• Handprint: Color Science
• CS 178, Spring 2014, Color 1: trichromatic theory

§ Color 1: Trichromatic Color Theory

§ Newton

• Figured out that white light (from the sun) can be split into a spectrum of colors.
• But wait, why isn't the light from the sun monochromatic, or randomly colored?
• something something about how the sun is a blackbody radiator, and thus emits a spectrum with components at all frequencies.
• humans can see the wavelengths from 400nm to 700nm, which is the visible spectrum.
• luckily for us, the light that is reflected can be computed by multiplying wavelength-by-wavelength the illumination spectrum with the reflectance spectrum of the object. Recall that these spectrum map wavelengths to power.
• We typically plot these as 'power distributions', which are normalized to clamp the maximum power to 1. Then, we can 'read off' what the color of the object is by looking at the peak of the power distribution.

§ Color Response Function Of Humans (Trichromatic Color Theory / Young-Helmholtz Three Component Theory)

• Each type of cone in the retina (for red, green, blue) have different 'spectral sensitivity distribution' curves, that map wavelength to the normalized sensitivity of the cone to that wavelength.
• If we multiply the input wavelength-by-wavelength power distribution with the spectral sensitivity distribution of each cone, and integrate, we get three numbers, which are the 'cone responses' of the human eye to that color at each area of the retina.
• Said more mathematically, take a stimulus spectrum $L_e(\lambda)$.
• Then, suppose we have divined the functions $\rho(\lambda)$ which gives us the normalized sensitivity of the red cone to wavelength $\lambda$.
• Then, $\rho = \int L_e(\lambda) \rho(\lambda) d\lambda$ is the response of the red cone to the stimulus spectrum $L_e(\lambda)$.
• So, overall, $(\rho, \gamma, \beta)$ equal $\int L_e(\lambda) (\rho(\lambda), \gamma(\lambda), \beta(\lambda)) d\lambda$.
• The function $L_e(\lambda) \mapsto (\rho, \gamma, \beta)$ is called the 'color response function of humans'.

§ Locus Of Spectral Colors

• Since the response to any spectrum $L_e(\lambda)$ is given by three values $(\rho, \gamma, \beta)$, we can consider the space of all possible $(\rho, \gamma, \beta)$.
• Moreover, since the computation of $\rho$ is linear in $L_e(\lambda)$, the mapping from $L_e(\lambda) \mapsto \rho$ is a linear function. Similarly for the others.
• What we can now do is to consider the span of the the color maping function $L_e(\lambda) \mapsto (\rho, \gamma, \beta)$.
• This gives us a convex shape in 3D. Let's call it the "gamut of percieveable colors". This will be convex hull of the "locus of spectral colors" or the "spectral locus".
• Since it's a convex shape, the boundary of the shape is formed by dirac delta functions at each wavelength. The boundary curve is called as the "locus of spectral colors".
• Also, see that the response to the diract delta at $\delta_{\lambda_0}$ recovers $(\rho(\lambda_0), \gamma(\lambda_0), \beta(\lambda_0))$.
• So, we can see that the locus of spectral colors is the curve traced out by $(\rho(\lambda), \gamma(\lambda), \beta(\lambda))$ as $\lambda$ varies from 400nm to 700nm.
• These are called the "tristimulus sensitivity functions" of humans, written as $\rho(\lambda), \gamma(\lambda), \beta(\lambda)$. (See that this matches the previous definition.
• We want the camera's sensitivity functions $R(\lambda)$ to be similar to that of the human's $\rho(\lambda)$, so we capture the same colors as humans do.

§ Reproducing Colors With 3 Primary Colors/ CMFs (Color Matching Functions)

• Colour and Vision Research Lab
• Given a target light $T$, we want to find a combination of the lights $P_1, P_2, P_3$ in order to reproduce the same color as $T$.
• Sometimes, we have to add some amount of, say, $P_1$ to $T$! (since $T$ was not saturated enough in the $P_1$ direction).
• So, in some sense, we're trying to solve for $T + k_1 P_1 + k_2 P_2 + k_3 P_3 = 0$. Naively, we would only imagine that $k_1, k_2, k_3 \geq 0$. However, we offer how to interpret negative values of $k_i$ as adding $P_i$ to the target light $T$!
• This takes the footing into pure abstraction.
• Now, the values of $k_1, k_2, k_3$ are called as the "color matching values" for a given target light $T$.
• The standard CIE 1931 RGB standard picks monochromatic primaries . Here, $P_1 = R = 700nm$, $P_2 = G = 546.1nm$, $P_3 = B = 435.8nm$.
• Then, the values $k_1, k_2, k_3$ used to match a target light $T = \delta_{\lambda_0}$ are written as $\bar \rho(\lambda_0), \bar \gamma(\lambda_0), \bar \beta(\lambda_0)$, and are called the "color matching functions" (CMFs).
• We picked 700nm red since it only excites the red cone basically, and is the most saturated.
• We picked the green and blue due to engineering convenience, since these happen to be spectra of a mercury lamp, if I got my reading right.
• Anyway, this gives us the functions $\bar \rho(\lambda)$, $\bar \gamma(\lambda)$, $\bar \beta(\lambda)$, which are the color matching functions of humans, with a chosen basis of the CIE standard.
• So, we can parametrize the true color $T$ by these values.
• The gamut that corresponds to our choice of basis is the set of all colors we can form with nonnegative combinations of the basis.
• Q. why don't we choose the basis to be the actual human response curves of our cones? A. Because the primaries have to be light we can actually generate, and we actually can generate pure wavelengths.

§ LMS Color space

• The name for the color space of the human matching functions.

§ Additive Color Mixing

• Just add wavelengths together, it's linear algebra.

§ Subtractive Color Mixing

• Multiply the transmittance spectra. The output $o(\lambda)$ equals $i_1(\lambda) \cdot i_2(\lambda)$ for input material transmittance spectra $i_1, i_2$.
• Subtractive mixing is used in printing, where all subsets of Cyan, Yellow, Magenta, blac K are used to create pigments of various shades. The colors of each spot have their spectra interpreted additively by the eye when viewed from a distance.
• Q. Why do we need the subtractive step in the first place? Can't we just use R, G, B?
• A. With just R, G, B, we can reproduce chromaticity (hue plus saturation), but not lightness.

§ Photometry

• Photometry is just radiometry (which cares about power and related concepts) integrated against the human visual system.
• So, I will describe the radiometric concept first, and the related photometric version.
• TODO: do the photometry thing here.

§ Color Appearance Terminology

• All taken from Fairchild's book, "Color Appearance Models".

§ Color

Attribute of visual perception consisting of any combination of chromatic and achromatic content. This attribute can be described by chromatic color names such as yellow, orange, brown, red, pink, green, blue, purple, etc., or by achromatic color names such as white, gray, black, etc., and qualified by bright, dim, light, dark, etc., or by combinations of such names.

§ Hue

Attribute of a visual sensation according to which an area appears to be similar to one of the perceived colors: red, yellow, green, and blue, or to a combination of two of them. Achromatic Color: Perceived color devoid of hue. Chromatic Color: Perceived color possessing a hue.

• Think of the hue circle as the correct mental model for 'hue'.

§ Brightness And Lightness

Brightness Attribute of a visual sensation according to which an area appears to emit more or less light.

Lightness (Relative Brightness) The brightness of an area judged relative to the brightness of a similarly illuminated area that appears to be white or highly transmitting

• A classic example to distinguish these is a piece of paper.
• When viewed indoors, it has some brightness, and is probably the lightest thing in the field of view.
• When viewed outdoors, it reflects a lot more light (and thus appears brighter), and is probably still the lightest thing in the field of view.
• Thus, the outdoor scene's paper has more brightness, but roughly equal lightness.
• Lightness is therefore approximately the same across luminances.

§ Colourfulness and Chroma

Colorfulness Attribute of a visual sensation according to which the perceived color of an area appears to be more or less chromatic.

Chroma Colorfulness of an area judged as a proportion of the brightness of a similarly illuminated area that appears white or highly transmitting.

• Color is thought to be three dimensional. Two of the dimensions (hue and brightness) have been defined already. The third dimension is colorfulness and chroma.
• Colorfulness is to chroma as brightness is to lightness.
• So, brightness : lightness :: colorfulness : chroma.
• chroma is "relative colorfulness".
• Colorfulness is the intensity of a hue in a given color stimulus.
• Acrhomatic colors have zero colorfulness (and thus, also zero chroma).
• As the amount of color content increases (holding brightness and hue constant), so does the colorfulness and chroma.
• Like Lightness, Chroma too tends to be approximately the same across luminances.

§ Saturation

Colorfulness of an area judged in proportion to its brightness.

• Saturation is a unique perceptual experience separate from chroma.
• Like chroma, saturation can be thought of as relative colorfulness.
• However, saturation is the colorfulness of a stimulus relative to its own brightness .
• Chroma is colorfulness relative to the brightness of a similarly illuminated area that appears white.
• In order for a stimulus to have chroma, it must be judged in relation to other colors.
• On the other hand, a stimulus seen completely in isolation can have saturation.
• An example of a stimulus that exhibits saturation, but not chroma, is a traffic signal light viewed in isolation on a dark night.
• The lights, typically red, yellow, or green, are quite saturated and can be compared with the color appearance of oncoming headlights whose saturation is very nearly zero (since they typically appear white).
• Saturation is sometimes described as a shadow series. = This refers to the range of colors observed when a single object has a shadow cast upon it. As the object falls into deeper shadow, it becomes darker, but saturation remains constant.

§ Five Parameters

• (1) Brightness
• (2) Lightness
• (3) Colorfulness
• (4) Chroma
• (5) Hue
• We don't need saturation, can be derived.

§ Color 2: Applications in Photography