Color Theory

1. Color Spaces
2. Color Harmony
3. Color Psychology
4. References

1. Color Spaces

1.1. CIE 1931 Color Space

It was defined in 1931 by Commission Internationale de l'éclairage, International Commission on Illumination.

Color is measured based on the effective value of stimulus in three kinds of cone cell, $ [L, M, S] $. It can be tranformed to other represention by matrices. In reality the color matching functions are used for calculation.

1.1.1. CIE XYZ Color Space

\begin{align*} X &= \int L_{\mathrm{e}, \Omega, \lambda}(\lambda) \bar{x}(\lambda)\,\mathrm{d}\lambda, \\ Y &= \int L_{\mathrm{e}, \Omega, \lambda}(\lambda) \bar{y}(\lambda)\,\mathrm{d}\lambda, \\ Z &= \int L_{\mathrm{e}, \Omega, \lambda}(\lambda) \bar{z}(\lambda)\,\mathrm{d}\lambda \\ \end{align*}

where $ L_{\mathrm{e}, \Omega, \lambda} $ is the spectral radiance, $ \bar{x}, \bar{y}, \bar{z} $ are the color matching functions.

$ Y $ is the luminance, $ Z $ is quasi-equal to blue of CIE RGB, and $ X $ is the the mix of the mix of red, green, and blue. This space is a linear transformation of the LMS space.

This is the space in which the color can be added directly, thereby forming a convex subset of three dimensional space.

1.1.2. CIE xy Chromaticity Diagram

\begin{align*} x&=\frac{X}{X+Y+Z}, \\ y&=\frac{Y}{X+Y+Z}. \end{align*}

This is the projection of the $ X + Y + Z = 1 $ plane onto the $ XY $ plane.

1.1.3. Optimal Solid

The solids whose points have less energy than certain threshold.

1.2. RGB Color Model

There are various RGB color spaces depending on the primary colors, transfer function (gamma), and the white point. The primary colors are often certain shade of red, green, and blue.

RGB is a great way of describing color since the convex combination of the primaries covers as much gamut as possible.

1.2.1. sRGB

Standard RGB

HP and Microsoft proposed it in 1996, and it became the official standard of the International Electotechnical Commision (IEC).¹ It is the standard color space for the web, and the default color profile of images.

Roughly speaking, sRGB uses $ \gamma = 2.2 $. This means that the color value stored means the perceived intensity. In order to convert to the physical intensity, red, green, and blue value need to be each exponentiated by $ \gamma $.

If $ \gamma = 1 $ the color space is called linear, and the colors can be added directly.

1.3. HSV, HSL, and HSI

HSV or HSL are different ways of representing the RGB color space. The specific color space that HSV refers to depends on the being discussed.

The RGB color coordinates $ R, G, B $ are assumed to range from 0 to 1,

1.3.1. Value

Brightness

\[ V := \max(R, G, B) \]

1.3.2. Lightness

\[ L := \operatorname{mid}(R,G,B) \]

1.3.3. Intensity

\[ I := \operatorname{avg}(R, G, B) \]

1.3.4. Chroma

\[ C := \operatorname{range}(R, G, B) \]

1.3.5. Saturation

\[ S_{\rm HSV} := \begin{cases} 0 & \text{if } V=0,\\ \frac{C}{V} & \text{otherwise} \end{cases} \]

\[ S_{\rm HSL} := \begin{cases} 0 & \text{if $L=0$ or $L=1$,} \\ \frac{C}{2L} & \text{if $L \le 1/2$,} \\ \frac{C}{2-2L} & \text{if $L > 1/2$.}\end{cases} \]

1.3.6. Hue

\[ H := 60\times \begin{cases} 0, & \text{if }C=0 \\ \frac{\displaystyle G-B}{\displaystyle C}\mod 6, & \text{if }V=R \\ \frac{\displaystyle B-R}{\displaystyle C}+2, & \text{if }V=G \\ \frac{\displaystyle R-G}{\displaystyle C}+4, & \text{if }V=B \end{cases} \]

1.4. Y'CbCr

Another representation of RGB color space, in which $ Y' $ is the luma, and $ C_B, C_R $ are the blue-difference and red-difference chroma.

It is simply a linear transformation of the non-linear RGB space, where the luma represents the perceived brightness. The transformation is given by: \[ \begin{bmatrix} Y' \\ P_B \\ P_R \end{bmatrix} = \begin{bmatrix} K_R & K_G & K_B \\ -\frac{1}{2}\frac{K_R}{1 - K_B} & -\frac{1}{2}\frac{K_G}{1 - K_B} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2}\frac{K_G}{1-K_R} & -\frac{1}{2}\frac{K_B}{1-K_R} \\ \end{bmatrix} \begin{bmatrix} R' \\ G' \\ B' \end{bmatrix} \] where $ K_R, K_G, K_B $ are predefined constants that sum to 1. Y'PbPr is then digitized in order to obtain Y'CbCr.

It is known that human is insensitivity to the change in Cb and Cr compared to the change in Y'. Consequently, the video and JPEG compression can throw more information away in the chroma.

1.5. Munsell Color System

Created in the first decade of 20th century by Albert H. Munsell. It is the first system that uses hue, chroma, and value to organize the colors in the perceptually uniform space.

It was adopted by United States Department of Agriculture as an official color space for soil in 1930s, and still used to this day.

A color is described as 10PB 7/8, where 10PB is the hue, 7 is the value, and 8 is the chroma.

1.5.1. Hue

5 principal hues:

Red
Yellow
Green
Blue
Purple

and 5 intermediate hues:

YellowRed
GreenYellow
BlueGreen
PurpleBlue
RedPurple

are each subdivided into 10 steps, each step gets a interger with 5 being the center, forming the entire hue circle.

5R -> 10R = 0YR -> 5YR -> 10YR -> …

1.5.2. Chroma

It tells the purity of the color which is roughly related to saturation.

1.5.3. Value

The value is subdivided into 11 levels (0~10), and it tells the lightness with 10 being the brightest.

1.6. CIE LAB Color Space

Color space defined by the International Commision on Illumination (CIE) in 1976. $ L^* $ is the perceptual lightness, $ a^* $ and $ b^* $ are value for the four unique colors.

\begin{align*} L^\star &= 116f(Y/Y_{\mathrm{n}}) - 16, \\ a^\star &= 500(f(X/X_{\mathrm{n}}) - f(Y/Y_{\mathrm{n}})),\\ b^\star &= 200(f(Y/Y_{\mathrm{n}}) - f(Z/Z_{\mathrm{n}})) \end{align*}

where \[ f(t) = \begin{cases} \sqrt[3]{t} & t > \delta^3 \\ \frac{1}{3}t\delta^{-2} + \frac{4}{29} & \text{otherwise} \end{cases}, \quad \delta = \frac{6}{29} \] and $ X_{\mathrm{n}}, Y_{\mathrm{n}}, Z_{\mathrm{n}} $ are predefined numbers about 100.

This takes into account that human differentiate darker shades more easily that lighter ones, to match the perceived difference in colors.

1.7. Oklab Color Space

It was introduced by Björn Ottosson in December 2020.

This space is more perceptually uniform. The gamma correction is done on the LMS space, so that it respects the biology more precisely.

\begin{align*} x_i = M_{2, ij}\sqrt[3]{M_{1,jk}X_k} \end{align*}

where $ x_i $ are the coordinates $ L, a, b $, and $ X_k $ are the CIE XYZ coordinates. The matrices are predefined.

2. Color Harmony

Figure 1: Traditional RYB Color Wheel

2.1. Analogous Colors

Colors right next to the other on the color wheel.

2.2. Complementary Colors

Opposite colors in the color wheel.

2.3. Triadic Colors

The set of three colors that form a triangle on the color wheel.

3. Color Psychology

3.1. Red

Passion
Danger

3.2. Orange

Energy
Creative

3.3. Yellow

Happy
Active

3.4. Green

Nature
Healthy

3.5. Blue

Calm
Wisdom

3.6. Purple

Wealth
Ambition

3.7. Black

Death
Power

3.8. White

Purity
Peace

4. References

Footnotes:

IEC 61966-2-1:1999.