# Origins of Hyperbolicity in Color Perception

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## Abstract

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## 1. Introduction

## 2. Yilmaz’s Experiments

#### 2.1. Notation and Nomenclature for Yilmaz’s Model

#### 2.2. Yilmaz’s Experiments

#### 2.2.1. The First Experiment

#### 2.2.2. The Second Experiment

#### 2.2.3. The Third Experiment

## 3. Recasting Yilmaz’s Model in a Mathematical Framework

- If $\Lambda \subset \mathbb{R}$ is the compact subset of $\mathbb{R}$ containing the visible wavelengths, typically $\Lambda =[380,780]$ nm, then a visible light can be identified with a finite energy non-negative function defined on $\Lambda $, i.e., an element of the space ${L}_{+}^{2}(\Lambda )=\{f:\Lambda \to {\mathbb{R}}^{+}\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\int}_{\Lambda}{\left|f\right|}^{2}<+\infty \}$;
- 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;
- $F,{F}^{\prime}$ will denote a visual stimulus provided by the visible light reflected by an object enlighted by the illuminant I or ${I}^{\prime}$, respectively. The object surface can be either the piece of white paper, and in that case we will write $W,{W}^{\prime}$, or a Munsell chip;
- $\tilde{R},\tilde{Y}$ 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.

#### 3.1. Coefficients from the First Experiment: The White Point Transformation

#### 3.2. Coefficients from the Second Experiment: The Red Point Transformation

#### 3.3. Coefficients from the Third Experiment: The Yellow Point Transformation

#### 3.4. Critical Issues in Yilmaz’s Model

## 4. Yilmaz’s Model and the Standard Formulation of Special Relativity

#### 4.1. Elements of Special Relativity

- 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.

- 3.
- the speed of light in vacuum has a constant value $c\in {\mathbb{R}}^{+}$ when measured in all inertial reference frames.

**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 $i=1,\cdots ,n$, then ${a}^{i}{b}_{i}:={\displaystyle \sum _{i=1}^{n}}{a}_{i}{b}_{i}.$ 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 $dx=(d{x}^{\mu})$, then the spacetime interval can be written as the (non positive-definite) quadratic form $d{s}^{2}=d{x}^{\mu}{\eta}_{\mu \nu}d{x}^{\nu}=d{x}^{t}\eta dx$, where $\eta =({\eta}_{\mu \nu})$ is the matrix $\eta =$ diag $(1,-1,-1,-1)$. The metric space $\mathcal{M}=({\mathbb{R}}^{4},\eta )$ is called Minkowski spacetime and $\eta $ is the matrix associated to the Minkowski quadratic form. The associated pseudo-norm, i.e., ${\parallel u\parallel}_{\mathcal{M}}^{2}={({u}^{0})}^{2}-[{({u}^{1})}^{2}+{({u}^{2})}^{2}+{({u}^{3})}^{2}]$ is called Minkowski norm of $u\in \mathcal{M}$.

#### 4.2. Similarities and Differences between Yilmaz’s Model and Special Relativity

- The Helson-Judd effect, see, e.g., [10], shows that human color perception experiences an incomplete adaptation to narrow-band illuminants, thus, in the previous table, the analogy between inertial frames and observers works only if they are adapted to broadband illuminants.
- While time t can be extended to the whole $\mathbb{R}$ with the identification of negative values of t as the ‘past’, a negative lightness is meaningless. So, only the upper part of the cone $\mathcal{C}$ makes sense in color perception. Moreover, and most importantly, this cone is not infinite: in fact, it is bounded from above by the glare limit defined by ${\gamma}_{max}$ and from below for two reasons: the first is the Purkinje effect [11] when we pass from photopic to scotopic vision via the mesopic range (in the photopic range the three retinal cones are activated, in the scotopic range only the retinal rods are, while in the mesopic both photoreceptors function simultaneously), and the second is the intensity threshold of the retinal rods. Thus, $\mathcal{C}$ is a truncated cone defined by the equation $\mathcal{C}=\{F=(\varphi ,\rho ,\gamma )\in [0,2\pi )\times {\mathbb{R}}^{+}\times [{\gamma}_{min},{\gamma}_{max}]\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}\rho <\Sigma \gamma \}$.
- While events in the Minkowski spacetime have four components, perceived colors have only three.

#### 4.3. The Issue of a Minkowski-Like Metric on $\mathcal{C}$

## 5. Relativistic Aberration and Yilmaz’s Third Experiment

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The color cone $\mathcal{C}$ for an ideal observer, no glare, nor visibility threshold is considered in this representation.

**Figure 5.**Depiction of Yilmaz’s first experiment with the notation established in Section 3.

**Figure 6.**Depiction of Yilmaz’s second experiment with the notation established in Section 3.

**Figure 7.**Depiction of Yilmaz’s third experiment with the notation established in Section 3.

Special Relativity | Yilmaz’s Color Perception Model |
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Homogeneous and isotropic spacetime | Homogeneous and isotropic color space |

Observer in an inertial frame | Observer adapted to a broadband illuminant |

Event $e=(t,\mathbf{x})\in {\mathbb{R}}^{4}$ | Perceived color $F=(\varphi ,\rho ,\gamma )\in \mathcal{C}$ |

Time coordinate $t\in \mathbb{R}$ | Lightness coordinate $\gamma \in {\mathbb{R}}^{+}$ |

Spatial coordinates $({x}^{1},{x}^{2},{x}^{3})\in {\mathbb{R}}^{3}$ | Chromatic coordinates $(\rho ,\varphi )\in {\mathbb{R}}^{+}\times [0,2\pi )$ |

Speed of light in vacuum c | Maximal perceived saturation $\Sigma $ |

Lorentz transformations (22) | Yilmaz transformations (13) |

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**MDPI and ACS Style**

Prencipe, N.; Garcin, V.; Provenzi, E.
Origins of Hyperbolicity in Color Perception. *J. Imaging* **2020**, *6*, 42.
https://doi.org/10.3390/jimaging6060042

**AMA Style**

Prencipe N, Garcin V, Provenzi E.
Origins of Hyperbolicity in Color Perception. *Journal of Imaging*. 2020; 6(6):42.
https://doi.org/10.3390/jimaging6060042

**Chicago/Turabian Style**

Prencipe, Nicoletta, Valérie Garcin, and Edoardo Provenzi.
2020. "Origins of Hyperbolicity in Color Perception" *Journal of Imaging* 6, no. 6: 42.
https://doi.org/10.3390/jimaging6060042