4 Color Vision
This chapter studies color vision. We will review two main retinal stages responsible for color vision: wavelength encoding by the photoreceptors and the opponent processes that take place post-receptorally. We discuss both the behavioral phenomena as well as the potential neural and physiological basis. That this chapter almost exclusively focuses on the retinal mechanisms should in no way be taken to downplay the significance of cortical mechanisms to color vision (Gegenfurtner 2003). We take this approach because: 1) the retinal mechanisms are much better understood and 2) many real-world applications such as color reproduction and detecting colored patterns could be adequately modeled by retinal mechanisms. This chapter concludes by briefly reviewing the evolution of color vision and deficient color vision.
4.1 Color Encoding at Photoreceptors
Newton presumably did the famous experiment where he showed that a beam of white light is really a mixture of photons at different wavelengths, and each wavelength gives a different color percept. Color is very much our subjective sensation. What is the physical reality is the spectral power distribution of light. In Newton’s words: “rays of Light in falling upon the bottom of the eye excite vibrations in the retina. Which vibrations, being propagated along the solid fibres of the optick Nerves into the Brain, cause the sense of seeing.” (Newton 1704).
4.1.1 From Light Spectrum to Cone Responses
As we have seen before, there are three classes of cones, each with a different spectral sensitivity function or a cone fundamental. We will now see how the cone fundamentals encode wavelength information that eventually gives rise to color vision.
The cone fundamentals we have seen in Figure 3.4 tell us the absolute spectral sensitivities of photoreceptors. It is customary to normalize the cone fundamentals to peak at unity. This normalization eliminates the differences at peak across photoreceptor types, but retains the relative spectral sensitivity within a particular type. Thus, this normalization is useful when we care only about comparing the sensitivity of different wavelengths of a particular type of photoreceptor, but not across different types of photoreceptor.
In addition, the cone fundamentals in Figure 3.4 are defined on an “equal-quantal” basis: the sensitivities at different wavelengths are given assuming each wavelength has the same amount of photons. Sometimes, especially in CIE standards, the cone fundamentals (and other functions related to cone fundamentals, such as luminous efficiency function and color matching functions, both of which we will discuss later) are defined based on “equal-energy”, assuming each wavelength has the same energy/power, not the same amount of photons. As we will see shortly, the equal-energy definition is practically useful since the spectrum of a light is defined as power/energy distribution, rather than quantal distribution, over wavelength.
Figure 4.1 compares the absolute, equal-quantal cone fundamentals with the normalized, equal-energy cone fundamentals. A normalized, equal-energy sensitivity function tells us the relative amount of photon absorption given a unit power at each wavelength. For instance, the normalized L cone response is 1 at 570 \(\text{nm}\) and 0.4 at 630 \(\text{nm}\). This means that given two lights that have the same power/energy, one with photons only at 570 \(\text{nm}\) and the other with photons only at 630 \(\text{nm}\), the fraction of photons absorbed in the 630 \(\text{nm}\) light is about 40% of that in the 570 \(\text{nm}\) light.
Critically, this also means if we have a 570 \(\text{nm}\) light at 1 \(\text{W}\) and a 630 \(\text{nm}\) light at 2.5 W, the two lights would cause the same amount of pigment excitations in L cones. If we had only L cones, these two lights would be seen as the exact same light, because the HVS will receive the exact amount of electrical responses — according to the Principle of Univariance. This explains why we could not see colors at night, when only rods are functioning.
In reality, of course, most humans have three classes of cones, so what is the signal we receive? Given the Spectral Power Distribution (SPD) of a light \(\Phi(\lambda)\), we can calculate the total number of photon absorptions for each cone type, given by:
\[ \begin{align} L &= \int_\lambda L(\lambda) \Phi(\lambda) d\lambda \\ M &= \int_\lambda M(\lambda) \Phi(\lambda) d\lambda \\ S &= \int_\lambda S(\lambda) \Phi(\lambda) d\lambda \end{align} \tag{4.1}\]
where \(L(\lambda)\), \(M(\lambda)\) and \(S(\lambda)\) represent the cone sensitivity functions. The fact that we can directly multiply \(\Phi(\lambda)\) with, say, \(L(\lambda)\) is a result of defining \(L(\lambda)\) on an equal-energy/power basis. The L/M/S values we calculate represent the total number of photon absorptions given an incident light. You would know why we care about photon absorption: it is equivalent to pigment excitation up to a constant scaling factor, and pigment excitations produce electrical signals that our brain actually receives. We sometimes simply call the L/M/S value the cone responses or tristimulus values of a light, but you should know that they do not represent the actual magnitude of the electrical responses of the cones, since the magnitude is not linearly proportional to absorption as we have discussed before.
In actual computation we discretize the spectra and perform summation rather than integration. We also limit the summation to within the [380 \(\text{nm}\), 780 \(\text{nm}\)] range, since the cone fundamentals are practically 0 beyond that range. Assuming that we are quantizing the spectra at a 1-\(\text{nm}\) interval, the cone responses are linearly related to the light spectrum by:
\[ \begin{align} \begin{bmatrix} L(380), L(381), \cdots, L(780)\\ M(380), M(381), \cdots, M(780)\\ S(380), S(381), \cdots, S(780)\\ \end{bmatrix} \times \begin{bmatrix} \Phi(380)\\ \Phi(381)\\ \vdots \\ \Phi(780)\\ \end{bmatrix} = \begin{bmatrix} L\\ M\\ S\\ \end{bmatrix} \end{align} \tag{4.2}\]
We can see that this is a huge dimensionality reduction. That is, our brain receives only the three-dimensional cone responses, not the actual spectrum of the light, which is of a much higher dimension. This is the basis of the trichromatic theory of color vision: color is a three-dimensional system. The theory was first proposed by Young (1802), who conjectured that there are three types of receptors, and later rediscovered, popularized, and extended by Hermann von Helmholtz in the later part of the nineteenth century.
The huge dimensionality reduction also means there are infinitely many lights (with different SPDs) that will be seen as having the same color, as long as they cause the same cone responses. One way to understand this is if we try to solve the system of linear equations in Equation 4.2 given \([L, M, S]^T\), with the constraint that the \(\Phi\) vector must be non-negative everywhere (since power cannot be negative), we would generally end up with infinitely many solutions, since it is an under-determined system. The fact that multiple physically different lights can end up having the same color is called metamerism, and these lights are called metamers of each other.
4.1.2 Cone Excitation Space, Spectral Locus, and HVS Gamut
The cone fundamentals essentially give us a color space, which we call the LMS cone space or cone excitation space. A color space allows us to geometrically interpret a color as a point in the coordinate system. In the cone space, the color of a light is interpreted as the amount of responses in each of the three cone classes produced by the light (as calculated by Equation 4.2).
The spectral locus is a curve on which each point represents the color of a spectral light at a wavelength. Figure 4.2 shows the spectral locus in the LMS cone space on the right and the cone fundamentals on the left. The L, M, and S cone responses of a spectral light at, for instance, 605 \(\text{nm}\) are 0.775, 0.265, and 0, which corresponds to the point [0.775, 0.265, 0] in the cone space. Connecting these points for all the spectral lights gets us the spectral locus in the LMS space.
We know a color corresponds to a point in the cone space, but does an arbitrary point in the cone space correspond to a real color? No. For instance, if a point has a negative coordinate it obviously could not be a color of a real light, since that a negative cone response would require negative power in the light. Also, [1, 0, 0] is also not a real color, since there is no real light that can produce only L cone response but no responses from M and S cones — if you examine the cone fundamentals carefully. We call these colors imaginary colors, since they cannot be produced by physically realizable lights, where the power must be non-negative at any wavelength.
In principle, an [L, M, S] point corresponds to a real color if Equation 4.2 has a non-negative solution for \(\Phi\). The total set of [L, M, S] points that have a non-negative \(\Phi\) solution corresponds to all the colors that humans can see, which is called the gamut of the human visual system. Geometrically, if a point in the cone space cannot be constructed through a positive, linear combination of the points on the spectral locus, it then is not a real color, since the SPD of a real light must be a positive, linear combination of the SPDs of the spectral lights.
For instance, the line segment connecting two points on the spectral locus contains real colors that can be produced by mixing some amount (i.e., positive linear combinations) of the two spectral lights. Of course we can apply this iteratively: once you get a real color through combining spectral colors, the color itself can then be used as a basic color to create other colors. Zhu (2022d) is an interactive tutorial that visualizes the HVS gamut in the cone space (and others), which you are invited to go through.
4.2 Trichromatic Color Matching
We can produce, in theory, any color by mixing three other colors, which we call the primary colors. Here is the mathematical intuition. Let’s say the SPDs of the three primary lights are \(R(\lambda)\), \(G(\lambda)\), \(B(\lambda)\). What is the power of each of the primary lights we need to produce the color of a target light \(\Phi(\lambda)\)? For the color of the mixed light to match that of the target light, their corresponding cone responses must match:
\[ \begin{align} \begin{bmatrix} \sum R(\lambda)L(\lambda),~ \sum G(\lambda)L(\lambda),~ \sum B(\lambda)L(\lambda)\\ \sum R(\lambda)M(\lambda),~ \sum G(\lambda)M(\lambda),~ \sum B(\lambda)M(\lambda)\\ \sum R(\lambda)S(\lambda),~ \sum G(\lambda)S(\lambda),~ \sum B(\lambda)S(\lambda)\\ \end{bmatrix} \times \begin{bmatrix} r\\ g\\ b \\ \end{bmatrix} = \begin{bmatrix} \sum\Phi(\lambda)L(\lambda)\\ \sum\Phi(\lambda)M(\lambda)\\ \sum\Phi(\lambda)S(\lambda)\\ \end{bmatrix}, \end{align} \tag{4.3}\]
where \(r, g, b\) represent the power of the three primary lights, respectively. This system in general has one unique solution because we have the same number of unknowns (\(r, g, b\)) as the number of equations. Each of the three equations constrains the cone-response matching of one class of cones. This means there is a single unique way to mix three primary lights to produce the color of an arbitrary target light.
What if we have more than three primary lights? We would end up with an under-determined system (e.g., three equations but four unknowns if given four primary lights), which means there are infinitely many ways to mix the primaries to produce the target color. If we have only two primaries, we end up with an over-determined system, where there is in general no solution.
4.2.1 Color Matching Experiments and Color Matching Functions
Equation 4.3 gives a mathematical explanation for trichromatic color matching, but it requires knowing the cone fundamentals, which, as we have seen before in Section 3.2, were not experimentally measured until the mid 20th century, first through microspectrophotometry (Marks, Dobelle, and MacNichol Jr 1964; Brown and Wald 1964; Dartnall, Bowmaker, and Mollon 1983) and then through suction electrode (Schnapf, Kraft, and Baylor 1987). But even without the cone fundamentals, nothing prevents us from performing an actual experiment to find the amount of primaries for producing a color. Thomas Young apparently had no interest in such an experiment (Mollon 2003). Maxwell (1857) is believed to be the first to undertake an actual color matching experiment in the 19th century, but he did the experiments using rotating discs painted with different colors, relying on the temporal integration of the HVS.
Modern color matching experiments started with Wright and Guild (W. Wright 1928, 1930; W. D. Wright 1929; Guild 1931). International Commission on Illumination (CIE) in 1931 standardized the color matching experiment and synthesized Wright’s and Guild’s data (without any additional experiments) to obtain what is now known as the CIE 1931 RGB Color Matching Functions. This process is discussed in detail in Broadbent (2004), Broadbent (2008), Service (2016), and Zhu (2020). We summarize the key elements here; the experimental setup is illustrated in Figure 4.3.
Observers are presented with a 2\(^{\circ}\) visual field. They are given three primary lights, which in the CIE 1931 standard are spectral lights (lights that have photons at only one single wavelength; also called monochromatic lights) at wavelengths 435.8 \(\text{nm}\), 546.1 \(\text{nm}\), and 700 \(\text{nm}\). The three primary lights are pointed at the same point on one side of the visual field. On the other side of the visual field is the target light. Their goal is to adjust the power of each of the three primary lights so that the colors from the two sides of the visual field match. CIE 1931 swept the entire visible spectrum for the target light at a 5-\(\text{nm}\) interval.
Color Matching Functions Require a Unit System and a White Point
The results obtained through the color matching experiments are shown in Figure 4.4 (left panel). The three curves are collectively called the CIE 1931 RGB Color Matching Functions (CMFs). Intuitively, the CMFs tell us the amount of primaries needed to match the color at each wavelength. But the devil is in the details. Let’s carefully walk through what this plot actually shows.
The \(y\)-axis represents the number of units required of each primary so that the mixture matches the color at a given wavelength at \(x\)-axis. What is a unit? The unit system is so defined that mixing the three primaries in equal units produces the color of the Equal-Energy White (EEW), whose SPD is a constant across the spectrum.
There are two judgment calls here. First, CIE 1931 decided that EEW was going to be the “white” color in their RGB color space. In general, however, there is no single color that we universally define as white, so if you were to design a color space you get to pick whatever color that you think is white in the color space. That said, an intuitive choice of white is one that is achromatic (colorless), a color that, subjectively, can only be described as having a certain level of gray but that has no apparent hue. Daylights at different times of a day are perceptually achromatic and could be used as the white point in a color space. The daylight colors are shown to be very similar to the colors of black-body radiation at different temperatures (Judd et al. 1964), shown in Figure 4.5.
You probably do not perceive most of the colors in Figure 4.5 as achromatic on the display right now, but when you are in an environment illuminated by one of these colors, e.g., outdoors at noon, you do perceive the illuminant as achromatic; this is because of chromatic adaptation, a topic we will discuss later in Section 7.3. Briefly, the human visual system is evolved to adapt to different daylight colors so that when you spend enough time under such an illuminant, you will see the illuminant as achromatic. The adaptation to other colors, however, is weak (or “incomplete” in chromatic adaptation parlance) 1, so it probably does not make much sense to pick other colors as the white point if you want your user to see your white as achromatic. CIE has standardized a set of what they call Standard Illuminants (D series), each of which approximates a different daylight color. For instance, the D65 standard illuminant approximates noon daylight and is similar to the color of a black-body radiation at a temperature of 6500 K. Many common color spaces, such as the sRGB color space, use D65 as the white point.
Second, CIE 1931 RGB space, and virtually all color spaces, define units so that white, however defined, must be produced by an equal-unit mixture of the primaries. This, again, is a judgment call. One could totally design a color space where white is produced by mixing, say, 2 units of red and 1 unit of green and blue each — nothing wrong with that. It is just more intuitive for most people that white is produced by equal amounts of the primaries.
The \(x\)-axis in Figure 4.4 is defined on an equal-energy/power basis. That is, the CMFs are interpreted as showing the amount (units) of the primaries needed to produce spectral lights of equal power. So if we actually mix the three primaries at each wavelength as indicated by the CMFs, we will get a set of spectral lights that have the same power.
What Does a Negative Unit Mean?
If you observe Figure 4.4 carefully, you will see that some CMFs are negative over certain ranges. For instance, the red CMF is negative at 500 \(\text{nm}\). This is perhaps a bit surprising, but mathematically it is entirely possible that some values in \([r, g, b]^T\) are negative when solving Equation 4.3. Physically, however, what does it mean to have a negative amount/power of primary light? The right panel in Figure 4.4 provides the intuition. It turns out that it is impossible to find a combination of the three primary lights to match the color of a spectral light at 500 \(\text{nm}\). What does provide a match is to add a little red primary to the target light, and then we can find a combination of the primaries such as the blue and green mixture has the same color as the target light and red primary mixture.
In fact, if you examine the CMFs, you will see that there is a negative contribution from a primary at all but three wavelengths — the only three exceptions are the wavelengths of the three primaries (where two of the primary contributions are zero and the other is positive). This means that no spectral light color (except the three special cases) can be physically produced by mixing the three primaries.
Representing Colors Using CMFs
Given a set of CMFs, we can describe the color of a light with a SPD \(\Phi(\lambda)\) using the following equation:
\[ \begin{align} \begin{bmatrix} \bar{r}(380), \bar{r}(381), \cdots, \bar{r}(780)\\ \bar{g}(380), \bar{g}(381), \cdots, \bar{g}(780)\\ \bar{b}(380), \bar{b}(381), \cdots, \bar{b}(780) \end{bmatrix} \times \begin{bmatrix} \Phi(380)\\ \Phi(381)\\ \vdots \\ \Phi(780)\\ \end{bmatrix} = \begin{bmatrix} R\\ G\\ B\\ \end{bmatrix} \end{align} \tag{4.4}\]
where \(\bar{r}(\lambda)\), \(\bar{g}(\lambda)\), and \(\bar{b}(\lambda)\) are the CMFs, and \(R\), \(G\), and \(B\) are the amounts of the three primaries needed to match the color of \(\Phi(\lambda)\).
The CMFs give us another color space, where the color of a light is interpreted as the amount of primary lights needed to match the color of the light. Of course, if we choose a different set of primary lights, we might end up with a new set of CMFs and a new RGB color space.
4.2.2 Connecting CMFs and Cone Fundamentals
CMFs and cone fundamentals both yield trichromatic color vision, so they must be inherently related, as they are just different ways of describing the same thing. We show the two are linearly related in theory, and the measurement data of the two match well, too.
Deriving Color Matching Functions From Cone Fundamentals
Given the cone fundamentals, we can derive the CMFs based on the linear system shown in Equation 4.3. The interactive tutorial by Zhu (2022a) walks through the process, which you are invited to go over, and we will describe the main steps here.
In order to construct the CMFs, we have to match the colors of all the spectral lights, which means we have to specify cone-response matching at each wavelength. Using the basic idea of Equation 4.3, we have:
\[ \begin{align} \begin{bmatrix} \sum R(\lambda)L(\lambda),~ \sum G(\lambda)L(\lambda),~ \sum B(\lambda)L(\lambda)\\ \sum R(\lambda)M(\lambda),~ \sum G(\lambda)M(\lambda),~ \sum B(\lambda)M(\lambda)\\ \sum R(\lambda)S(\lambda),~ \sum G(\lambda)S(\lambda),~ \sum B(\lambda)S(\lambda)\\ \end{bmatrix} \times \begin{bmatrix} r(380),\cdots,r(780)\\ g(380),\cdots,g(780)\\ b(380),\cdots,b(780)\\ \end{bmatrix} = \begin{bmatrix} L(380),\cdots,L(780)\\ M(380),\cdots,M(780)\\ S(380),\cdots,S(780)\\ \end{bmatrix}, \end{align} \]
where \(L(\lambda)\), \(M(\lambda)\), and \(S(\lambda)\) are the cone fundamentals; \(L(\lambda_0)\) is the L cone response of the spectral light at a particular wavelength \(\lambda_0\); \([r(\lambda_0), g(\lambda_0), b(\lambda_0)]^T\) represents the (to-be-solved-for) power of each primary needed to match the color of the spectral light at \(\lambda_0\); \(R(\lambda)\), \(G(\lambda)\), and \(B(\lambda)\) are the SPDs of the primary lights used in the CIE 1931 color matching experiment. The first matrix is a constant matrix given a particular set of CMFs, and we will denote it as the \(\mathbf{M}\) matrix. We can solve the system of equations by inverting the first matrix:
\[ \begin{align} \begin{bmatrix} r(380),\cdots,r(780)\\ g(380),\cdots,g(780)\\ b(380),\cdots,b(780)\\ \end{bmatrix} = \mathbf{M}^{-1} \times \begin{bmatrix} L(380),\cdots,L(780)\\ M(380),\cdots,M(780)\\ S(380),\cdots,S(780)\\ \end{bmatrix}. \end{align} \]
To get the CMFs, however, we need to turn the power measure into a unit measure. Recall the requirement that white must be produced by equal units of the primaries. We calculate the power of each primary needed to produce the EEW; let’s denote the solution \([r_w, g_w, b_w]^T\):
\[ \begin{align} \begin{bmatrix} r_{w}\\ g_{w}\\ b_{w}\\ \end{bmatrix} = \mathbf{M}^{-1} \times \begin{bmatrix} L_{w}\\ M_{w}\\ S_{w}\\ \end{bmatrix}, \end{align} \]
where \([L_w, M_w, S_w]^T\) denotes the total L, M, and S cone responses of EEW. For the so-calculated \([r_w, g_w, b_w]\) to represent equal units, the last step is to scale \([\bar{r}(\lambda), \bar{g}(\lambda), \bar{b}(\lambda)]^T\) at each \(\lambda\) by \([r_w, g_w, b_w]\):
\[ \begin{align} \begin{bmatrix} \bar{r}(380), \cdots, \bar{r}(780)\\ \bar{g}(380), \cdots, \bar{g}(780)\\ \bar{b}(380), \cdots, \bar{b}(780) \end{bmatrix} &= \begin{bmatrix} r_w,~0,~0\\ 0,~g_w,~0\\ 0,~0,~b_w \end{bmatrix} \times \begin{bmatrix} r(380),\cdots,r(780)\\ g(380),\cdots,g(780)\\ b(380),\cdots,b(780)\\ \end{bmatrix}\\ &= \begin{bmatrix} r_w,~0,~0\\ 0,~g_w,~0\\ 0,~0,~b_w \end{bmatrix} \times \mathbf{M}^{-1} \times \begin{bmatrix} L(380),\cdots,L(780)\\ M(380),\cdots,M(780)\\ S(380),\cdots,S(780)\\ \end{bmatrix}\\ &= \mathbf{T}_{lms2rgb} \times \begin{bmatrix} L(380),\cdots,L(780)\\ M(380),\cdots,M(780)\\ S(380),\cdots,S(780)\\ \end{bmatrix}, \label{eq:cone2cmfsub} \end{align} \tag{4.5}\]
where \([\bar{r}(\lambda), \bar{g}(\lambda), \bar{b}(\lambda)]^T\) gives us the unit measure, i.e., the values of the CMFs, at each \(\lambda\).
Cone Space and RGB Space are Related by a Linear Transformation
The rightmost matrix in Equation 4.5 is the cone fundamentals written out in the matrix form, and the leftmost matrix in Equation 4.5 is the CMFs written out at discrete wavelengths. So Equation 4.5 essentially describes a linear transformation from the cone fundamentals to the RGB CMFs, where the transformation is dictated by \(\mathbf{T}_{lms2rgb}\). We can look at this in two ways. One, we can think of the cone fundamentals as the CMFs in the cone space: they tell us how much of each cone response we need to match the color of a spectral light. Two, just like how we can construct the spectral locus from cone fundamentals, the RGB CMFs also give us a way to construct the spectral locus — in the RGB space. \(\mathbf{T}_{lms2rgb}\) essentially transforms these two representations of the spectral locus, and this is visualized in Figure 4.6.
There is something deeper: \(\mathbf{T}_{lms2rgb}\) not only transforms the spectral locus, it transforms the entire coordinate system from the cone space to the RGB space. In other words, it transforms every single color in the LMS space to its corresponding coordinates in the CIE 1931 RGB space. The way to think about this is to ask: given that the cone space and the CIE 1931 RGB space provide two ways to represent the color of a light \(\Phi\), how are the cone-space representation \([L_c, M_c, S_c]\) and the RGB-space representation \([R_c, G_c, B_c]\) related? Using Equation 4.2, Equation 4.4, and Equation 4.5, it is easy to see that they are related by a linear transformation through \(\mathbf{T}_{lms2rgb}\):
\[ \begin{align} \begin{bmatrix} R_c\\ G_c\\ B_c \end{bmatrix} = \mathbf{T}_{lms2rgb} \times \begin{bmatrix} L_c\\ M_c\\ S_c \end{bmatrix}. \end{align} \]
Cone Responses Fully Explain Psychophysical Color Matching
The CMFs can be both experimentally measured and calculated if we know the cone fundamentals (through a linear transformation), but do the mathematical estimation and the measurement data match? If so, we can say that the physiological process of encoding light power as cone responses can fully account for the color matching experiments in psychophysics.
Baylor, Nunn, and Schnapf (1987) performed one such comparison and showed the two sets of data matched very well. The results are shown in Figure 4.7, where the smooth curves are from W. Stiles and Burch (1955), which uses a different set of primaries and white point than those used in the CIE 1931 RGB CMFs. The markers are the predicted CMFs through a linear regression from the cone fundamentals measured from macaques, after accounting for ocular and macular absorptions 2.
In fact, the modern versions of the cone fundamentals are constructed so that they are precisely a linear transformation away from some RGB CMFs. For instance, the CIE 2006 “physiologically-relevant” LMS functions (based on Stockman, Sharpe, and Fach (1999) and Stockman and Sharpe (2000)) are constructed by 1) first experimentally measuring the cone fundamentals in psychophysics (from color-vision deficient observers), 2) calibrating the results with a set of RGB CMFs in W. S. Stiles and Burch (1959) (which uses a different set of primary lights from the CIE 1931 RGB CMFs) to derive a best-fit linear transformation, and 3) applying the linear transformation to the CMFs to derive a “clean” set of cone fundamentals.
4.3 Post-Receptoral Color Encoding: Opponent Processes
Cone-response encoding can perfectly explain the trichromatic theory of color vision, where any color can be mixed from three other colors. The trichromatic theory of color has a perfect neural basis: the human visual system has three classes of cones, so color is a three-dimensional system. But the trichromatic theory is not concerned with our subjective experience of color that we encounter on a daily basis. Here are two examples that highlight the difference between perceptual color experience and physical color mixing.
First, when we see an orange color, we feel that it has a little bit of yellow in it and a little bit of red in it. Even though there are many ways to produce orange, some of which do not require mixing yellow and red lights, we cannot help but perceptually feel that orange combines yellow and red. Second, when we mix a red light with a green light, we get yellow, but perceptually, if we stare at yellow, most people would not say that yellow has contributions from red or green.
Hering (1878) 3 hypothesized that, perceptually, there are four primary hues, which form two opposing pairs. Opposing hues cannot co-exist, perceptually, in a color. Any hue can be produced by combining two non-opposing hues. The four hues are: the Yellow and Blue opposing hues and the Red and Green opposing hues. Hering also considered light-dark as another opposing pair: no color can be simultaneously light and dark. In his theory, color vision is still a three-dimensional system, where the three axes are: Yellow-Blue axis, Red-Green axis, and light-dark axis. Any color, a point in this 3D space, is produced by mixing some amount of Red or Green, some amount of Yellow or Blue, and some level of lightness.
The opponent theory seems to contradict the trichromatic theory, which was dominant for the most part of the history — because it has both a solid psychophysical and neural basis. First, the color matching experiment quantitatively shows that, behaviorally, humans could match a color by mixing three other colors. In contrast, Hering had only a qualitative description of perceptual mixing. His description was something like “after this blue comes blue of increasing redness…(blue violet, red violet, purple red), until the last trace of blueness vanishes in a true red.” (Hering 1964, p. 41). To Hering’s theory’s rescue, Jameson and Hurvish performed a now-famous experiment, called the hue cancellation experiment, providing the first quantitative, psychophysical evidence of the opponent processes (Jameson and Hurvich 1955; Hurvich and Jameson 1957).
Second, the trichromatic theory has a clear neural and physiological basis (i.e., wavelength encoding by cone responses), and the physiological data match the behavioral data very well, as shown before. So a natural question is: are there neural mechanisms that can account for the opponent processes and, if so, how does that mechanism relate to the encoding mechanisms by the cone photoreceptors?
It turns out that we do need a set of new neural mechanisms to start accounting for the opponent processes. Not only do these new mechanisms not contradict the cone encoding mechanisms, they build on top of the cone encodings and operate post-receptorally. Schrödinger (1925) 4 synthesized the earlier zone theory by Kries (1905) and argued that the trichromatic theory and the opponent processes were nothing more than different stages of color encoding in the visual system. That said, while these new neural mechanisms seem to have what it takes to form the basis for the behavioral opponent observations, they do not fully explain those observations yet; the link between the two is still very much an open research question.
The rest of this section will discuss the hue cancellation experiment and the quest for a neural and physiological basis in more detail.
4.3.1 Hue Cancellation Experiment
In a landmark study, Jameson and Hurvich (1955) (while working for Eastman Kodak in Rochester) quantitatively measured the perceptual color opponency using a behavioral experiment. The participant is given a test light and is asked to first judge whether the light appeared blue-ish or yellow-ish. If the test light is judged to be blue-ish, the participant is then given a yellow-ish cancellation light (e.g., a spectral light at 588 \(\text{nm}\)) and is asked to adjust the intensity of the cancellation light so that the mixture of the test and cancellation light perceptually appears neither blue nor yellow. If the test light is judged to be yellow-ish, the participant is then asked to adjust the power of a blue-ish cancellation light (e.g., a spectral light at 467 \(\text{nm}\)) so that the test-cancellation mixture is again neither blue nor yellow. We sweep the spectrum from about 400 \(\text{nm}\) to 700 \(\text{nm}\) for the test light of equal energy, and record the energy of yellow or blue cancellation light needed at each step.
The result for one subject is shown in Figure 4.8 (a), where the \(y\)-axis is showing the intensity of the yellow and blue cancellation light, i.e., the strength of blue-ness and yellow-ness of the test light. For the reference, we attached a colorbar showing roughly the color of the test light between 400 \(\text{nm}\) and 700 \(\text{nm}\), but take this color visualization as a huge grain of salt, since it is almost certain that your display will not be able to actually render the colors of the spectral lights.
Unsurprisingly, we get two peaks, one in the blue range and the other in the yellow range, indicating, respectively, that the participant needs a lot of the yellow and blue cancellation lights in those two regions. The test light at about 500 \(\text{nm}\) requires no cancellation light, indicating light there, which roughly has a green-ish color is yellow-blue neutral: it naturally looks neither blue nor yellow.
Jameson and Hurvich then repeated the same experiment, but this time measuring the red-green opponent process, where the two cancellation lights are a 700 \(\text{nm}\) red-ish light and a 490 \(\text{nm}\) green-ish light. The results are in Figure 4.8 (b), where the \(y\)-axis indicates the amount of red-ness and green-ness in the test light. Two observations are worth noting. First, while it is unsurprising that long-wavelength lights have a strong red component, it is perhaps surprising that short-wavelength lights appear red-ish too. That, however, becomes less surprising when we realize that short-wavelength lights (shorter than pure blue) appear violet, which perceptually is a red-ish blue. Second, because of the two red-ish regions over the spectrum, the entire red-green curve has two zero-crossings, one at about 470 \(\text{nm}\) and the other near 570 \(\text{nm}\): pure blue and pure yellow look neither green nor red.
Figure 4.8 (c) summarizes the two sets of data by inverting the blur section of the curve in (a) and the green section of the curve in (b). That way, the \(y\)-axis can be simply interpreted as the relative strength of red-ness and yellow-ness over the spectrum.
4.3.2 Light-Dark Mechanism and Luminous Efficiency Function
Hurvich and Jameson (1957) also performed a measurement of the white-black (light-dark) opponent process, asking participants to assess the “whiteness” of spectral lights between 400 \(\text{nm}\) and 700 \(\text{nm}\) of equal power. %Intuitively, the measurement tells us the perceived brightness of lights at different wavelengths. A more modern method to measure the luminance mechanism is heterochromatic flicker photometry, where we alternate between a test light and a fixed reference light at a frequency of, say, 25 Hz. We adjust the intensity of the test light so that the alternation produces no visual flickering, at which point we say the two lights produce the same level of luminance (Sharpe et al. 2005, 2011). We again sweep the entire visible spectrum for the test light and record the relative intensity at each step. The so-obtained function is called the luminance efficiency function (LEF). The dashed gray curve in Figure 4.9 shows a modern version of the photopic LEF (the so-called CIE 2008 “physiologically-relevant” 2-deg function) 5.
The way to interpret the LEF is that the \(y\)-axis is inversely proportional to the light power at each wavelength needed to produce the same level of perceptual brightness. The photopic LEF at 509 \(\text{nm}\) is about 0.5, half of that at 555 \(\text{nm}\). It means we need twice as much power at 509 \(\text{nm}\) to produce the same level of brightness as that at 555 \(\text{nm}\). It also explains the word “efficiency” in the name: if a wavelength needs less power to produce a criterion level of brightness, the wavelength is more efficient in its use of power. The way LEF is obtained, however, does not permit us to interpret the result as the relative brightness at different wavelengths. That is, 555 \(\text{nm}\) is not twice as bright as 509 \(\text{nm}\). This is similar to our interpretation of the cone fundamentals.
For comparison, the gray curve in Figure 4.9 is the scotopic LEF. The CIE 1951 scotopic LEF synthesizes the psychophysical measurements from Wald (1945) and Crawford (1949). Both used a threshold method where they measured the light intensity at each wavelength needed to produce a just detectable flash. Note that the photopic LEF peaks at about 555 \(\text{nm}\) and the scotopic LEF peaks at about 507 \(\text{nm}\).
As a result, the relative brightness of longer-wavelength colors and shorter-wavelength colors is inverted when our vision transitions from the cone-mediated photopic vision to the rod-mediated scotopic vision. This phenomenon is called the Purkinje shift. In the words of Glassner (1995, p. 21), “When the sun is still above the horizon, your cones are active, and the yellow flower will appear lighter than the leaves because yellow is closer to peak of the photopic sensitivity curve than dark green. When the sun has set and light levels are lower, your rods are the principal sensors. The scotopic sensitivity curve is more responsive in the shorter wavelengths, so the green leaves will now appear relatively lighter than the yellow flower, though both will of course be much darker due to the lower amount of incident light.”
Combining the light-dark (luminance efficiency) curve with the two opponent curves in Figure 4.8 (c), we again have three spectral sensitivity functions. Figure 4.10 puts the three opponent measurements in one plot (the two plots are for two separate participants). Compare this plot with the cone fundamentals in Figure 4.1. Once again, a light with its SPD can be reduce to three-dimensional point, using Equation 4.1, except 1) instead of the three cone fundamentals we use the three opponent functions and 2) instead of getting the three cone responses we get the strength of the three opponent mechanisms. Effectively, the hue cancellation curves and the light-dark curve construct a new three-dimensional color space. We call this the hue-opponent space, and we will return to this space in Section 4.4.3 and discuss how this space relates to the colorimetric spaces we have discussed so far.
4.4 Neural and Physiological Basis of Opponent Processes
The hue cancellation experiment solidifies Hering’s opponent theory at the level of psychophysics. But recall Figure 1.2; any behavioral responses measured through psychophysics are fundamentally the result of the underlying neural and physiological mechanisms. So the next natural step in the scientific quest is to understand what underlying neural and physiological mechanisms can account for the behavioral opponent processes.
4.4.1 Spectrally-Opponent and Non-Opponent Neurons
There are RGC and LGN neurons that show opponent properties. G. Svaetichin (1953), G. Svaetichin (1956), and G. Svaetichin and MacNichol Jr (1958) are the first to identify opponent neurons in a fish retina; they recorded from horizontal cells. R. De Valois et al. (1958) and R. L. De Valois, Abramov, and Jacobs (1966) measured the responses of LGN neurons in macaques using monochromatic lights, and found spectral opponent neurons, which get excited or inhibited depending on the wavelengths. (A – D) in Figure 4.11 show the recordings of four classes of opponent cells. (A) shows a class of LGN cells whose firing rate exceeds the spontaneous rate under long-wavelength, red-ish lights and whose firing rate drops below the spontaneous rate under short-wavelength, blue-is lights. These cells are denoted +R-G (red-ON/green-OFF) cells. (B), (C), and (D) show that there exists +G-R, +B-Y, and +Y-B cells, respectively.
R. L. De Valois, Abramov, and Jacobs (1966) also identified non-opponent cells, whose responses are universally inhibited or excited across the spectrum, as shown in (E) and (F) in Figure 4.11, respectively. These neurons are still wavelength-sensitive, but their responses are either universally excited or universally inhibited across the spectrum, unlike the spectrally-opponent neurons whose responses change polarity across the spectrum.
4.4.2 Potential Neural Circuitries
What are some of the underlying visual pathways that could potentially give rise to these spectral tuning curves? Recall that LGN cells/RGCs have antagonistic Receptive Fields (RFs), and the antagonism seems to be a perfect mechanism to implement the opponent process. This suggests that in order to understand the opponent cells we must study their RF structures.
Much of the early work is done by Wiesel and Hubel (1966). While De Valois and his collaborators used diffuse lights to illuminate a large visual field, Wiesel and Hubel (1966) used both small spot lights that stimulated the center of the RF and larger lights that covered the entire RF. By comparing the responses under these two stimuli across different wavelengths (and white), they suggested potential RF structures of both opponent and non-opponent cells in macaque LGN. Derrington, Krauskopf, and Lennie (1984) designed a clever experiment that explicitly tied cone responses to LGN cell responses and thus more directly revealed the RF structure.
Before getting into the details, it is worth reminding ourselves that studying the LGN cells and studying the RGCs are equivalent (Section 2.5.1), since different classes of RGCs project to distinct LGN layers with virtually the same RFs: midget RGCs project to the Parvocellular layers (P cells) in the LGN (forming the P pathway/stream), parasol RGCs project to the Magnocellular layers (M cells) in the LGN (forming the M pathway/stream), and bistratetified RGCs project to the Koniocellular layers (K cells) in the LGN (forming the K pathway/stream).
Y-B Opponent Cells
The visual pathway for the Y-B opponent cells seems to be clear. Derrington, Krauskopf, and Lennie (1984) showed that some LGN cells receive antagonistic inputs from S cone vs. L and M cones. Dacey and Lee (1994) later identified that the small bistratified RGCs (which project to the K cells in the LGN) are responsible for carrying such signals. The small bistratified RGCs are excited by S cone responses and inhibited by L and M cone responses (or vice versa). Since blue-ish lights produce strong S cone responses and red/green lights produce strong L/M cone responses (recall red + green is yellow), it stands to reason that if a cell is excited by S cones and inhibited by L and M cones, it would give a vigorous on-response under blue lights and a vigorous off-response under yellow lights, producing the kind of yellow-ON/blue-OFF spectral tuning curve that we see in Figure 4.11 (C).
Figure 4.12 (A) illustrates the potential Receptive Field (RF) of a yellow-ON/blue-OFF small bistratefied cell, and (B) shows the neutral circuitry that gives rise to such an RF (but also see Field et al. (2007)). The small bistratefied RGC have a center-only RF, which receives excitatory responses from a S-cone bipolar cell and inhibitory responses from another class of bipolar cells that are connected to L and M cones. Dacey and Lee (1994) records both the membrane potential and the spiking rate of a small bistratified RGC, shown in (C), under periodic, out-of-phase blue and yellow (red+green) lights. The cell’s responses are the strongest under maximum yellow light (maximum excitatory S cone responses) and minimum blue lights (minimum inhibitory L and M cone responses).
R-G Opponent Cells
Derrington, Krauskopf, and Lennie (1984) showed that most of the midget RGCs (and thus P cells in LGN) are either excited by L cone responses and inhibited by M cone responses (L-ON/M-OFF) or the other way around. Given that, loosely, L cones are excited by red-ish lights but not so much by green-ish lights and M cones behave oppositely, it stands to reason that L-ON/M-OFF cells produce vigorous on-responses (above spontaneous rate) under red lights and vigorous off-responses (below spontaneous rate) under green lights, giving a spectral tuning curves shown in Figure 4.11 (A).
The actual RF structure of these cells takes two forms (Wiesel and Hubel 1966). Some of these cells have a center-surround RF, so there are four combinations: L+/M- (L center-ON/M surround-OFF), L-/M+, M-/L+, and M+/L-. Other midget RGCs have no center-surround arrangement. The excitatory and inhibitory regions have the same spatial extent. Either way, signals from the L cones and M cones are antagonistic in these cells.
Non-Opponent Cells
Finally, the parasol RGCs (and thus M cells in LGN) seem to be the most probable source for the luminance opponent mechanism (B. Lee, Martin, and Valberg 1988). These cells do have a center-surround RF but the L cones and M cones contribute to both the center and the surround (Wiesel and Hubel 1966); S cones seem to be contribute little, if any, to these cells (Lennie, Pokorny, and Smith 1993). When the total excitation by the L and M cones to the center out-weighs the total inhibition to the surround, the entire cell appears to be excited by L and M cone responses, giving a broadband, non-opponent spectral tuning curve in Figure 4.11 (E); otherwise we see a tuning curve like Figure 4.11 (F).
4.4.3 A Cone-Opponent Model for Color-Opponent Mechanisms
It is clear that there are cells that receive opponent cone signals; the spectral tuning curves of these cells seem to largely account for the perceptual opponent mechanisms. Based on these observations, Derrington, Krauskopf, and Lennie (1984) proposed a cone-opponent color space, which is now commonly used (in color science and, to a large extent, visual neuroscience) to give a first-order approximation of the perceptual color-opponent processes. The color space is now famously known as the DKL color space 6.
- The Y-B channel is given by \(a\)S-(\(b\)L+\(c\)M), where \(a\), \(b\), and \(c\) are all positive values representing the contributions of the S, L, and M cones to the Y-B opponent process. It is generally said that this signal is delivered by the Koniocellular pathway.
- The R-G channel is given by \(d\)L-\(e\)M, where \(d\) and \(e\) are all positive values representing the contributions of the L and M cones to the R-G opponent process. This opponent signal is generally said to be delivered by the Parvocellular pathway.
- The Light-Dark or luminance channel is given by \(f\)L+\(g\)M, where \(f\) and \(g\) are all positive values representing the contributions of the L and M cones to the luminance channel. This luminance channel is meant to represent the LEF (Section 4.3.2), which generally is believed to be delivered by the Magnocellular pathway.
The DKL space operates not on raw cone responses but on response contrasts with respect to a perceptually neutral/achromatic color. The inherent assumption is that the achromatic color should have no strength in any of the three cone-opponent channels and be the origin in the cone-opponent space. The achromatic color depends on an observer’s state of chromatic adaptation, a topic we will discuss later in Section 7.3. People usually fit data to regress the values of the free parameters, and the exact values depend on which cone fundamentals are used and the normalization convention. Brainard (1996) describes one such procedure.
Since the cone-opponent model operates on (contrast of) cone responses, a common theory of color vision is that it is a two-stage process: the wavelength encoding by cone photoreceptors followed by opponent encoding of cone responses post-receptorally. While the cone response encoding can perfectly explain the color matching experiments as we have see earlier, the cone opponent encoding is only an approximation of the hue cancellation experiments, as we will see next.
4.4.4 There are Many Inconvenient Truths
The cone-opponent model is a good approximation for behavioral color-opponent mechanisms, but there are many inconsistencies between these two. Reconciling the two and thus elucidating how humans perceptually code opponent hues is still an open research question.
P and K Pathways Do Not Fully Account For R-G and Y-B Opponent Processes
The opponent neurons clearly have what it takes to start accounting for the perceptual opponent processes, but the spectral tuning curves of those neurons have only a weak correlation with the hue cancellation curves. Thus, it is unlikely that excitation and inhibition in opponent neurons cause our perception of red-green and blue-yellow opponency.
The most jarring difference appears in the R-G process. The R-G hue cancellation curve (Figure 4.8 (C)) shows two perceptually neutral colors, as there are two zero-crossings. However, the spectral tuning curve of the R-G neurons (Figure 4.11 (A–B)) shows only one zero-crossing. These neurons do not predict the R-G neutral color in the short-wavelength range and, by extension, cannot explain the fact that short-wavelength violet-ish lights appear to have a red hue. Derrington, Krauskopf, and Lennie (1984) (also see Wandell (1995, fig. 9.18)) shows a great deal of variation of the spectral tuning property within P cells, making them even less certain as the sole candidate for R-G opponent mechanism.
In fact, people have shown that the perceptual R-G hue cancellation data can be fit by \(a'\)L-\(b'\)M+\(c'\)S, where \(a'\), \(b'\), and \(c'\) are cone contributions (Poirson and Wandell 1993; Bäuml and Wandell 1996). Intuitively, the contribution by S cones in the short-wavelength range could give rise to a positive response there. However, there is no physiological evidence that L cone and S cone responses combine at some point in the visual pathway, suggesting the phenomenological nature of these models.
Even though the K pathway clearly shows the capability of carrying S vs. L+M signals, the latter do not accurately predict Y-B neutral signals and, thus, do not fully account for the Y-B hue opponency. That is, a color that leads to a null response (no significant increase or decrease compared to the spontaneous response rate) in the L-M channel is not perceptually pure yellow or pure blue (Shevell and Martin 2017, fig. 4f). Similarly, a color that causes a null response in the S-(L+M) channel is not perceptually pure red or pure green. That is, null-response colors in the DKL cone-opponent space are not perceptually neutral in the hue-opponent space, implying fundamental discrepancies between cone-opponent and hue-opponent spaces.
M Pathway Does Not Fully Account For Luminance
The Magnocellular pathway (starting from the parasol RGCs) is said to be responsible for the dark-light opponent cells, but that poses a dilemma. We know that parasol RGCs have large RFs. A large RF is equivalently to applying an aggressive low-pass filter to the optical image; as a result, the M pathway has a low spatial acuity. So if the M pathway is fully responsible for mediating our luminance perception, we should be insensitive to spatial blurring (low-pass filtering) in the luminance signal. But the result is the opposite: our vision is very sensitive to spatial blurring in in the luminance channel (but relatively insensitive to blurring in the two color opponent channels).
This is illustrated in Figure 4.13, where we take an image, decouple it into three channels: luminance, red-green, and blue-yellow. We then spatially blur one of the channels while keeping the other two channels unchanged and then reconstruct the image. Our vision is much more sensitive to spatially blurring in the luminance channel (a) than is to blurring in the red-green channel (b) and in the blue-yellow channel (c); in fact, this is the basis of chroma subsampling, a key step in modern image and video compression algorithms. This suggests that the M pathway alone cannot be exclusively responsible for our luminance perception.
Gouras and Zrenner (1979) also shows that P cells, which are ordinarily thought of as L-M spectrally-opponent, could also give a LEF-like spectral tuning curve as if it acts as the luminance channel. The reason is that the surround signals reach a cell later than do the center signals, so at a high frequency the out-of-phase center-surround signals can actually come in the same phase.
Hue-Opponent Space is Not a Linear Transformation from Cone Space
It is perhaps not surprising, by now, that if there is a color space that can fully account for the perceptual coding of opponent hues, it is never going to be a linear transformation from the LMS space (or any other space that is a linear transformation away from the LMS space, e.g., the CIE 1931 XYZ space or the DKL cone-opponent space).
As we have seen above, for instance, the DKL space (Derrington, Krauskopf, and Lennie 1984), which is a linear transformation from the LMS cone space, does not fully account for the perceptual opponent processes, e.g., does not predict any unique hue. People have shown that one can construct a linear transformation from the LMS space that can accurately predict three of the four unique perceptual hues by fitting data from psychophysical measurements that do not presuppose the existence of opponent mechanisms (Poirson and Wandell 1993; Bäuml and Wandell 1996), but they cannot predict the fourth unique hue. Schrödinger (1925) also estimated a linear transformation between the cone response space and the hue-opponent space based on the four unique hues, but the transformation could not accurately predict the achromatic color (also see the commentary by Zaidi in Schrödinger (1994)).
The reason is that perceptually unique red and green hues are not collinear with white, the achromatic color that is perceptually neutral in both the Y-B and R-G channel, i.e., does not appear yellow, blue, red, nor green 7. That is, red, white, and green do not lie on a line. Why is this significant? Assuming there was a linear transformation \(T\) from the cone responses to the strengths of the hue-opponent mechanisms:
\[\begin{align} \begin{bmatrix} \text{Y/B}\\ \text{R/G}\\ \text{Lum} \end{bmatrix} = T \times \begin{bmatrix} L\\ M\\ S \end{bmatrix} \end{align}\]
Both unique red hue (\([L_R, M_R, S_R]\)) and unique green hue (\([L_G, M_G, S_G]\)) have no yellow (or blue) hue, so their response in the Y-B channel response would be 0:
\[\begin{align} \begin{bmatrix} \text{0}\\ |\\ | \end{bmatrix} = T \times \begin{bmatrix} L_R\\ M_R\\ S_R \end{bmatrix},~~~ \begin{bmatrix} \text{0}\\ |\\ | \end{bmatrix} = T \times \begin{bmatrix} L_G\\ M_G\\ S_G \end{bmatrix} \end{align}\]
Therefore, any mixture of the unique red hue and the unique green hue would not appear to have a yellow hue either:
\[\begin{align} \begin{bmatrix} \text{0}\\ |\\ | \end{bmatrix} = T \times \begin{bmatrix} a L_R + b L_G\\ a M_R + b M_G\\ a S_R + b S_G \end{bmatrix}, \end{align}\]
where \(a\) and \(b\) are contributions of red and green to the mixed color. However, we know that when we mix red with green colors we get yellow. The fact that two colors without any yellow hue can generate a color that does have a yellow hue means the hue-opponent space cannot be a linear transformation from the LMS cone space.
Figure 4.14 illustrates this point with some real data. The empty markers are three sets of perceptually unique hues (which do not have to be spectral colors) measured psychophysically in Bäuml (1993). When we fit a straight line across each set of unique hues and extrapolate the line we can estimate what spectral colors are unique hues (blue Ⓑ, green Ⓖ, and yellow Ⓨ). No spectral color is seen as a unique red hue (all spectral red-ish colors appear to have a yellow hue), which requires a mixture of unique blue hue and a spectral red to cancel the yellow percept (Dimmick and Hubbard 1939b; Larimer, Krantz, and Cicerone 1975) (and also see the commentary by Zaidi in Schrödinger (1994)). Dimmick and Hubbard (1939a) measured that unique red hues Ⓡ are complementary to a spectral light at 494 \(\text{nm}\); that is, spectral light at 494 \(\text{nm}\), white Ⓦ, and unique red hues should fall on a straight line 8.
Bäuml and Wandell (1996) constructed a linear transformation from the cone space to the hue-opponent space that can accurately predict the unique spectral hues of blue, green, and yellow. It is comforting, and corroborates others (Larimer, Cicerone, et al. 1974), that blue Ⓑ, white Ⓦ, and yellow Ⓨ are collinear, as would be required by a linear transformation from the cone space to the hue-opponent space: mixing colors that have no green or red hue will not give a color that does. But clearly the predicted red hue Ⓡ’ deviates significantly away from red hue Ⓡ from actual measurements. If we connect the unique green hue Ⓖ and unique red hue Ⓡ, the line would not across Ⓦ. This suggests that a simple linear transformation does not exist; at least the Y-B null-response axis is not linear with respect to the cone responses. Non-linear models have been proposed (Larimer, Krantz, and Cicerone 1975; Shevell and Martin 2017).
4.5 Evolution of Color Vision
See Lamb (2013), Lamb (2020), Lamb (2022), Lamb, Collin, and Pugh (2007), Shichida and Matsuyama (2009), Jacobs (2009), Bowmaker (2008), and Bowmaker (1998) for comprehensive discussions. We provide a concise summary of what is relevant to our discussions in this chapter.
4.5.1 The Rise of Trichromacy
Proto-Vertebrates Had Four Cone Opsin Genes
When studying the evolution of vision and performing comparative studies of vision across species, we must understand the differences in all the molecules that participate in phototransduction and the subsequent neural circuitry. The photopigment/opsin itself is one of the most important components (e.g., pigments differ in their peak absorption wavelength), and its evolution is the most well-understood. It is the main focus of our discussion.
A primordial opsin gene that existed in ancestral bilateria (before the separation of protostomes and deuterostomes), through duplications and mutations, gave rise to many opsin genes, some of which are even expressed in non-visual opsins (e.g., ipRGCs and RPEs). One of them is called the Long-Wavelength Sensitive (LWS) opsin gene. The LWS opsin gene was duplicated multiple times, each of which went through mutations, and eventually four opsin genes existed in proto-vertebrates. The sequence of duplications most likely went like the following. The LWS gene was duplicated, and the duplicated copy evolved to be Short-Wavelength Sensitive and is called the SWS opsin gene. The SWS gene was duplicated again. One of the copies is called the SWS1 gene. The other was duplicated once again. Of its two copies, one is called the SWS2 gene, and the other is called the RHL gene. RHL means “rod like”; it is given the name because its duplicate would later evolve to be expressed in rhodopsin (photopigment in rods).
As a result, proto-vertebrates (primitive animals from which the vertebrates evolved; think of them as ancestors to vertebrates but are not vertebrates themselves) possessed 4 opsin genes: LWS, SWS1, SWS2, and RHL. These genes were expressed in pigments that mediate photopic vision, just like modern cones, so we will simply call them cone opsins, but keep in mind that proto-vertebrate “cones” are most definitely different from modern vertebrate cones.
Generally, a modern form of a gene is different from its ancestral form, but for simplicity, we usually call them by the same name. This might be a common source of confusion when studying evolutionary biology. For instance, modern vertebrates all have LWS opsin genes (that are slightly different); they are all evolved from, but most definitively different from, the ancestral LWS gene at the time of duplication that gave rise to the SWS branch. It is perhaps more rigorous to say, “the ancestral LWS was duplicated; one copy evolved to become the modern LWS genes, and the other evolved to become the ancestral SWS gene, which was duplicated again; its two copies are the ancestral SWS1 gene and the ancestral SWS2 gene”, but you can see how cumbersome this would be. Even with this it is sometimes unclear at which point in the evolution does “ancestral” refer to.
Most Vertebrates are Tetrachromatic and Most Mammals are Dichromatic
Then something remarkable happened. During the Cambrian epoch, at around 530 to 500 million years ago (Mya), there were two rounds of Whole Genome Duplication (WGD), a.k.a., polyploidy. Each WGD made a complete copy of all the genes. So two rounds of WGD would quadruple the number of genes. Each WGD was followed by a relatively short period of genome instability with extensive loss of genes. As a result, not all four copies (a result of two rounds of WGD) of a gene were retained. WGD in general is a major source of speciation, and the two rounds of WGD were responsible for the explosion of new species at the end of the Cambrian epoch (Cambrian explosion). WGDs increased the number of genes and made more genes available for mutation. As a result, there was a sudden radiation of species and diversification of life. Perhaps most importantly to human evolution, vertebrates appeared after the first round (1R) WGD.
As far as the four cone opsin genes are concerned, all their copies were lost except two copies of RHL, which are now called RH1 and RH2. So after the two rounds of WGD, our ancestral vertebrates possessed five cone opsin genes: SWS1, SWS2, RH1, RH2, and LWS. RH1 evolved to be expressed in rhodopsins (pigments for rods) to mediate scotopic vision in vertebrates; the other four evolved too but retained their ability to mediate photopic vision. This is why most vertebrates are tetrachromatic. From the sequence of duplications, we can also deduce that the rod pigment evolved after all four classes of cone pigment were already present (Okano et al. 1992). In fact, rod signals largely piggyback onto the pre-existing circuitry for cone signaling (Lamb 2016).
Early mammals arose at the age of dinosaurs and were nocturnal, so they had no need for strong color vision and lost two of the four cone opsin genes. Only LWS and SWS1 genes were retained. This is why most mammals are dichromatic.
Figure 4.15 shows the phylogenetic tree of vertebrate visual opsins, of which there are five classes. I say “class” of opsin genes here because there are variations between, say, the SWS1 gene in humans and that in mice. Circles represent gene duplications. The first duplication on the ancestral LWS gene was local and gave rise to the SWS branch. The other duplications within the SWS branch are part of the two rounds of WGDs (which, recall, were followed by gene losses, which are omitted in the diagram). The local duplication within the LWS branch gave humans trichromacy, which we will discuss next.
LWS Gene Duplication in Catarrhini Gave Human Trichromacy
Then primates evolved from some mammals, first to prosimians, and then anthropoids (higher primates) split off from prosimians. Anthropoids include Platyrrhini (broad noses) and Catarrhini (narrow noses). Platyrrhini evolved into modern New World (South America) monkeys. Catarrhini is the common ancestor of Old World (Africa and Asia) monkeys, apes, and humans.
Trichromacy emerged in an early Catarrhini through gene duplication. The LWS gene is duplicated. One copy is evolved to be expressed in modern L cones, and the other evolved to be expressed in modern M cones. Since the duplication occurred “only” about 30 Mya, which is fairly recent from an evolutionary standpoint, the L and M cone pigments are very similar. 96% of the amino acids in the L cone opsin and M cone opsin are the same (Nathans, Thomas, and Hogness 1986): they simply have not had much time to be mutated enough yet. With the SWS1 gene being expressed in S cones, all Catarrhini (including humans) are trichromatic. Figure 4.16 illustrates the duplication and the spectral sensitivities before and after the duplication.
What was the evolutionary pressure for the duplication that gave rise to the trichromacy? Since the duplication of the LWS gene gave us medium-wavelength pigments that peak at green-ish lights, one interesting theory is that the duplication offered the ability to distinguish red-ish colors from green-ish colors in order to select ripe from unripe fruit (or ripe fruits against the background of green leaves), which had an obvious evolutionary advantage (Hunt et al. 1998; Bowmaker 1998, p. 544).
4.5.2 The Rise of Scotopic vs. Photopic Vision
Only Jawed Vertebrates Have a ``Modern’’ Scotopic Vision
Vertebrates arose after 1R. 1R is important for the rod-cone duplex vision in today’s vertebrates. RH1, which eventually is evolved to be expressed in rhodopsins, appeared after 1R. 1R also duplicated other non-opsin genes important for phototransduction (e.g., G proteins, PDEs, cGMPs, etc.) (Lamb 2020). These genes evolved to be expressed in distinct isoforms of many of the molecules that participate in the rod vs. cone phototransduction cascades.
Jawless and jawed vertebrates (from which humans evolved) split after 1R, so they both possessed distinct scotopic and photopic vision. But there are differences in the scotopic vision between the two. Why? Because the second round (2R) WGD happened only to jawed vertebrates. 2R introduces a few new genes that allow jawed vertebrates to possess ``true’’, modern rod photoreceptors, where rod pigments are more thermally stable (lower dark noise) than cone pigments and regenerate faster than cone pigments (Lamb, Collin, and Pugh 2007; Lamb 2022). In jawless vertebrates, rods also have a cone-like anatomical structure, but 2R changed the morphology of rods in jawed vertebrates so that their rods look different from cones, e.g., having sealed-off discs. Interestingly, morphology apparently is not important for scotopic vision: northern hemisphere lampreys (a kind of jawless vertebrate) have rods that look like cones but physiologically behave like rods, e.g., they are very sensitive to lights, so they do mediate scotopic vision (Morshedian and Fain 2015). Whether you call the these photoreceptors rods or cones is largely a matter of definition.
Scotopic Vision Predates Vertebrates
It is also interesting to note that proto-vertebrates, likely some chordate ancestors of ours, also possessed distinct photopic and scotopic vision way before vertebrates, except those ancestral scotopic photoreceptors utilized pinopsin (Okano, Yoshizawa, and Fukada 1994), rather than rhodopsin, as their photopigments (Lamb 2020; Sato et al. 2018). Pinopsin evolved prior to 1R and, in fact, was duplicated along with LWS cone opsin in proto-vertebrates from a ciliary opsin (C-opsin).
After 1R, presumably RH1-expressed rhodopsins had better performance under scotopic conditions, so they gradually superseded pinopsins to be the pigments expressed in scotopic photoreceptors. Mammals do not possess pinopsins anymore, but some vertebrate species still use pinopsins for scotopic vision (Sato et al. 2018), and many vertebrates use pinopsins for non-vision functions such as regulating circadian rhythm (as they are expressed in the pineal organ) (Takanaka et al. 1998).
4.6 Color Vision Deficiencies
Normal color vision is trichromatic in that there are three classes of cone photoreceptors on the retina. If the retina is deprived of the functions of one of two cone classes, the color vision is no longer trichromatic. Instead of being represented in a three-dimensional space, a color would now be expressed in a two-dimensional or one-dimensional space. Individuals with two functioning classes of cones are dichromatic, and those with one functioning cone class are monochromatic. Dichromatic vision is further classified into three types: Protanopia, where L cones are missing; Deuteranopia, where M cones are missing; and Tritanopia, where S cones are missing. An interesting form of dichromatic vision is called small-field dichromacy. At the central 20 arcmin of the fovea there are no S cones, so human vision is effectively dichromatic there. There are also individuals who have no functioning cones, only rods, and those people have rod monochromacy.
In addition to strict dichromatic vision, another important form of CVD is anomalous trichromatic vision, where individuals have all three cone types, but the spectral sensitivity of a cone type deviates from normal. Protanomaly, Deuteranomaly, and Tritanomaly are the names given to the three types of anomalous trichromatic vision. Figure 4.17 illustrates how the cone fundamentals change under different anomalous trichromacy compared to normal trichromacy. For protanomalous and deuteranomalous individuals (by far the most common anomalous trichromats), their L and M cone fundamentals are closer than normal, so the L and M cone excitations are very similar. In theory their color vision is still three-dimensional, but the L and M dimensions are very correlated, which weakens their color discrimination ability.
Finally, some females with anomalous trichromacy may have four cone classes on the retinal mosaic owing to random X chromosome inactivation. However, it is unclear whether their color vision is four-dimensional or they have a stronger color discrimination ability (Jordan et al. 2010; Simunovic 2010).
We will primarily focus on dichromatic vision, building a phenomenological model for simulating dichromacy, and then discuss the genetic basis of CVD now that we have a good understanding of the evolution of color vision.
4.6.1 Models of Deficient Color Vision
Perhaps one of the most important questions in CVD is this: what exactly do individuals with CVD actually see? In other words, how do we simulate a particular CVD? It is incredibly difficult to answer, since color is a subjective experience, and we can never be so certain of one another’s subjective experience. With the help of some luck (yes, some luck, from evolution) and some psychophysics, there is at least a consensus on how to model dichromatic vision. Taking Deuteranopia as an example, the interactive tutorial Zhu (2022c) walks through a commonly used model. We will give the main intuitions here.
Confusion Lines
Dichromatic individuals see colors only in a 2D space, because they lack (the functionality of) one cone type. For instance, Deuteranopes lack the M cones and, thus, any color is encoded only in the L and S cone excitations, resulting in a 2D color space. Therefore, any colors that differ only in the M dimension are seen as the same color by a Deuteranope. A line parallel to the M dimension in the LMS space is called a Deuteranopic confusion line, as all colors on that line look the same to a Deuteranope. The left panel in Figure 4.18 plots two such confusion lines for Deuteranopia, although it is easy to see that there are infinitely many confusion lines.
The confusion lines are more commonly visualized in the CIE 1931 xy-chromaticity diagram, as illustrated in the three diagrams in Figure 4.18 to the right. We will discuss the xy diagram in the colorimetry lecture soon, but briefly, the xy diagram is a perspective projection from the LMS space that provides a useful 2D representation of colors by discarding the luminance dimension. Therefore, parallel confusion lines in the LMS space converge in the xy diagram.
Not all colors on a dichromatic confusion line are confusing to the corresponding anomalous trichromatic individuals. From a modeling perspective, a typical approach is to restrict the confusing colors to a small segment on a confusion line (Flatla et al. 2015). Figure 4.19 illustrates a model to reason about this using Protanomaly as an example. C1 and C2 are two colors that differ only in the L cone response. For simplicity, Figure 4.19 shows only the L and M dimensions. Since the L and M cone fundamentals are very similar, the L and M cone responses are similar too (the extreme case being L = M). So C1 and C2 are mapped to C1’ and C2’ in a protanomalous color space, where both colors are pulled toward the L=M line. C1’ and C2’ are never going to exactly coincide, because the L and M cone fundamentals do not exactly overlap, but when C1’ and C2’ are sufficiently close, they can still be perceptually undiscriminable. This is not unique to CVD individuals: even for normal trichromats the color discrimination is not perfect, as has been demonstrated extensively through psychophysics Krauskopf and Karl (1992), which we will discuss in Section 5.7 soon.
Isochromes
We know that all colors on the same confusion line are perceptually the same, but still we do not know, for all those colors on a confusion line, exactly what is the color a dichromate sees. The key to answering this question is the notion of isochromes, which are colors perceived correctly by a dichromate. The question is, how do we find the isochromes?
It is impossible to find isochromes by simply querying trichromats and dichromates. Imagine we have a normal trichromat and a Protanope looking at a color; even if they have the same color sensation, how would they communicate with each other about it? You might be tempted to find isochromes by asking a dichromate whether two colors appear the same.
Remarkably, there is an exceedingly rare form of CVD called unilateral dichromacy, where an individual has one dichromatic eye and another trichromatic eye. Color matching between the two eyes by a unilateral dichromate would allow us to identify isochromes, assuming, of course, that the dichromatic eye and the trichromatic eye are similar to those of a “normal” dichromatic and trichromatic eye, respectively. This is a remarkable luck we get from nature; without unilateral dichromats, we might never be able to quantitatively study dichromats’ color vision. There are a handful of studies reported on unilaterial dichromates. Judd (1949) meticulously summarized data from prior studies, where only 8 had quantitative data that were useful. Sloan and Wollach (1948), Graham and Hsia (1958), and MacLeod and Lennie (1976) reported results for unilateral Protanopes/Deuteranopes, while Alpern, Kitahara, and Krantz (1983) reported results for a unilateral Tritanope.
Such studies show that monochromatic lights at 475 \(\text{nm}\) and 575 \(\text{nm}\) are isochromes for protanopes and deuteranopes (i.e., no significant difference between these two types when it comes to isochromes, but of course their confusion lines are different), and for tritanopes isochromes are found at 485 \(\text{nm}\) and 660 \(\text{nm}\).
Brettel et al. Model
There have been two main ways to build a model for dichromatic color vision: that of a phenomenological nature such as Brettel, Viénot, and Mollon (1997), Viénot, Brettel, and Mollon (1999), and Meyer and Greenberg (1988) and that based on first principles of the visual pathway, such as Jiang, Farrell, and Wandell (2016) and Rodriguez-Pardo and Sharma (2011). We will primarily focus on the phenomenological model described in Brettel, Viénot, and Mollon (1997). Like all other models, this model is not perfect. It makes assumptions that are not experimentally validated on unilateral dichromats, but it is a popular model that seems to work well in practice. Zhu (2022c) discusses cases where this model falls apart.
Brettel, Viénot, and Mollon (1997) assumes that Equal-Energy White (EEW) is also an isochrome. In fact, they assume that the entire plane that contains EEW, 475 \(\text{nm}\), 575 \(\text{nm}\), and Black is an isochrome plane, on which all colors are isochromes for deuteranopes and protanopes. Similarly, the plane that contains EEW, 485 \(\text{nm}\), 660 \(\text{nm}\), and Black is an isochrome plane for tritanopes. Two caveats must be noted here. First, this is not validated against unilateral dichromats; it is just an assumption. Second, the isochrome “plane” is not an actual plane. It is more like two half-planes that share a border. For Protanopia and Deuteranopia, the two half-planes are almost parallel so that they look like part of one plane. The left panel in Figure 4.20 shows the two half planes (with distinct colors) for Protanopia.
Assuming that the isochrome plane assumption by Brettel, Viénot, and Mollon (1997) is true, we can then reason about the colors that dichromats actually see: the intersection between the isochrome planes and a confusion line is the color a dichromat actually sees for all the colors on that confusion line. The left panel in Figure 4.20 visualizes this, where the trichromatic spectral locus is projected to the isochrome planes along the direction of the confusion lines. The resulting locus lies completely on the isochrome planes and represents how the spectral colors will actually be perceived by a Protanope.
The two isochrome half-planes become two line segments in the xy-diagram. The right panel in Figure 4.20 shows how the Brettel, Viénot, and Mollon (1997) model predicts a Deuteranope’s color perception in the xy-diagram.
Modeling anomalous trichromacy is “easy” as long as we know the new set of cone fundamentals. Assuming their subsequent neural processing is the same as that of normal trichromacy, we can then calculate the cone responses in an anomalous trichromatic color space given any light spectrum.
4.6.2 CVD Assistive Technologies
There are many assistive techniques attempting to enhance the color vision of CVD individuals. Zhu et al. (2024, sec. 3) provides a good review.
Re-Coloring Helps Color Discrimination
By far the most commonly used technique is called “re-coloring”; it is available on most Operating Systems in laptops, PCs, and smartphones. The idea is to apply a (usually linear) transformation to colors in an image (colloquially referred to as color filters) so that initially confusing colors become distinct (i.e., no longer on a confusion line).
The main limitation of re-coloring is that, while initially confusing colors might be distinguishable after a transformation, these colors will inevitably be confused with others. Fundamentally, a dichromat’s color vision is still two-dimensional, and the confusion lines are still there. Usually the transformation is designed so that a set of ecologically relevant, confusing colors (colors that we commonly encounter in everyday life and are important to discriminate, e.g., red flowers and green leaves) become distinct.
Re-coloring can also be done optically. Many commercially available glasses for CVD individuals, such as those from EnChroma and VINO, use identical spectral “notch filters” for both eyes. The filter eliminates light from a narrow spectral band, where the human L and M cone sensitivities overlap the most. Recall that for Protanomalous and Deuteranomalous individuals, their L and M cone fundamentals are closer than normal, so the L and M cone excitations are very similar. The notch filters act to pull the two cone fundamentals apart and amplify the difference between L cone and M cone excitations. Therefore, in principle, the method can enhance color discrimination for anomalous trichromats but provides no benefit for strict dichromats. In comparative tests, the functional effectiveness of such glasses for anomalous trichromats is also not definitive (Gómez-Robledo et al. 2018; Patterson et al. 2022).
Color Recognition By Reconstructing the Missing Dimension
Re-coloring methods, thus, do not help with color recognition and naming (“pass me that pink marker” or “look at the person in a red shirt”), which is a common user complaint found in a user study (Geddes, Flatla, and Connelly 2023). For color recognition, we must restore a three-dimensional color space for dichromats. So the key is to somehow reconstruct the missing dimension.
One idea is to introduce binocular color disparity, where the stimuli are differentially altered for the two eyes. The idea was originated by Maxwell (1857) and then later revived by Cornsweet (1970). Maxwell conjectured that the disparity across the two eyes would essentially introduce a new dimension of perception, which would augment the existing 2D percept of a dichromat, providing 3D color perception. Knoblauch and McMahon (1995) shows that the improved color discrimination afforded by binocular filters might have limited use for Protanopes and Deuteranopes.
Zhu et al. (2024) introduces a smartphone App to reconstruct the missing dimension through temporally modulating colors. The idea is that as a user swipes their finger in the App, they apply a color-space transformation such that originally confusing colors undergo distinct color shifts. The combination of the initial 2D color precept with the induced temporal shifts reconstructs a new 3D space for the user. By spending time interacting with our system, users then learn to associate different color names in this new 3D space, thereby recognizing colors.
4.6.3 Genetic Basis of CVD
L and M cone genes are on the X chromosome. Because the L and M genes are created from a duplication, they are tandemly arrayed (spatially adjacent in a head-to-tail manner) and, thus, are subject to crossovers during recombination of meiosis, which might lead to deletion of a whole gene or hybrid genes in a X chromosome. This is the genetic basis of Protanopia, Deuteranopia, Protanomaly, and Deuteranomaly. Onishi et al. (2002) shows that only two in a sample of over 3000 macaque monkeys (an Old World monkey) have CVD, much lower than the CVD rate in humans, indicating that the crossovers might be recent (B. B. Lee 2008).
Intergenic Crossovers Give Protanopia and Deuteranopia
Figure 4.21 illustrates a crossover that can potentially lead to Deuteranopia. The two X chromosomes, due to an intergenic crossover, either lose or gain an M gene after recombination. One of the new X chromosomes has only the L cone gene, so an individual inheriting that X chromosome from the mother would get Deuteranopia. Interestingly, while the other X chromosome gets an additional M open gene, only the first two genes are sufficiently expressed. The fact that the L and M cone genes are in an X chromosome means that biological females are less vulnerable to Protanopia and Deuteranopia than biological males, simply because there are two X chromosomes in females. Even if one of the inherited X chromosomes has only, say, a L cone gene, the other inherited X chromosome, if normal, can still be sufficiently be expressed to give both L and M cones.
Intragenic Crossovers Give Protanomaly and Deuteranomaly
Intragenic crossover might also occur during recombination, and this would lead to anomalous trichromacy. Figure 4.22 shows an intragenic crossover that might give rise to Deuteranomaly and “Protanopia”. The second X chromosome after recombination has a normal L opsin gene and another gene that is a mixture of the L cone gene and the M cone gene. The spectral sensitivity of such a hybrid pigment is in between that of an L cone and an M cone (Sharpe et al. 1999), as illustrated in the left panel in Figure 4.23. This is the source of anomalous trichromats.
Intergenic Crossovers Can Give Abnormal Protanopia and Deuteranopia
It is interesting to observe that intragenic crossovers can also give a dichromatic vision, although in a somewhat abnormal form. Observe in Figure 4.22 that the first X chromosome after recombination also has a hybrid gene. If the L cone gene dominates, that gene, when inherited, will be expressed in a pigment that has a spectral sensitivity that is closer to that of a L cone. If you want, you can still say that the inheriting individual has “Protanopia”, but its L cone sensitivity is different from that of a normal L cone.
The right panel in Figure 4.23 illustrates another scenario where dichromatic vision can arise from intragenic crossovers. If the two hybrid genes in the chromosome are sufficiently similar, they will be expressed in two pigments that are sufficiently similar, and the individual effectively has a dichromatic vision (Sharpe et al. 1999).
Deuteranomaly is the Most Common CVD
Tritanopia and tritanomaly are much rarer than the other forms of CVDs. The gene for S cone opsin is in a nonsex chromosome (chromosome 7), which is not subject to L/M gene crossovers. Gene mutations causing changes in S-cone pigment are much more rare. Among CVDs that are caused by the crossovers, anomalous trichromacy is more common than strict dichromats, and Deuteranomaly is the most common (about 4.9% in caucasians) (Wyszecki and Stiles 1982, Table 1(5.4.2), p. 464), but the statistics certainly vary across ethnic groups (Birch 2012).
After all, artificial lights are a very recent thing in the scale of evolution, so our HVS has not had a chance to adapt to non-daylight colors yet, if ever.↩︎
One subtlety is that Baylor, Nunn, and Schnapf (1987) used suction electrode to measure electrical responses (Section 3.2.2), so they obtained only the relative absorbance not the absolute absorption of the pigments. So what they actually ended up doing is to use the psychophysical CMFs to fit the peak axial absorption and calculate the cone fundamentals, and show that the regressed CMFs from the so-obtained cone fundamentals match that from psychophysics.↩︎
In later research by Jameson and Hurvich, their white-black function was made equal to the CIE 1924 luminous efficiency function (Hurvich and Jameson 1955, p. 604), which is known to have severe flaws at low wavelengths and which is later corrected by Judd (1951) and Vos (1978). Compared to the Judd and Vos corrections, the function shown here has the advantage of being “physiologically relevant” in that the LEF is a linear combination of the cone fundamentals, whereas both the CIE 1924 LEF and its later corrections are not intentionally designed to be linear combinations of anything.↩︎
named after the three authors; the L is Peter Lennie, who was twice on the faculty at University of Rochester and served as the Provost↩︎
Again, what is considered achromatic depends on the observer’s adaptation state; there is no single achromatic color.↩︎
Of course, it is conceivable the result might vary in population and depend on the adaptation state (i.e., what is considered white/achromatic).↩︎