Essential Color Science

The retina contains specialized cells known as rods and cells. Rods react mainly to light intensity, while cones react to certain wavelengths in different intensities.

Humans with normal thrichromatic vision¹ contain three types of cone cells, named long (L), medium (M) and short (S) after their peak response wavelengths. These responses don't correspond directly to colors as we normally understand, also depending on the encoding by the visual cortex and brain, and determined through experimentation. The discrete L,M,S responses to a certain light spectrum stimulus are known as tristimulus values.

Given the light spectrum (an infinite-dimensional space) is projected to tristimulus values (3-dimensional), different light spectra may map to the same tristimulus. These indistinguishable light spectra are known as metamers.

Hermann Graßmann (Grassmann) presented in "On the Theory of Color Mixing" (1853)² an algebra to describe light mixing, obeying the following laws:

Additivity

Given lights \(\{a, b, z\}\), if \(a = b\) (metamers), then \(a + z = b + z\) (also metamers).

Proportionality

Given lights \(\{a, b\}\) and a constant \(m\), if \(a = b\) (metamers), then \(a * m = b * m\) (also metamers).

Transitivity

Given lights \(\{a,b,c\}\), if \(a = b\) and \(b = c\), then \(a = b = c\) (all colors are metameric).

Future color experiments were based on this theory, performed by Hermann von Helmholtz and James Clerk Maxwell in the 1850's, and the experiments by Wright (1928) and Guild (1931).

In these experiments, an observer sees a bipartite white screen through an aperture, onto which the light of different lamps will be projected. At one side, a target light \(T\) generates a certain color impression in the observer, while another side is illuminated by three monochromatic lights \(R, G, B\) that mixed together generate the color impression \(C\). The objective is to match the color impresions from both sides by adjusting the power of the three monochromatic lights, or in some cases, mixing one of the monochromatic lights with the target.

The human visual system responds to luminance approximately logarithmically. The Weber-Fechner law states that the perceived change in a stimulus is proportional to the ratio of the change to the initial intensity, not to the absolute difference. That is the same as stating that stimulus increases have to follow a geometric progression for the perception to follow an arithmetic progression.

Photographic film exhibits a strikingly similar response. The Hurter-Driffield (1890) curve modeled how the optical density of a developed emulsion is approximately proportional to the logarithm of the exposure (the product of luminance and time). The curve has three regions: a toe where the emulsion barely responds to low exposures, a straight-line section where density rises proportionally with log exposure, and a shoulder where the emulsion saturates and additional light produces diminishing returns. In the straight-line portion, the film acts as a logarithmic encoder of scene luminance.

The slope of the straight-line segment is known as the film's gamma. A gamma greater than 1 increases contrast; less than 1 compresses it. Different film stocks and development processes yield different gamma values. A high-contrast copy film might have a gamma of 2.0 or more, while a negative film intended for pictorial work might sit around 0.6, relying on the print stage to restore contrast.

The CIE RGB is the colorspace based on the standardized color matching functions \(\bar{r}(\lambda), \bar{g}(\lambda), \bar{b}(\lambda)\) obtained using three monochromatic light sources (primaries) at standardized wavelengths of 700 nm (red), 546.1 nm (green) and 435.8 nm (blue) ⁴.

The distribution of spectral energy is normalized and adding all color matching functions equals 1. Negative values in the graph indicate primary colors have to be mixed with the target color before color matching.

The CIE rg is a two-dimensional space derived from RGB that determines the chromaticity coordinates in a plane and discards the luminance. A color is represented by the proportion of red, green, and blue in the color, rather than by the intensity of each.

The CIE XYZ tristimulus values are a device-invariant representation of color ⁵. It serves as a standard reference against which many other color spaces are defined.

It is defined as a conveninent linear transformation of the CIE 1931 RGB colorspace, such that:

1. \(\bar{x}(\lambda), \bar{y}(\lambda), \bar{z}(\lambda) \in \mathbb {R} _{>0}\)
2. \(\{(x, y) \in \mathbb{R}^2 \mid x \geq 0,\ y \geq 0,\ x + y \leq 1\}\); so the gamut of colors will lie inside a unit triangle
3. \(\bar{x}(E) = \bar{y}(E) = \bar{z}(E) = 1/3\); where \(E\) is the equal-energy white point

In the CIE 1931 model, Y is the luminance, Z is quasi-equal to blue (of CIE RGB), and X is a mix of the three CIE RGB curves chosen to be nonnegative. Setting Y as luminance has the useful result that for any given Y value, the XZ plane will contain all possible chromaticities at that luminance.

The color matching functions \(\bar{r}(\lambda), \bar{g}(\lambda), \bar{b}(\lambda)\) are negative at certain wavelengths to allow for any monochromatic sample to be matched. This is why in the rg chromaticity diagram the spectral locus extents into the negative r direction and ever so slightly into the negative g direction. Avoiding negative color coordinate values prompted the change from to rg to xy.

Three dimensional color can be divided into two parts: brightness and chromaticity. For example, the color white is a bright color, while the color grey is considered to be a less bright version of that same white. In other words, the chromaticity of white and grey are the same while their brightness differs. The CIE xyY color space was deliberately designed so that the Y parameter is also a measure of the luminance of a color. The chromaticity is then specified by the two derived parameters x and y, two of the three normalized values derived from the tristimulus values X, Y, and Z.

When Y = 1 is defined as the brightest white, the corresponding chromaticity values for x and y can then be inferred using the standard illuminants.

Illuminants A, B, and C were standardized in 1931, with the intention of respectively representing average incandescent light, direct sunlight, and average daylight. Illuminants D (1967) represent variations of daylight, illuminant E is the equal-energy illuminant, while illuminants F (2004) represent fluorescent lamps of various composition.

By choosing different RGB primaries (basis vectors in 3-dimensional space) and white-point other than CIE 1931 RGB's, new volumes can be defined spanning a different gamut⁸ of colors. This allows trading-off color reproduction accuracy and data encoding sizes, depending on the intended application.

When a RGB colorspace specifies a display device or media , it's also known as output space. When a RGB colorspace is used as an intermediate step in a image manipulation pipeline, it can also be named working space. Some examples are standard RGB (sRGB)

Early examples of RGB color spaces came with the adoption of the NTSC color television standard in 1953 across North America, followed by PAL and SECAM covering the rest of the world. These early RGB spaces were defined in part by the phosphor used by CRTs in use at the time, and the gamma of the electron beam. While these color spaces reproduced the intended colors using additive red, green, and blue primaries, the broadcast signal itself was encoded from RGB components to a composite signal such as YIQ, and decoded back by the receiver into RGB signals for display.

HDTV uses the BT.709 color space, later repurposed for computer monitors as sRGB. Both use the same color primaries and white point, but different transfer functions. The gamut of these spaces is limited, covering only 35.9% of the CIE 1931 gamut ⁹. While this allows the use of a limited bit depth without causing color banding, and therefore reduces transmission bandwidth, it also prevents the encoding of deeply saturated colors that might be available in an alternate color spaces. Adobe RGB and ProPhoto, which are intended as working spaces, are designed with expanded gamuts to address this issue, however this does not mean the larger space has 'more colors". The numerical quantity of colors is related to data encoding bit depth and not the size or shape of the gamut. There's a fundamental precision trade-off between a wide gamut space encoded as low bit depth. More recent color spaces such as Rec. 2020 for UHD-TVs define an extremely large gamut covering 63.3% of the CIE 1931 space ¹⁰. This standard was forward-compatible, since it wasn't realizable with LCD panel technology at the time.

If the camera's raw RGB values are known, one may use the 3x3 diagonal matrix:

\[\begin{bmatrix}R\\G\\B\end{bmatrix}=\begin{bmatrix} R'_m/R'_w & 0 & 0 \\ 0 & G'_m/G'_w & 0 \\ 0 & 0 & B'_m/B'_w \end{bmatrix}\begin{bmatrix}R'\\G'\\B'\end{bmatrix}\]

Where \(R'_m, G'_m, B'_m\) are the maximum values (saturation points) in the camera RGB space, and \(R'_{w}, G'_{w}, B'_{w}\), are the components of a pixel which is believed to be a white surface in the image before color balancing. This relies on the property that the coordinate \(RGB = (1, 1, 1)\) should encode the white point of the output media (display or print).

Performing balance in camera RGB space has been found to be superior than doing the same after a conversion to display RGB or XYZ.¹²

Johannes von Kries's theory motivated the method of converting color to the LMS color space, representing the effective stimuli for the Long-, Medium-, and Short-wavelength cone types that are modeled as adapting independently. A 3x3 matrix converts RGB or XYZ to LMS, and then the three LMS primary values are scaled to balance the neutral; the color can then be converted back to the desired final color space:

\[\begin{bmatrix}L\\M\\S\end{bmatrix}=\begin{bmatrix}1/L'_{w}&0&0\\0&1/M'_{w}&0\\0&0&1/S'_{w}\end{bmatrix}\begin{bmatrix}L'\\M'\\S'\end{bmatrix}\]

Where \(L'_{w}, M'_{w}, S'_{w}\), are the tristimulus values of an object believed to be white in the un-color-balanced image.

Matrices to convert to LMS space were not specified by von Kries, but can be derived from CIE color matching functions and LMS color matching functions when the latter are specified; matrices can also be found in reference books.