From Trichromacy to Color Opponency Timothy Klitz Final Project, Psy 5036

History of Trichromacy

à The Young-Helmholtz Model of Color Vision The first organized theory to attempt to explain color vision was developed in the 19th century by Young and Helmholtz. Their explanation of simple color discrimination performance in the vision system is based upon the responses of three color mechanisms in the eye, which we now know as the three types of cone photoreceptors. Each of the three cones have a characteristic spectral absorption curve, i.e. each cone type absorbs the wavelengths of visible light in a specified way. The three channels are classified as red (also called long), green (medium), and blue (short) wavelength channels. The red channel does not absorb only red light, but it's maximal sensitivity is at wavelengths that correspond to red light. The same applies for the blue and green channels.

à Physiological Measurements of Cone Absorption It was not until the middle of the 20th century that physiological measurements of the sensitivities of the photoreceptors were made. Simply, samples of each type of photoreceptor were extracted from the eyes of humans or monkeys, and the absorption characteristics of the photoreceptors, as a function of wavelength of light impinging on the photoreceptor, were measured systematically. Each photoreceptor contains a different make-up of light-absorbing chemicals, explaining the fact that each of the types of photoreceptors have different absorption characteristics, and thus different peak sensitivities (red = 559 nm, green = 531 nm, blue = 419 nm).

Ÿ Procedures Used in Physiological Measures by Microspectrophotometry In 1983, Dartnall, Bowman & Mollon measured the spectral absorption curves for both rods and cones, using seven eyes from subjects who had their eyes surgically removed for various reasons. In five of the seven cases, the eyes were tested within 90 minutes of removal so that post-mortem changes in sensitivity of the photoreceptors did not occur. The technique of microspectrophotometry was used to examine the eyes. Individual photoreceptors were isolated, and then exposed to a range of wavelengths of light, from 350 to 700 nm. Absorption was measured for each type of cell at intervals of 10 nm. The plot of the original data, seen in most textbooks, depicting the sensitivity of each of the types of photoreceptor, is below.

Ÿ Data from Dartnall, Bowman, and Mollon

Rods = N[{{400,38.4},{410,40.1},{420,42.1},{430,45.7},{440,51.5}, {450,60.5},{460,71.8},{470,83.0},{480,92.4},{490,98.2},{500,99.5}, {510,95.0},{520,85.5},{530,72.5},{540,58.1},{550,43.4},{560,31.5}, {570,22.7},{580,15.8},{590,11.0},{600,7.3},{610,4.7},{620,3.7}}] The following plot represents the mean relative absorbance of each type of photoreceptor, as measured by Dartnall, Bowman, and Mollon. 2 SampleFinalProjectFormatted.nb

Combined = Show[RodPlot,BluePlot,GreenPlot,RedPlot,PlotRange->{ {350,690},{0,100}}, AxesLabel->{"Wavelength(nm)","Absorbance"}]

Absorbance 100

0 Wavelength(nm) 400 450 500 550 600 650

à Psychophysical Measurements of Cone Sensitivity In addition to the physiological information above, Vos and Walraven also derived the relationship between the sensitivity of the photoreceptors and the input wavelength to those receptors. Their measures are derived from the standard CIE primaries in 1931. Note that the spectral response curves indicated below are in different units (sensitivity) than the physiological data (absorbance)

Ÿ Derivation of Vos and Walraven

VWBlue = N[{{400,0.00150},{410,0.00308},{420,0.00572}, {430,0.00777},{440,0.0086},{450,0.00788},{460,0.00689}, {470,0.00596},{480,0.00404},{490,0.00239},{500,0.00141}, {510,0.000826},{520,0.000409},{530,0.00022},{540,0.000106}, {550,0.000047},{560,0.0000205},{570,0.000011},{580,0.00000862}, {590,0.00000572},{600,0.00000631},{610,0.0000024}, {620,0.00000118},{630,0.000000577},{640,0.00000023}, {650,0.000000103},{660,0.0000000358},{670,0.0000000126}, {680,0.00000000337}}] The following plot represents the spectral response curve for each type of photoreceptor(sensitivity multiplied by the standard luminous efficiency function V, as discussed by Vos and Walraven, that defines the standard observer). Note also that this plot is on a log scale. SampleFinalProjectFormatted.nb 3

VWCombined = Show[VWBluePlot, VWGreenPlot, VWRedPlot,AxesOrigin-> {370,-3},PlotRange->{{370,710},{-3,0}},AxesLabel->{"Wavelength( nm)","Sensitivity"}]

Sensitivity 1 0.5

0.1 0.05

0.01 0.005

0.001 Wavelength(nm) 400 450 500 550 600 650 700

Development of Color-Opponent Theory

à The Need for a New Theory of Color Vision It became apparent in the late 1800's that trichromacy theory could not account for all of the color vision phenomena known at that time. In particular, there are many types of color contrast effects that cannot be explained simply with three primary color channels, the main example being the complementary relationships between red and green, blue and yellow, and black and white (i.e a hue can be described in terms of it's redness or greenness, but not by a combination of the two, and a hue can be described in terms of it's blueness or yellowness, but again, not as a combination of the two. These relationships are independent though, in that a color can be described in terms of red-yellow, or blue-green). It was clear that a new theory was needed to explain these phenomena.

Hering, and many others later on, developed a theory of color opponency. In this theory the red, green, and blue channels are transformed at some point in the visual system such that three new channels, based on the additive and subtractive combinations of the three primary channels, are formed. These three channels can then account for the complementary color relationships and other observations made that required a different approach than trichromacy. 4 SampleFinalProjectFormatted.nb

+ R+G+B+ + B G R

Achromatic Channel

+ - + + Cone Channels + - Color Opponent Chann

B+Y- R+G- It was many years before there was any solid physiological or psychophysical evidence to support this idea, but there was eventually evidence. The following discussion describes a mathematical model for deriving color- opponent channels from the three primary channels, and a comparison of these results with psychophysical and physiological data.

à The Buchsbaum and Gottschalk Model of Color Opponency

Ÿ Rationale for the Model The theory of Buchsbaum and Gottschalk is based on the assumption that the visual system acts like an information processing mechanism, in which information must be transmitted from the retina to the brain in an efficient manner. The information in the photoreceptors at the retina must transformed such that the pathways leading from the retina to the brain can pass on that information with the least amount of redundancy as possible.

It is found by examining the spectral sensitivity curves presented above, that the three cone channels have significant spectral overlap, particularly in the case of the red and green cones. Clearly, if information is transmitted to the brain in this way, efficiency will be very poor (i.e. there will be a great deal of redundancy). Buchsbaum and Gottschalk propose that the signals from these highly correlated channels can be transformed into another set of channels that are either uncorrelated or significantly less correlated. In this way, the available channel capacity (3 channels) can be most efficiently used. SampleFinalProjectFormatted.nb 5

Ÿ General Theory and Description of the Model Buchsbaum and Gottschalk cite mathematical evidence that there exists a single transformation that will decorrelate the red, green, and blue channels into a set of three channels such that there is a maximal compaction of the information in the initial signals. In addition, the transformation process also allows a ranking of the three resultant channels based on the amount of the original signal energy that is transmitted through each of the channels. Buchsbaum and Gottschalk explain this by saying that resources can then be allocated such that the channel that transmits the largest portion of the signal energy gets a majority of the transmission resources available.

This entire process of decorrelating the initial cone responses, such that a new set of channels is formed, is called principal components analysis (PCA). It can also be called eigenvector transformation or the Karhunen-Loeve analysis. Following is a description of this procedure for the color channel transformation. 6 SampleFinalProjectFormatted.nb

Principal Components Analysis for Color Channels (Buchsbaum and Gottschalk)

à General Description The response curves for the three cones, as a function of wavelength, can be given as: R(λ), B(λ), and G(λ). An input signal to these cones is given by S(λ). As a result of an input signal impinging on the three cones, each cone has a particular output response given by the following equations:

b = ∫S(λ)B(λ) dλ g = ∫S(λ)G(λ) dλ r = ∫S(λ)R(λ) dλ

These three equations are typically integrated over all wavelengths, i.e. assuming a white nature of light. This is not a necessary assumption.

The key element in this principal components analysis is the covariance matrix, C, of the output responses of the three cones.

The covariance is a measure of how the values of two particular signals are related to each other. For instance,

Crb is a measure of how the red and blue signals covary, i.e. how closely correlated their outputs are given a particular input to those channels. If any two of the channels are related, then the covariance of those two channels is 0. Dividing the covariance of two variables by the product of their standard deviations gives the familiar statistical value of the correlation coefficient. It is the goal of principal components analysis to get the covariance between different channels to be zero, i.e. they are independent (redundancy reduction).

The individual values of the covariance matrix can be written generally as Cxy = E[xy] - E[x]E[y], where E[x] and E[y] are the expectations of the two variables x and y, in this case the expected values of the red, green, and blue channels.

From this covariance matrix are derived it's eigenvectors, of which there are three. These eigenvectors constitute an orthogonal set, i.e. a set of vectors that describe the transformation from the three correlated cone channels to the three opponent-process channels that are uncorrelated. If W is a 3x3 matrix whose columns are the three eigenvectors, then it can be derived that W' (the transpose of W) will operate on the three primary cone channels in such a way that the result will be a vector representing the output responses of the three new uncorrelated opponent-process channels. In addition, the eigenvalues corresponding to the eigenvectors of C represent the variance of the three new channels, which gives us information regarding the amount of signal energy distributed in each of the channels. SampleFinalProjectFormatted.nb 7

If a, p and q are the output responses of the three new channels A(λ), P(λ), and Q(λ), then

|a| |r| |p|=W'|g| |q| |b|

As mentioned previously, a, p and q are uncorrelated, which means that information in each of the channels is not repeated in the other channels. Through some mathematics, it can be found that the values in the covariance matrix C can be represented by the following forms:

2 λ λ) 2 λ λ) Crr = Integral(R ( ) d Cgg = Integral(G ( ) d 2 λ λ) λ λ λ) Cbb = Integral(B ( ) d Crb = Integral(R( )B( ) d λ λ λ) λ λ λ) Crg = Integral(R( )G( ) d Cgb = Integral(G( )B( ) d

Buchsbaum and Gottschalk, and others normally integrate these equations for a white source of light, i.e. they are integrated over the full range of visible wavelengths. As mentioned previously, this is not necessarily the case in general.

Finally, the result of the transformation is in the following form:

|A(λ)| | R(λ)| |P(λ)| = W'| G(λ)| |Q(λ)| | B(λ)|

Once W' is determined by finding the eigenvectors of the covariation matrix C, and since R, G, and B are known from physiological or psychophysical data, then A, P, and Q can be determined. These are the three color opponent channels.

In summary form, here is what the Principal Components Analysis gives us: 1> Three orthogonal channels representing the color opponent channels in the visual system. The outputs of these channels are uncorrelated. 2> The signal energy in the channels is given by the ratio of the eigenvalues of the covariance matrix C, which correspond to the following ratio: Integral(A2(λ) dλ) : Integral(P2(λ) dλ) : Integral(Q2(λ) dλ). 3> The mathematics of the eigenvector transformation insure that only one of the resultant channels will be an all positive linear combination of the three cone channels, i.e. there will only be one achromatic channel ( A(λ) ), and it will have the highest signal energy. 4> The second channel ( P(λ) ) has only one zero crossing of the wavelength axis, and has the second highest signal energy. 5> The third channel ( Q(λ) ) has two zero crossings of the wavelength axis, and has the lowest energy.

à Specific Derivation for the Dartnall et al data The following steps derive the opponent channel formulation via principal components analysis for the physiological data given above by Dartnall et al. Their data represents the three primary cone channels R, G, and 8 SampleFinalProjectFormatted.nb

B as a function of wavelength as given above by Blue, Green and Red. The results of the derivation give a set of three curves (and equations) representing the three opponent channels A, P and Q as a function of wavelength.

The following is the extraction of the absorbance values for the Dartnall data:

The first step in the Buchsbaum-Gottschalk derivation is to find the covariance matrix for the three cone channels. We shall assume here that the equations given earlier for the elements of the covariance matrix are integrated over all of the wavelengths.

These integrations can be done numerically with the cone sensitivity data that is given. For example, Crr = Integral(R2(λ) dλ) can also be written as the dot product of R(λ) and R(λ). The same principle holds true for the rest of the covariances.

CovarianceMatrix = {{Crr, Crg, Crb},{Crg, Cgg, Cgb},{Crb, Cgb, Cbb}} So, now we have the covariance matrix for the Dartnall data. Now we need to find the eigenvectors and eigenvalues of this matrix.

CEigenValues = Eigenvalues[CovarianceMatrix] CEigenVectors = Eigenvectors[CovarianceMatrix] In the discussion previously, it was determined that the matrix of eigenvectors with the eigenvectors as the three rows of the matrix is W'. This is what CEigenVectors represents here. In addition, the eigenvalues give the ratio of the signal energy in each of the three new channels.

W' = CEigenVectors DisplayW' = TableForm[W']

0.68999 0.666461 0.282389 0.684981 -0.727302 0.0428098 -0.233913 -0.163892 0.958344

So, when the matrix W' is multiplied by the vector of cone sensitivity functions, we will obtain the equations for the three opponent channels. The three channels will be called Achromatic, RedGreen, and BlueYellow. The graphs for these three channels are given below. à Specific Derivation for the Vos-Walraven derived primaries As with the Dartnall et al data, the following is a derivation of the three opponent channels given the derived primary channels of Vos and Walraven, following the same process of principal components analysis.

The Vos-Walraven primary cone channel sensitivities are given previously in the form of three tables, VWBlue, VWRed, and VWGreen. These tables are pairs in the form of {wavelength, sensitivity}. Since all three cones have data for all of the wavelengths from 400 to 680nm, we are interested only in the sensitivity data for those cones. SampleFinalProjectFormatted.nb 9

CovarianceMatrix = {{Crr, Crg, Crb},{Crg, Cgg, Cgb},{Crb, Cgb, Cbb}} So, now we have the covariance matrix for the Vos-Walraven data. Now we need to find the eigenvectors and eigenvalues of this matrix.

W' = CEigenVectors DisplayW' = TableForm[W'] So, when the matrix W' is multiplied by the vector of cone sensitivity functions, we will obtain the equations for the three opponent channels. The three channels will be called VWAchromatic, VWRedGreen, and VWBlueYellow. The graphs for these three channels are given below.

Ÿ Numerics 10 SampleFinalProjectFormatted.nb

VWBYPlot = ListPlot[VWBlueYellow, AxesOrigin->{400,0},PlotJoined-> True, AxesLabel->{"Wavelength","Sensitivity"}]

Sensitivity

0.008

0.006

0.004

0.002

Wavelength 450 500 550 600 650

VWRGPlot = ListPlot[VWRedGreen, AxesOrigin->{400,0},PlotJoined-> True, AxesLabel->{"Wavelength","Sensitivity"}]

Sensitivity

0.1

0.05

Wavelength 450 500 550 600 650 -0.05

-0.1 SampleFinalProjectFormatted.nb 11

VWAchromaticPlot = ListPlot[VWAchromatic, AxesOrigin-> {400,0},PlotJoined->True, AxesLabel->{"Wavelength","Sensitivity"}]

Sensitivity

0.7 0.6 0.5 0.4 0.3 0.2 0.1 Wavelength 450 500 550 600 650

Psychophysical Data for Opponent-Channels

à Method It is important that the results derived above by mathematical means be compared to psychophysical data. Without presentation of the data or graphical information, it will be stated here that the achromatic channel derived by Buchsbaum and Gottschalk fits the experimental data very well. Here, we will concentrate on the more interesting cases of the the two color opponent channels. It was not until 1955 that Jameson and Hurvich developed the psychophysical methods to measure these opponent processes. These measures are the standards to which mathematical models of opponent processes are compared.

The method used by Jameson and Hurvich was based on the fact that the sensation of hue has an opponent nature. In other words, hue can be described based on its redness or greeness and its blueness or yellowness. The process is used to examine this is called hue cancellation. A monochromatic test stimulus is shown, along with one of four cancellation stimuli (R,G,B,Y). The subject is allowed to adjust the cancellation stimulus given until the mixture satisfies one of two constraints: Either the subject will report a Y-B cancellation so there is no sensation of yellowness or blueness, or the subject will report a R-G cancellation, so there is no sensation of redness or greenness. For example, to measure the strength of redness, they measured the energy of a green light required to cancel the redness. Similarly for blue and yellow.

à Psychophysical Results The following data and graphs are from the Jameson and Hurvich study. This first set of graphs are the measures they obtained in running the hue cancellation task discussed above. The graphs correspond to the responses of the two types of color opponent channels, measured in "chromatic valence", which is just another way of drawing a chromatic response function. Below are the data and indivdual graphs for each section of the curves.

Ÿ Numerics 12 SampleFinalProjectFormatted.nb

The Jameson and Hurvich plot for their hue cancellation task is below.

CombinedJHExpt = Show[ JHExptBluePlot,JHExptYellowPlot,JHExptRed1Plot,JHExptGreenPlot,JHE xptRed2Plot, PlotRange->{{400,720},{-1.2,1.2}},AxesLabel-> {"Wavelength(nm)","Chromatic Valence"}]

Chromatic Valence

0.5

0 Wavelength(nm) 450 500 550 600 650 700

-0.5

-1

-Graphics-

à Derived Results The data and graphs below are for opponent channel curves that are derived from the color channel primaries by Jameson and Hurvich, instead of being obtained psychophysically.

The Jameson and Hurvich plot for their hue cancellation task is below. SampleFinalProjectFormatted.nb 13

CombinedJHDer = Show[ JHDerBluePlot,JHDerYellowPlot,JHDerRed1Plot,JHDerGreenPlot,JHDerRe d2Plot, PlotRange->{{400,720},{-1,.6}},AxesLabel->{"Wavelength( nm)","Chromatic Valence"}]

Chromatic Valence

0.4

0.2

0 Wavelength(nm) 450 500 550 600 650 700 -0.2

-0.4

-0.6

-0.8

-1

Comparison of PCA Derived Opponent Channel Responses and Psychophysically Obtained Opponent Channel Responses

The following set of graphs summarizes all of the graphs displayed previously (excluding those for the achromatic channel). The graphs are assembled here for comparison. It can be seen here that the data for the R-G channel matches the results obtained by using the principal components analysis in a very close fashion. However, it can also be seen that the psychophysical curves obtained for the B-Y channel do not match so well with the Buchsbaum and Gottschalk principle components analysis B-Y channel, even though the general shape is similar.

The graphs are as follows ... they are not combined on one graph so comparisons can be made more easily. Remember that each of the graphs has a different measure for the y-axis!! NOTE also that the BY curves for Dartnall and Vos-Walraven, to match the BY curves of Jameson, should be mirrored around the x-axis in order to get the proper orientation of the graph. This is due to the properties of the signs of eigenvectors and the fact that for each color opponent type of cell (R-G and B-Y), there are two possible combinations (A+B- and A-B+), where A or B can be excitatory or inhibitory.

Dartnall R+G- Dartnall B+Y- Vos-Walraven R+G- Vos-Walraven B+Y- Jameson R+G- & B-Y+ (Expt) Jameson R+G- & B-Y+ (Derived) 14 SampleFinalProjectFormatted.nb

CombineAll = Show[Graphics[{Rectangle[{0,0}, {2,1},CombinedJHExpt],Rectangle[{2,0}, {4,1},CombinedJHDer],Rectangle[{0,1},{2,2},VWRGPlot],Rectangle[ {2,1},{4,2},VWBYPlot],Rectangle[{0,2},{2,3},RGPlot],Rectangle[ {2,2},{4,3},BYPlot]}]]

Absorbance Absorbance 80 20 60 10 40 Wavelength 20 400 450 500 550 600 650 Wavelen 400 450 500 550 600 650 -10 -20 -20 -40 Sensitivity Sensitivity

0.008 0.1 0.006 0.05 0.004 Wavelength 450 500 550 600 650 0.002 -0.05 Wavelen -0.1 450 500 550 600 650 Chromatic Valence Chromatic Valence 1 0.4 0.2 0.5 0 Wavelengt 0 Wavelength(nm) -0.2 450500550600650700 450500550600650700 -0.4 -0.5 -0.6 -0.8 -1 -1

-Graphics-

Note that although the graphs for the BY and RG color opponent mechanisms are very similar for all the methods used and displayed above, there is one unexplained difference that comes out of the principal components analysis. That is the ratio of the signal processing energy for each of the channels. In the case of the Buchsbaum and Gottschalk derivation for the Vos-Walraven primaries, the ratio is 4.22: 0.12: 0.0003 corrsponding to the channels: Achromatic: RedGreen: BlueYellow. However, the B & G derivation for the Dartnall physiological data show a different story. The ratio for that case is as follows: 205014: 5471: 54626 corresponding to the same channels as above. So, it seems, for some reason, that the BY and RG channels are reversed in importance in the two cases. An explanation for this has not yet been formulated. SampleFinalProjectFormatted.nb 15

Neural Support for a Color Opponent Mechanism

Carlson cites evidence that ganglion cells in the human retina respond to pairs of primary colors, with red opposing green and blue opposing yellow. As an example, ganglion cells may be organized in a center/surround fashion such that the center is excited by red and inhibited by green, while the surround is inhibited by red and excited by green

Carlson describes the following scenario. Red light stimulates the red cones, which excites the R-G ganglion cells, so red is signalled by excitation of R-G cells. Green light stimulates the green cones, which inhibit the R-G ganglion cells, so green is signalled by inhibition of R-G cells. Yellow light stimulates the red and green cones equally. But both the red and green cones excite the Y-B ganglion cells. But the red and green stimulation of the R-G ganglion cells cancels out, since red is excitatory, and green is inhibitory. So, the visual system signals blue when the Y-B firing rate increases, while the R-G firing rate remains stable.

Carlson also cites evidence that primate LGN cells also encode color in the same way as the retinal ganglion cells. These cells, like the ganglion cells, respond in a center/surround manner, as described above. The responses of single neurons to various wavelengths of light result in a set of curves much like the color opponency response curves described earlier.

There is also some evidence that opponent processing cells are found in the visual cortex, but this evidence is not as well developed.

Summary

The early history of color vision work concentrated on the Young-Helmholtz trichromacy description of color, in which the responses of three cones, each with a particular spectral sensitivity function, accounted for color phenomena. However, as more color experiments were done, it was found that the trichromacy theory was not sufficient to describe all the experimental results. As a result, a new theory, called opponent-process theory was proposed by Hering. Since that time, much of the work in color vision has concentrated on developing this theory and finding evidence to support it. Physiological evidence in the form of ganglion and LGN cells that respond in a color opponent fashion, and psychophysical evidence in the form of hue cancellation studies provide experimental evidence of the existence of such a color opponent process in the visual system.

One of the major questions is why such a system exists. Buchsbaum and Gottschalk set out to answer that question in terms of an information processing analysis. The goal of information processing study is to find ways to transmit information such that inefficiency is minimized. The three cone channels lend themselves perfectly to such an analysis, because their spectral response curves overlap a great deal, and if information were transmitted in this way, it would be very inefficient. Through the process of principle components analysis, Buchsbaum and Gottschalk are able to derive a set of channels that would transmit color information in such a way that the redundancy of the three cone channels is greatly reduced. These channels are the two color opponent channels (R- G and Y-B) and an achromatic channel that transmits light information over a broad spectrum of wavelengths. The results of Buchsbaum and Gottschalk for the achromatic channel and the R-G color opponent channel match very closely the results obtained physiologically and psychophysically. The B-Y channel does not match quite as well, but the general form is close, and further work may be able to enhance that relationship. 16 SampleFinalProjectFormatted.nb

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