresponding to the tristimuli values of a light, has its
colorfulness given by (x, y) = (X/(X + Y + Z), Y/(X + Y + Z))
coordinates in the chromaticity diagram.

Conversely, projective planes (and thus the chromaticity di-
agram) can be used as a projective subspace P<fnref n="2" /> to construct a
three-dimensional projective space P<fnref n="3" /> (P<fnref n="3" /> ⊃ P<fnref n="2" />). Many projec-
tive spaces can be constructed from a projective subspace. They
are defined by five points forming a projective frame<fnref n="1" /> [18], and
a projection π: R⁴ − {0} → P<fnref n="3" /> which relates the (quadruplet)
coordinates of points in R⁴ − {0} to the (triplet) coordinates of
points in P<fnref n="3" /> ≃ R<fnref n="3" /> ∪ {∞}<fnref n="2" />. The choice of a frame is arbitrary, but
the frame-projection pair completely defines a projective space
by a projective invariant [19].

In this paper, the formalism of projective geometry in three
dimensions is described. Application to the construction of
the chromaticity diagram and Yilmaz’s color solid is shown. A
spectral model based on Yilmaz’s basis functions is developed.
Modification of the geometry due to light adaptation is also
discussed.

PROJECTIVE COLOR SPACES

The space E = P<fnref n="3" />(R) ≃ R⁴ − {0} / ∼ is constructed on the
vector space R⁴ (deprived of the vector {0 = (0,0,0,0)} with
an equivalence relation ∼. The equivalence relation ∼ is the
collinearity relation and the origin of the projective space corre-
sponds to the origin of the vector space [18, 19].

Alternatively, the projective space E ≃ R<fnref n="3" /> ∪ {∞} is the set
of all projective lines in R⁴ (projective lines are lines passing
through 0), which is isomorphic to the three dimensional vector
space R<fnref n="3" /> added with a infinite plane ∞. Usual color spaces are
considered as three-dimensional affine subspaces of R⁴. To show
this, the first step is to add to R<fnref n="3" /> a homogeneous variable w ≠ 0:

$$
x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \in \mathbb{R}^3 \xrightarrow{\pi^{-1}} \hat{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ w \end{pmatrix} = \begin{pmatrix} x \\ w \end{pmatrix} \in \mathbb{R}^4, \quad (1)
$$

for which the collinearity equivalence relation ~ is written:

$$
\begin{align*}
& \hat{x}, \hat{y} \in \mathbb{R}^4 - \{0\}, \\
& \hat{x} \sim \hat{y} \Leftrightarrow \hat{x} = k\hat{y}, \quad k \in \mathbb{R} - \{0\}. \tag{2}
\end{align*}
$$

Two collinear vectors belong to the same projective line when
placed at the origin, they belong to the same equivalence class,
[ˆx] = [ˆy] [19]. Projective space P<fnref n="3" /> is represented in the three-
dimensional space R<fnref n="3" /> ∪ {∞} by setting the hyperplane w = 1
as the projection hyperplane which induces the following trans-

formation:

$$
\begin{align*}
\hat{x} \sim \hat{y} \Rightarrow \hat{y} &= \frac{\hat{x}}{w} = \begin{pmatrix} x_1/w \\ x_2/w \\ x_3/w \\ 1 \end{pmatrix} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ 1 \end{pmatrix} \in \mathbb{R}^4 - \{0\} \\
\xrightarrow{\pi} y &= \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} \in \mathbb{P}^3 \simeq \mathbb{R}^3 \cup \{\infty\}. \quad (3)
\end{align*}
$$

The vector ŷ is the vector chosen to be the representative
of the equivalence class [ŷ] and corresponds to a point in the
hyperplane P₁ = {x̂ = (x, w) ∈ ℝ⁴ | w = 1}. All the vector lines
belonging to the hyperplane P∞ = {x̂ = (x, w) ∈ ℝ⁴ | w = 0}
could not be represented as a point in P₁ in this affine map
representation, they belong to the infinite hyperplane.

Equations 1 and 3 provide a correspondence for constructing
a representation of the three-dimensional projective space P<fnref n="3" />
inside the vector space R<fnref n="3" />. As stated earlier we are interrested
on the relationship between the projective subspace P<fnref n="2" /> and the
projective space E = P<fnref n="3" />. By adding a linear relation to the four
variables that reduce the dimension, a projective subspace P<fnref n="2" /> is
defined as the intersection of two hyperplanes in R⁴. Through
the linear relation, the hyperplane P₁ is represented as a plane in
R<fnref n="3" /> which is the image of the projection plane. The hyperplane
P∞ is represented in R<fnref n="3" /> as a plane parallel to the projection plane
passing through zero.

**Example 1: The CIExy chromaticity diagram**

In this example, the space E is built on the vector space CIEXYZ<fnref n="3" />,
and we call x = (X, Y, Z) ∈ R<fnref n="3" /> the CIEXYZ tristimulus coordi-
nates of a light. The vector x̂ ∈ R⁴ is defined by x̂ = (X, Y, Z, w).
The projective subspace is obtained by adding a relation con-
necting the four variables, such as the linear form w − v¹x = 0.
For v = (1, 1, 1) the relation defines a projective hypersurface
being the hyperplane {x̂ | w = X + Y + Z} in R⁴. The equa-
tion w = v¹x = X + Y + Z is also the equation of the plane
orthogonal to the vector v and of level w in R<fnref n="3" />. The chromaticity
diagram CIExy is defined by the following projection:

$$
\hat{x} = \begin{bmatrix} X \\ Y \\ Z \\ X + Y + Z \end{bmatrix} \sim \hat{y} = \frac{\hat{x}}{w} \xrightarrow{\pi} y = \begin{bmatrix} \frac{X}{X+Y+Z} \\ \frac{Y}{X+Y+Z} \\ \frac{Z}{X+Y+Z} \end{bmatrix}, \quad (4)
$$

with vector y ∈ R<fnref n="3" /> is a representation of vector ŷ ∈ R⁴ belong-
ing to the same equivalence class than ˆx. Fixing the representa-
tion of E to the two hyperplanes P₁ and w − v¹x = 0, we fix the
projection surface in R<fnref n="3" /> by the plane orthogonal to v and of level
1. The figure 1 shows the projection of the spectrum locus (the
set of points x = (X, Y, Z), corresponding to monochromatic
lights) in the chromaticity diagram.

<footnote n="1">¹Here a projective frame $\mathcal{R}_a = (A, \vec{B}, \vec{C}, \vec{D}, \vec{E})$ is an ordered quintuplet. It is such that the first object $A$ is the coordinate of the origin of the projective space, the vectors $\vec{B}, \vec{C}, \vec{D}$ are the basis of the vector space, and $\vec{E}$ is the projection direction.</footnote>

<footnote n="2">²Here ∞ is an arbitrary symbol corresponding to the infinity hyperplane.
See [19][p. 211-212] for further explanations on the construction of P³.</footnote>

<footnote n="3">³Here vectors X, Y and Z are considered the canonical basis of R³ even if
they are not necessarily orthogonal for the spectral scalar product (see section
relationship with spectra).</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Spectral cone in CIEXYZ color space. (a) Spectral functions XYZ along the wavelength with a third of their sum W. (b) Spectral locus in color space XYZ (black curve) and its projection on the chromaticity diagram (magenta curve). The two curves define the same (deformed) conical surface as the union of all vectors from the origin to the curve. The chromaticity diagram is the projective plane, a projective subspace of E defined by $w - v^t x = 0$ with $v = [1, 1, 1]^t$. Defining a representation of E as the hyperplane $w = 1$, we show that the projective plane is the plane orthogonal to $v$ and of level $v^t x = 1 = w$ in $\mathbb{R}^3$, which contains the point W such that $\overrightarrow{OW} = [1, 1, 1]^t/3$. The projective frame is $\mathcal{R}_{XYZ}(O, \overrightarrow{O^X}, \overrightarrow{O^Y}, \overrightarrow{O^Z}, \overrightarrow{OW})$. These vectors have their correspondences as spectral functions in (a). The point O corresponds to zero spectral function.</figcaption></figure>

### Projection onto a quadratic surface

Rather than projecting onto a plane in $\mathbb{R}^3$, we can consider projecting onto a quadratic surface. In this case, we choose that the equivalence class representative belongs to a quadratic surface. For this, the relation that links the four variables is based on a quadratic form $\hat{q}(x, w) = w^2 - x^t M x = 0$. The relation defined a three dimensional quadratic form $q(x) = x^t M x = w^2$, with M is the metric of the projective surface in $\mathbb{R}^3$. If the signature of the eigenvalues of M is $(+, +, +)$ the surface is an ellipsoid while for the signature $(-, -, +)$ the surface is a hyperboloid of two sheets<fnref n="4" />. The case $(-, +, +)$ that corresponds to hyperboloid of one sheet is not considered here.

The vector $\hat{x} \in \mathbb{R}^4$ and $y \in \mathbb{R}^3 \cup \{\infty\}$ represented by the hypersurface $P_1$ in $\mathbb{R}^4$ are written:

$$ \hat{x} = \begin{bmatrix} x \\ \sqrt{x^t M x} \end{bmatrix} \xrightarrow{\pi} y = \frac{x}{\sqrt{x^t M x}}, \qquad (5) $$

which guarantee $y^t My = 1$ and that any point M such that $\overrightarrow{OM} = y$ belongs to the projective quadratic surface given by the metric M and of level 1 in $\mathbb{R}^3$. The entire space covered by choosing another value for $w$, such that $w = k, k > 0$ is a set of elliptic balls $E = \{y \in \mathbb{R}^3 | y^t My = k^2\}$ of radius $k$ if the signature is $(+, +, +)$. Or the interior of the isotropic cone $C = \{y | y^t My > 0\}$ whose envelope is $\delta C = \{y | y^t My = 0\}$ and covered with hyperboloid surfaces $H_k = \{y | y^t My = k^2\}$ of radius $k$ if signature is $(-, -, +)$ [19, 20].

There is considerable literature describing color space as being a hyperbolic metric space [14, 15, 21–26] and the geometrical properties of this space have recently been evaluated experimentally on human observers [27]. It was shown that the change of Weber ratio for different color can be compensated for by the use of the individual measured hyperbolic norm [27][Table 2 and Supplementary results S2]. For those reasons we will focus on hyperbolic projective geometry for the remaining of the paper and show how projective geometry of dimension three formalizes it.

### Example 2: Yilmaz’s color solid

Yilmaz [10–12, 26, 28] had developed a perceptual model for an object’s color relative to the illuminant. Yilmaz’s color solid is delimited by a cone of revolution symmetric around the illuminant axis. Because object’s radiance function is lower than illuminant radiance function (object’s reflectance is bound between 0 and 1), color domain is also limited in intensity. The set of all possible reflectance spectra is then included in the color solid defined by the interior of the cone limited by the illuminant vector.

Let’s consider the canonical basis $\psi = (\psi_1, \psi_2, \psi_3)$ of $\mathbb{R}^3$ given by $\psi_1 = (1, 0, 0)$, $\psi_2 = (0, 1, 0)$ and $\psi_3 = (0, 0, 1)$ in which any tristimulus value $x = (x_1, x_2, x_3)$ of a light is defined by $x = x_1\psi_1 + x_2\psi_2 + x_3\psi_3$. In this section, the illuminant radiant vector is fixed at $\psi_3$; adaptation will be discussed later. Spectral cone is the isotrope surface of a quadratic form $q(x) = x^t J x$, which is the locus of monochromatic lights (quadratic form is associated with a bilinear polar form $\phi(x, y) = x^t J y$ and a norm $\|x\| = \sqrt{q(x)}$). Tristimulus values are restricted to the interior of the spectral cone of equation $q(x) > 0$ because no light can be seen equal or more saturated than monochromatic lights. Projective space E corresponding to the spectral cone oriented symmetrically around $\psi_3$ is given by:

$$ \left\{ \begin{array}{l} \hat{q}(x, w) = w^2 - x^t J x \\ \mathcal{R} = (0, \psi_1, \psi_2, \psi_3) \\ \pi(\hat{x}) = \frac{x}{\|x\|} = \frac{x}{\sqrt{x^t J x}} \end{array} \right. \qquad J = \frac{1}{2} \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{bmatrix}. \qquad (6) $$

The fifth vector of the projective frame $\mathcal{R}$ is intentionally chosen to correspond to the axis of symmetry of the spectral cone, here $\psi_3$. It is called the projective direction of the projective space. Figure 2 shows the configuration described here.

Yilmaz associated the perceptual attributes of color with the spectral cone, namely hue $\theta = \text{atan}(x_2, x_1)$, saturation $\sigma = (x_1^2 + x_2^2)^{1/2}/x_3$, and intensity $x_3$. The conical surface $\delta C$ is the maximum saturation limit $\Sigma$ for which $q(x) = 0$ ($\Sigma = \sqrt{2}$ in our formulation). The projective surface is the unit hyperboloid $H_1 = \{x | q(x) = 1\}$ which serves as a hyperboloid model for hyperbolic geometry. This surface can be considered as the surface of unitary perceived lightness. In Yilmaz’s color solid, the coloring of a point p of coordinates $(x_1, x_2, x_3)$ is given either by the position of the projective line in the unit hyperboloid (point M) or by the position on the disk (point m). The disk is the intersection of the plane $P_1$ with the interior of the cone and corresponds to the Klein disk $K$ [26].

Let’s fix another projective frame by choosing five points. The point of origin $O = (0, 0, 0)$, three other points $\chi_i, i = 1..3$ corresponding to three vectors belonging to the conical surface. The fifth point W is defined by $(\vec{O}\chi_1 + \vec{O}\chi_2 + \vec{O}\chi_3)/\sqrt{3} =$

<footnote n="4">⁴The other possible signatures of the eigenvalues $(-, -)$ and $(+, +)$ are equivalent to the former respectively $(+, +)$ and $(-, -)$ because one can always write a quadratic form and its negative for a given quadratic surface. [p. 148, 19]</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 2. Yilmaz's color solid.** The volume inscribed into the cone is the locus of the color of objects seen under the white point W representing the reflectance of the perfect diffuser or the radiance of the illuminant. The conical surface $\delta C$ represents the infinite surface for E, defined for $w = 0$. While the unitary hyperboloid $H_1$ represents the projective surface for $w = 1$. Using $H_1$ as the hyperboloid model for hyperbolic geometry, the Klein disk $\mathbb{K}$ is obtained as the intersection of the plane $P_1$ with the cone. Any point $p$ corresponding to a color tristimulus of an object is coded as a point $m$ in the Klein disk and a point $M$ in the hyperboloid model. Its colorfulness is coded by the deviation apart from the white point W.</figcaption></figure>

$\overrightarrow{OW}$. The same projective space in this projective frame writes:

$$ 
\left\{ 
\begin{aligned}
\hat{q}_{\chi}(x, w) &= w^2 - x^t \mathcal{J}_{\chi} x \\
\mathcal{R}_{\chi} &= (0, \chi_1, \chi_2, \chi_3, W) \\
\pi_{\chi}(\hat{x}) &= \frac{x}{\sqrt{x^t \mathcal{J}_{\chi} x}}
\end{aligned}
\right. 
\quad \mathcal{J}_{\chi} = \frac{1}{2} \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} . \quad (7)
$$

This latter projective frame is closer to the one described in Example 1 because the fifth vector in the frame is the sum of the three others multiplied by a constant.

## RELATIONSHIP WITH SPECTRA

Many authors have proposed that the space of light spectral functions can be formalized as a Hilbert<fnref n="5" /> vector space [21] called here $\vec{F}$. Vector color spaces are considered three-dimensional vector subspaces [29] given by choosing three functions that form a Hilbert basis.

Hilbert color space considers a spectral function as an infinite vector. Spectral functions correspond either to the radiance spectrum of lights or the spectrum sensitivity of a human color mechanism along a variable $\zeta$. Variable $\zeta$ is related to the hue by $\zeta = \theta/2\pi$, with a given range in the visible domain $\zeta \in [-1/2, 1/2]$. Hue $\theta \in [-\pi, \pi]$ is related to wavelength $\lambda \in [\lambda_m, \lambda_M]$ by $\theta = 2\pi(\lambda - \lambda_m)/(\lambda_M - \lambda_m) - \pi$ with $\lambda_m$ and $\lambda_M$ are respectively the lower and upper limits of the visible domain. A spectral function $f: [-\frac{1}{2}, \frac{1}{2}] \to \mathbb{R}$ can be represented as a vector $|f\rangle$ belonging to the real vector space of infinite dimension by:

$$ 
\begin{aligned}
|f\rangle &= (f(-1/2), \dots, f(0), \dots, f(1/2)) = [f(\zeta_0)] \in \vec{F} \\
&= \left[ \int_{\zeta} \delta_{\zeta_0} f(\zeta) d\zeta \right] = (|\delta_{\zeta_0} \rangle \langle \delta_{\zeta_0}|) |f\rangle,
\end{aligned} 
\quad (8)
$$

where $\delta_{\zeta_0} = \delta(\zeta - \zeta_0)$ is the Dirac distribution shifted to $\zeta_0$. The spectral sampling operator $|\delta_{\zeta_0}\rangle\langle\delta_{\zeta_0}|$ is the orthogonal projection operator of the Hilbert space [30]. The scalar product is given by $\langle f|g\rangle = \int_{\zeta} f(\zeta)g(\zeta)d\zeta = \langle f|\delta_{\zeta_0}\rangle\langle\delta_{\zeta_0}|g\rangle$. The norm is induced by the scalar product, $||f|| = \sqrt{\langle f|f\rangle}$.

Notice that the scalar product above is idealized and seems arbitrary. Alternatively, a more sophisticated scalar product of kind $\int_{\zeta} A(\zeta)f(\zeta)g(\zeta)d\zeta$, in which $A(\zeta)$ represents a weighted spectrum function [22, 31], can be used to account for the elliptical shape of the spectrum locus in Figure 1. In projective geometry $A(\zeta)$ can be interpreted as a modification of the observer's adaptation state (and the quadratic form) along $\zeta$. But this modification may only apply to physiological measurements. It is unlikely that it will apply to the color perception space, for which the idealized scalar product would be used. See section Discussion for more details.

## Canonical vector basis for E

Suppose, as Yilmaz did, that the vector subspace in which E is constructed, is given by a Hilbert basis composed by the three first components of the Fourier series. We called this basis $\langle\psi|$ and defined it as a matrix composed by a vector of bra functions<fnref n="6" /> as follows:

$$ \langle\psi| = \begin{bmatrix} \langle\psi_1| \\ \langle\psi_2| \\ \langle\psi_3| \end{bmatrix} = \begin{bmatrix} \langle\sqrt{2} \cos 2\pi\zeta| \\ \langle\sqrt{2} \sin 2\pi\zeta| \\ \langle 1| \end{bmatrix} . \quad (9) $$

The vector space spanned by this basis corresponds to periodic and band-limited functions such that any vector writes:

$$ |g\rangle = g_1|\psi_1\rangle + g_2|\psi_2\rangle + g_3|\psi_3\rangle = |\psi\rangle\langle\psi|g\rangle. $$

The operator $|\psi\rangle\langle\psi|$ is the orthogonal projection operator for the three-dimensional subspace of the Hilbert space. Any vector $|f\rangle$ can be approximated as a vector $|g\rangle$ belonging to the vector space defined by $\langle\psi|$ by the following operator called Cohen's R matrix [29]:

$$ 
\begin{aligned}
\mathrm{R:} \quad \vec{F} &\to \vec{F} \\
|f\rangle &\mapsto |g\rangle = \mathrm{R}|f\rangle = |\psi\rangle (\langle\psi|\psi\rangle)^{-1} \langle\psi|f\rangle.
\end{aligned} 
\quad (10) $$

In our case, the operator $\mathrm{R}$ can be simplified as $\mathrm{R} = |\psi\rangle\langle\psi|$ because the functions $\langle\psi_i|$ are orthogonal functions $\langle\psi|\psi\rangle = \mathbb{I}$, $\mathbb{I}$ being the identity matrix. $\mathrm{R}$ is a projector because idempotent, $\mathrm{R}\mathrm{R} = \mathrm{R}^2 = \mathrm{R}$. Its rank is 3, so it leaves $\infty - 3$ dimensions invisible, projecting onto 0 called the metameric black [17] (see Figure 3(a) for an illustration of spectral approximation by operator $\mathrm{R}$ and an example of metameric black).

## Metamerism projection

The operator $\mathrm{R}$ defines an equivalence relation $\approx^R$ between spectra, which is not collinearity but projection identity, called metamerism. Two vectors $|f\rangle$ and $|g\rangle$ are equivalent if they project onto the same point or belong to the same equivalence class according to the operator $\mathrm{R}$. This is written:

$$ |f\rangle \stackrel{R}{\approx} |g\rangle \iff \mathrm{R}|f\rangle = \mathrm{R}|g\rangle. \quad (11) $$

The subspace defined by the image of spectral functions through $\mathrm{R}$ is isomorphic to Euclidean space $\mathbb{R}^3$ because the

<footnote n="5">⁵For a Hilbert space, the space of functions should be considered as the space of square integrable functions which is the case for vision because high energy functions would burned the eye far below the square integrable constraint (i.e. direct sun or laser view). Convergence of Cauchy series is not considered here.</footnote>

<footnote n="6">⁶We use the braket notation [30] here to distinguish between light spectra as ket and human sensitivity spectra as bra.</footnote>
scalar product is the standard scalar product of $\mathbb{R}^3$ inherited from the Hilbert space scalar product. Let us defined the linear application $\alpha$ from $\vec{F}$ to $\mathbb{R}^3$ as follows:

$$ \alpha: \vec{F} \to \mathbb{R}^3 \qquad (12) \\ |f\rangle \mapsto f = \alpha(|f\rangle) = \langle\psi|f\rangle. $$

We have $f \cdot g \stackrel{def}{=} f^t g = \alpha(|f\rangle)^t \alpha(|g\rangle) = \langle f|\psi\rangle \langle\psi|g\rangle = \langle f|g\rangle$.

## Projective frames of E

It is worth noting that the image through $\alpha$ of the basis $|\delta_{\xi_0}\rangle$ of $\vec{F}$ is the set of vectors covering a conical surface:

$$ \alpha(|\delta_{\xi_0}\rangle) = \langle\psi|\delta_{\xi_0}\rangle = \begin{bmatrix} \psi_1(-1/2), \dots, \psi_1(1/2) \\ \psi_2(-1/2), \dots, \psi_2(1/2) \\ \psi_3(-1/2), \dots, \psi_3(1/2) \end{bmatrix} = \delta C. $$

This conical surface is exactly the same as the one described in Example 2. Indeed, we have $\psi = \alpha(|\psi\rangle) = \mathbb{I}$ and $\psi$ is the same basis as in Example 2. The cone swept by the monochromatic light through the linear application $\alpha$ corresponds exactly to the spectral cone defined by the projective space E of Example 2 in the reference frame $\mathcal{R}$ (Equation 6) with the fourth function or projective direction given by $\langle\psi_3|$. We can now deduce what would be the corresponding functions $\langle\chi|$ in the second projective frame $\mathcal{R}_\chi$ (Equation 7). We have:

$$ \langle\chi| = P\langle\psi| = \frac{1}{\sqrt{3}} \begin{bmatrix} \langle 1 + 2 \cos 2\pi(\xi + 1/3) | \\ \langle 1 + 2 \cos 2\pi\xi | \\ \langle 1 + 2 \cos 2\pi(\xi - 1/3) | \end{bmatrix}, \quad (13) $$

$$ \begin{align} P &= \frac{1}{\sqrt{3}} \begin{bmatrix} \alpha(|\delta_{-\frac{1}{3}}\rangle) & \alpha(|\delta_0\rangle) & \alpha(|\delta_{\frac{1}{3}}\rangle) \end{bmatrix} \nonumber \\ &= \frac{1}{\sqrt{3}} \begin{bmatrix} \psi(-\frac{1}{3}) & \psi(0) & \psi(\frac{1}{3}) \end{bmatrix} \nonumber \\ &= \frac{1}{\sqrt{3}} \begin{bmatrix} -\frac{1}{\sqrt{2}} & \sqrt{2} & -\frac{1}{\sqrt{2}} \\ -\frac{\sqrt{6}}{2} & 0 & \frac{\sqrt{6}}{2} \\ 1 & 1 & 1 \end{bmatrix}. \tag{14} \end{align} $$

The fourth function corresponds to $(1/\sqrt{3}) \sum_i \langle\chi_i|$ and is a constant function of $\xi$. The rotation matrix $P$ provides a relation on the metrics of projective surfaces expressed in either frame, $\mathbb{J}_\chi = P^t \mathbb{J} P$.

## Physical space

Physical space<fnref n="7" /> spectral functions should be positive since they relate to photon absorption. But $\langle\psi|$ and $\langle\chi|$ are not positive and cannot be physical spectral functions. Moreover they should be correlated to maximize the absorption of photons regardless of their wavelength in the visible spectrum. The physical basis

functions called $\langle\varphi|$ are defined by:

$$ \langle\varphi| = \Lambda\langle\chi| = \frac{2}{\sqrt{3}} \begin{bmatrix} \langle 1 + \cos 2\pi(\xi + 1/3) | \\ \langle 1 + \cos 2\pi\xi | \\ \langle 1 + \cos 2\pi(\xi - 1/3) | \end{bmatrix}, \quad (15) \\ \Lambda = \frac{1}{3} \begin{bmatrix} 4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4 \end{bmatrix}. $$

These positive functions correspond to LMS function but in Yilmaz color solid. The associated projective frame is $\mathcal{R}_\varphi(0, \varphi_1, \varphi_2, \varphi_3, 1/(2\sqrt{3}) \sum_i \varphi_i)$ with the fourth function defined as $1/(2\sqrt{3}) \sum_i \langle\varphi_i|$ is a constant function of $\xi$. It corresponds to a change of frame with correlation/coupling applied to the orthogonal axis $\chi_i$. The projection $\pi_\varphi$ is written $\pi_\varphi(\hat{x}) = x / \sqrt{x^t \mathbb{J}_\varphi x}$, with $\mathbb{J}_\varphi = \Lambda P^t \mathbb{J} P \Lambda$.

The three different frames $\mathcal{R}_\varphi$, $\mathcal{R}_\chi$ and $\mathcal{R}$ and their associated projections correspond respectively to the physical, physiological and perceptual color space based on Yilmaz's color solid.

## Barycentric coordinates

A function $f(\xi)$ or equivalently a vector $|f\rangle$ in $\vec{F}$ can be seen as a curve $\gamma[f](\xi)$ in $\mathbb{R}^3$ defined as follows:

$$ \begin{align} \gamma[f]: [-\frac{1}{2}, \frac{1}{2}] &\to \mathbb{R}^3 \tag{16} \\ \xi &\mapsto \gamma[f](\xi) = \alpha(|\delta_{\xi_0}\rangle\langle\delta_{\xi_0}|f\rangle) \nonumber \\ &= \langle\psi|\delta_{\xi_0}\rangle\langle\delta_{\xi_0}|f\rangle \nonumber \\ &= \begin{bmatrix} \langle\psi_1(\xi)f(\xi)| \\ \langle\psi_2(\xi)f(\xi)| \\ \langle\psi_3(\xi)f(\xi)| \end{bmatrix} \nonumber \end{align} $$

By definition, the curve is inscribed into the conical surface because it corresponds to point multiplication of the function $\psi_i(\xi)$ by the value of the function $f(\xi)$. Scalar product in $\vec{F}$ can be defined as barycentric coordinate [32] of the points of the curve, $\langle\psi|f\rangle = \int_{\xi} \gamma[f](\xi)d\xi$. Barycentric coordinates make a direct correspondence between curves lying into the conical surface and vectors in $\mathbb{R}^3$ (Figure 3).

## Example 3: The double cone of color perception

Transformation between $\langle\varphi|$ and $\langle\chi|$ can also be seen as a translation instead of decorrelation. We have:

$$ \langle\chi| = \langle\varphi| - \frac{1}{\sqrt{3}}v \quad (17) $$

$$ \langle\psi| = P\langle\varphi| - \mathbb{I} $$

where $\mathbb{I}$ is the unit vector aligned to the third dimension $\psi_3$.

The translation can be interpreted as a change of the origin of projection. In basis $\psi$ the translation is $-\mathbb{I} = -\psi_3$, meaning that one can see the projective space E as having origin $-\mathbb{I}$ instead of O. It is the same to see points in $\varphi$ from 0 than points in $\chi$ from -1. In the hyperboloid model of hyperbolic geometry, projection from $-\mathbb{I}$ onto the plane $\psi_3 = 0$ is the definition of the Poincaré disk. An illustration of those different concepts is presented in Figure 3.

<footnote n="7">⁷Call it bio-physical if preferred, because it is the image of physical light by cone photoreceptors.</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 3. The double cone of color perception.** (a) Display of the different basis functions of variable $\zeta$ that represents the visible domain in the range [-1/2, 1/2]. A radiant spectrum $|f\rangle$ of a light along with its approximation $|g\rangle$ by basis functions $\langle\psi|$ is also shown. Their difference, $|f - g\rangle$, projects onto the metameric back. (b) The invariant of color perception for a fixed adaptation state of the observer is the envelope of the double cone (red surface) of equation $x^lJx = 0$ in the base ($\psi_1, \psi_2, \psi_3$). A spectral functions $f(\zeta)$ can be drawn as a curve on the double cone surface with the application $\gamma[f]$. The barycentric coordinate of a curve is a vector (the vector $\varphi_1$ corresponds to the barycentric coordinate of the green curve $\gamma[\varphi_1](\zeta)$, $\chi_1$ corresponds to the red curve and $\psi_1$ to the black curve). Green cone represents the physical space that corresponds to light spectrum sampling using $\langle\varphi|$. Its aperture is small because of the correlation between the positive sensitivity functions $\langle\varphi|$. The unit hyperboloid (not shown) of the double cone is the hyperboloid model of the Poincaré disk when considering the projection from the point $-1$.</figcaption></figure>

## COLOR ADAPTATION

So far, Yilmaz's color solid is only considered for a fixed illumination of the scene whose illuminant vector is equal to $\psi_3$, corresponding to constant spectrum function. Yilmaz [10–12, 26] had described a situation where the vector basis is fixed and only the projection direction changes with the illuminant. Let

<figure><img src="image_6.png" /><figcaption>**Fig. 4. Adaptation.** (a) When the illumination suddenly changes at $W'$, another projective frame whose projection axis is aligned with $W'$ is considered. An instantaneous hyperbolic rotation along the unit hyperboloid realizes a correspondence between the two projective frames. It adjusts the coordinates of the points, providing an instantaneous representation of perceptual constancy. A typical example of such adaptation is illustrated with simultaneous contrast (c). (b) When the observer is held under the updated illumination, the four axes of the projective frame can change. This is a consequence of the change in the sensitivities of photoreceptors and/or post-receptoral mechanisms. In that case the quadratic form of the projective space is changed too. As a result of the slow change in the four axes, a color after effect emerges (d).</figcaption></figure>

us call $y$ the coordinates of the updated illuminant on basis $\psi$. Yilmaz showed that the transformation $\Omega$ that changes the point $W(x)$ into $W'(y)$ is given by [26]:

$$ \overrightarrow{OW} = (0, 0, 1) = x, \quad \overrightarrow{OW'} = y, \quad y = \Omega x, \\ \Omega = k' R_{\zeta} H_s R_{-\zeta}, \qquad (18) $$

$$ R_{\zeta} = \begin{bmatrix} C & -S & 0 \\ S & C & 0 \\ 0 & 0 & 1 \end{bmatrix}, H_s = \begin{bmatrix} Ch & 0 & \sqrt{2}Sh \\ 0 & 1 & 0 \\ \frac{1}{\sqrt{2}}Sh & 0 & Ch \end{bmatrix}, $$

$$ C = \cos 2\pi\zeta, \quad S = \sin 2\pi\zeta, \quad Ch = \cosh s, \quad Sh = \sinh s, $$

$$ k' = \sqrt{y^l J y}, \quad \zeta = \frac{1}{2\pi} \tan \frac{y_2}{y_1}, \quad s = \operatorname{arctanh} \frac{\sqrt{y_1^2 + y_2^2}}{\sqrt{2}y_3}, $$

where $s = \operatorname{arctanh}(\sigma/\sqrt{2})$ is called saturability. When saturation $\sigma$ is limited by the maximum saturation $\sigma < \Sigma = \sqrt{2}$ in our formulation, saturability is limited to infinity which is a better variable to describe the envelop of the spectral cone as a projective infinite surface [26].

The transformation (Equation 18) is a Lorentz transformation that leaves the quadratic form and the unit hyperboloid invariant. However, it changes the projective direction in the projective frame according to the hyperbolic rotation $\Omega$. This rotation aligns the old illuminant $W(\psi_3)$ with the new illuminant $W'(y)$ (Figure 4(a)). This kind of adaptation typically corresponds to what is called local color contrast and is likely to be defined locally at the surface of the retina (Figure 4(c)). This
transformation can be called functional because it is instantaneous or at least below the time scale needed to be conscious. Indeed in local color contrast (Figure 4(c)), the observer does not see the time course of adaptation of the two center circles induced by the two surrounding circles.

When the observer is immersed in the new illumination for a long period of time, one can think that the whole projective frame has moved. A general transformation would allow a change of origin, vector space and projective direction, thereby changing the projective quadratic form. In that case, a three-dimensional rotation and/or scaling are applied, transforming the cone symmetric to W into another cone symmetric to W' (Figure 4(b)). The transformation can be defined as follows:

$$T = k'[u \ v \ i] \qquad (19)$$

$$k' = \sqrt{y^T y}, \quad i = \overrightarrow{OW'}/k', \quad v = \frac{\psi_1 \wedge i}{\|\psi_1 \wedge i\|}, \quad u = i \wedge v,$$

with ∧ is the vector product in $\mathbb{R}^3$.

Figure 4(d) is a typical example of such adaptation. Observers are asked to fix on the left cross in the middle of the four colored squares for a few seconds (typically 10s) and then move their gaze toward the right cross. There are then four after-effect squares on the right side filled with the opponent colors of the left configuration [33]. This second kind of adaptation transformation is slower than the first. This is probably because it implies a change in the photoreceptor and/or post-receptoral sensitivities and consequently the projective invariant. Observers can notice this color adaptation. It becomes conscious as the after effect decays slowly.

## DISCUSSION

Considering color spaces as projective spaces of dimension three $E = \mathbb{P}^3$ offers many perspectives on color representation and transformation, as illustrated by the few examples above. Human color vision can be considered a projective space of dimension three with Minkowski's quadratic form as projective invariant. This is in accordance with Yilmaz's color solid and a recent experimental finding which shows that surfaces of constant perceived lightness are hyperboloids sheets [27]. In human vision, color adaptation has two sides. Both are spatially local and require frame modification. Instantaneous adaptation modifies the projective direction only, leaving the spectral cone and unitary hyperboloid fixed. Long term adaptation is a general transformation of the projective space, with a modification of the quadratic form. This is induced by a change in the visual system sensitivities. The difference between objective and subjective color measurement lies in color adaptation. The two measures can be linked by a homography in $E$.

In this view, it remains to explain why Yilmaz's color solid is an ideal symmetrical cone (Figure 2 and Figure 3) whereas measurements on a human observer show a deformed cone cut by the purple plane (Figure 1). There is no objective reason to believe that the space of perceived colors is not closed and circular. The reason for this is that purple colors are also perceived as other colors, and all colors in a scene can be ordered symmetrically around white. Discrepancy should then be found in the manner in which spectral sensitivity functions are measured. An explanation could be that the color matching experiment is not based upon an observer's fixed adaptation state. In that case, the resulting functions would correspond to color matching under different adaptation states of the observer (and consequently different quadratic forms) for each tested wavelength, breaking

the cone's symmetry. The match on the two border sides of the visible spectrum would then be for two distinct observers' internal states. The metric would not be the same on either side, producing the purple plane.

Projective space $\mathbb{P}^3$ can be decomposed as three projective lines $\mathbb{P}^3 = \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. A projective line is isomorphic to a circle if considering the antipodal relation of equivalence instead of collinearity [19], $\mathbb{P}^1 \simeq \mathbb{S}^1 / \{x, -x\}$. In addition, one of the projective lines must be isomorphic to a hyperbola, since the projective surface is a hyperboloid with constant perceived lightness. The last projective line is the intensity that fixed the level of the hyperboloid. The three dimensional projective space of color can be written $E = S^1 \times \mathbb{H}^1 \times \mathbb{P}^1$, where $S^1$ is parametrized by an angle, the hue $2\pi\zeta$, $\mathbb{H}^1$ is parametrized by a hyperbolic angle, the saturability $s$ and finally $\mathbb{P}^1$ is parametrized by a factor $k$ which is the perceived intensity of a light. In this conception, color vision is a succession of two projections, orthogonal then central. The first is the metameric projection that construct an orthogonal subspace of the infinite dimensional space of light spectral functions. The second is a decomposition into colorfulness (hue and saturability) and intensity.

The first three Fourier basis functions $\langle\psi|$ which are used to construct the metameric subspace tend to be the only functions with the required properties. They transform into positive functions $\langle\varphi|$ when correlated, providing fundamental sensitivity for the visual system. They form an orthogonal basis of $\mathbb{R}^3$, allowing to conceive the color space as an orthogonal subspace of spectral functions. They are periodic, producing a perfectly symmetric cone (without being cut by the purple plane). And they can design a hyperboloid projective surface (belonging to the isotropic surface of the quadratic form for $\langle\chi|$). This surface can be used as the hyperboloid model, showing that colorfulness can be understood as coordinates on the Klein or Poincaré disk. They are also plausible functions for the color matching experiment (for $\langle\chi|$) because when one function is at its maximum equal to one, the other two cross at zero, similarly to Thornton's primary wavelength colors [34]. Fourier basis functions would favor undulatory light conception for color vision. As Yilmaz said, the visual system would be a poor Fourier analyzer if it were restricted to only three components. However, due to adaptation, light representation changes depending on the scene and illumination configuration.

## CONCLUSION

Colorimetry is widely used in industry and in multimedia technologies. In order to match human color vision, colorimetry objective spectral sensitivity functions are derived from the color matching experiment. Considering that human color measurement is subjective and adaptive, complex transformations are required to determine color appearance based on color matching. Here, the three-dimensional color space is defined as a projective space instead of a vector space. Using projective geometry, the human color spaces are unified as different representations of the same projective space. The formalism also accounts for the change of representation induced by adaptation to light.

The fourth vector dimension implied by the projective space of dimension three is the key to modulating the three-dimensional color vector space with observer adaptation. Nonlinearity in color processing by the visual system is resumed by the choice of a quadratic form as projective invariant. While the concept of Hilbert space would favor an Euclidean geometry for three-dimensional color space, the visual system operates rather
on Minkowski's geometry for light decomposition into intensity and colorfulness. Among the unification of the human-based color space there is also a potential unification of the description of colors in terms of lightness or brightness. This would then be related by a change of the color vector norm from Euclidean norm to Hyperbolic norm. In the same way, the eye's spectral efficiency function matches a yellowish color far apart from the apparent white direction. Here the visual system can perform a projective transformation along the visual pathway. This makes the appearance aligned with the white whereas physiology measurement is aligned to the eye's efficiency function.

If one generalizes the concept of projective geometry for color vision to all the general concepts of human cognition, this study recalls previous works that described hyperbolic geometry as a general concept for human vision [14, 25, 35–38]. In this view, cognition would be a reduced subspace of the infinitely wide physical space. Projecting to a subspace defines a projective infinite, a limit, determined by neural processing efficiency (for instance, the spectral cone in color vision or the speed limit in motion perception). Nevertheless, this limit is not fixed everywhere, but can change based on human adaptation.

**Funding.** This study have been conducted with laboratory proper funding.

**Acknowledgment.** This paper is dedicated to Denis Bichsel and Jeanny Hérault for their time and effort spent on the author's education.

**Disclosures.** The author declare no conflicts of interest.

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