# Using Information Theory to Explain Color Bias

In a previous post, I showed how information theory can be used to explain why it is that human beings perceive changes in luminosity logarithmically, and not linearly, as a function of actual luminosity (see, “Using Information Theory to Explain Color Perception“). In short, I showed that if you assume that what human beings perceive as luminosity is actually a measure of information content, then perceived luminosity should be a logarithmic function of actual physical luminosity, which is indeed the case (see, for example, Fechner’s Law). In this post, I’ll show how we can use similar concepts to explain why it is that human beings perceive the colors near the boundary of the green and blue portion of the spectrum as brighter than the colors near the red portion of the spectrum.

Luminosity, Energy, Information, and Perception

In a very long paper posted below (that you absolutely do not need to read to understand what follows), I showed that assuming that light literally contains information has a lot of interesting consequences, and even implies that time-dilation will occur. In this post, I’ll take a slightly different approach that nonetheless assumes that the environment with which we interact can be thought of as literally containing information. This doesn’t require us to assume that we’re living in The Matrix, but rather, that at a fundamental level, all of our perceptions require representations of external stimuli to be generated, which means that our bodies are quite literally generating information. Because information is measured in bits, this view implies that we can measure the information content of external stimuli in bits.

Specifically, we can measure the sensory information triggered by a single photon landing on a human eye by making use of some basic principles from contemporary physics. For those that are unfamiliar with Planck’s equation, $E = hf$, in short, it gives us the energy of a single “parcel” of light $(E)$, known as a photon, as a function of its frequency $(f)$. The constant $h$ is known as Planck’s constant, given that Max Planck penned the equation, though Einstein also had a role in popularizing the view that light is “quantized” into photons through his work on the photoelectric effect.

Returning to our analysis, as noted above, human beings perceive changes in luminosity logarithmically as a function of actual physical luminosity. So, for example, if a light source has a luminosity of $L$, then the level of luminosity perceived by a human observer of that source is given by $\log(L)$. We showed below that we can explain this phenomenon, as well as other related phenomena, by assuming that what is perceived as luminosity is actually a representation of the amount of information observed. That is, the perceived brightness of a light source is not a direct representation of the luminosity of the light, but is instead a representation of the amount of information required to represent the luminosity of the source. In crude (and imprecise) terms, if the luminosity of a light source is 10, then we need $\log(10)$ bits to represent its luminosity, meaning that what we perceive as brightness should actually be measured in bits.

In more precise terms, luminosity is also a measure of energy, but it is not a measure of the energy of the individual photons ejected by a light source (like Planck’s equation). Luminosity is instead a measure of the total energy ejected by a light source. As a result, when we take the logarithm of the luminosity of a given light source, we are taking the logarithm of an amount of energy. Generalizing upon this, it should also be the case that the energy of an individual photon incident upon a human eye is also perceived logarithmically. As it turns out, assuming that this is the case again leads to equations that are consistent with known results as to how human beings perceive colors.

The Perceptual Information Content of a Photon

Summarizing the discussion above, my hypothesis is that what is perceived as light is actually a visual representation of the number of bits required to encode the amount of energy observed. In the case of measuring the luminosity of a light source, this hypothesis works out perfectly, since it implies that perceived luminosity is given by the logarithm of actual physical luminosity (see below for the relevant analysis). In the case of an individual photon, it implies that the perceived luminosity of the photon should be given by $log(E) = log(f) + C$.

However, we also know that at a certain point on either side of the visible spectrum of light, perception falls off, meaning that as the frequency of a source increases past blue, our response vanishes, making the light invisible. Similarly, as the frequency of a source decreases below red, the light will again eventually become invisible. This implies that we’ll need to adjust our equation to account for the fact that at both ends of the visible spectrum of light, the perceived luminosity should approach zero. We can accomplish this by adding a “weight” given by, $W = \sin[\pi (f - f_{min}) / (f_{max}- f_{min})]$,

where $f$ is the frequency of the photon in question, $f_{min}$ is the minimum perceptible frequency (i.e., the minimum visible frequency of red) and $f_{max}$ is the maximum perceptible frequency (i.e., the maximum visible frequency of blue/violet). Combining these two equations (but dropping the constant), we have the following equation relating the frequency of a photon to its perceived luminosity: $L_P = \log(f) \sin[\pi (f - f_{min}) / (f_{max}- f_{min})]$.

Plugging in the reasonable values of $f_{min} = 350$ THz and $f_{max} = 850$THz,  we obtain the attached graph. Note that this equation implies that perceived luminosity is maximized around a frequency of 607 THz, right at the boundary of the the green and blue portions of the visible spectrum, which is consistent with known results as to how human beings perceive differences in luminosities across the color spectrum.

In summation, whether or not we’re living in The Matrix, assuming that human beings perceive the world around us using encoded representations implies equations that are consistent with known results as to how we actually perceive the external world. Specifically, if we assume that what we perceive is information itself, and that the information content of a perceptual signal is given by the logarithm of the physical stimulus energy, then it seems that, as a general matter, we produce equations that are consistent with experimental data as to how human beings react to external stimuli.