Statistical properties of color matching functions
aa r X i v : . [ q - b i o . N C ] J u l Statistical properties of color matching functions
María da Fonseca , , and Inés Samengo Instituto Balseiro, CONICET, and Department of Medical Physics, Centro atómicoBariloche, Argentina. Center for Brain and Cognition, and Department of Information and Communica-tion Technologies, Universitat Pompeu Fabra, Barcelona, Spain.
Keywords:
Photon absorption, color perception, color matching functions.
Abstract
In trichromats, color vision entails the projection of an infinite-dimensional space (theone containing all possible electromagnetic power spectra) onto the 3-dimensional spacedetermined by the three types of cones. This drastic reduction in dimensionality givesrise to metamerism, that is, the perceptual chromatic equivalence between two differentlight spectra. The classes of equivalence of metamerism is revealed by color-matchingexperiments, in which observers equalize a monochromatic target stimulus with the su-perposition of three light beams of different wavelengths (the primaries) by adjustingtheir intensities. The linear relation between the color matching functions and the ab-sorption probabilities of each type of cone is here used to find the collection of primariesthat need to be chosen in order to obtain quasi orthogonal, or alternatively, almost-always positive, color-matching functions. Moreover, previous studies have shown thatthere is a certain trial-to-trial and subject-to-subject variability in the color matchingfunctions. So far, no theoretical description has been offered to explain the trial-to-trial variability, whereas the sources of the subject-to-subject variability have been as-sociated with individual differences in the properties of the peripheral visual system.Here we explore the role of the Poissonian nature of photon capture on the wavelength-dependence of the trial-to-trial variability in the color matching functions, as well astheir correlations.olor vision has limitations. If we are instructed to provide objective measures ofthe percept produced by a chromatic stimulus, our responses are endowed with somedegree of trial-to-trial variability. In a previous paper (da Fonseca and Samengo, 2016),we showed that although there are many putative sources of variability in the visualpathway, the Poissonian nature of photon absorption by cones suffices to explain a largefraction of the variance in discrimination experiments (MacAdam, 1942). Those exper-iments reported the trial-to-trial fluctuations in color-matching experiments in some ofthe popular systems of coordinates employed to report color, as well as in a so-called“natural” system (da Fonseca and Samengo, 2018). In this letter, we derive the effect ofphotoreceptor Poisson noise in the color-matching functions (CMFs). These functionsare extensively used in colorimetry to represent color (see below). Quite unfortunately,the scientific community working in color, and often dealing with the needs of industry,only seldom talk to and are addressed by neuroscientists studying vision, more focusedon principled descriptions. Our analytical description of the trial-to-trial variability ofCMFs based on a probabilistic description of cone functioning is an attempt to facilitatethe dialogue between the two fellowships.When a light beam of spectrum I ( λ ) impinges on the retina, the three types of color-sensitive photoreceptors, cones of type S , M and L absorb k = ( k s , k m , k ℓ ) t photonswith probability distribution (Zhaoping et al., 2011) P [ k | I ( λ )] = Y i ∈{ s , m ,ℓ } Poisson ( k i | α i ) , (1)where each Poisson factor readsPoisson ( k | α ) = e − α α k k ! , with mean and variance α i = β i Z I ( λ ) q i ( λ ) d λ, i ∈ { s , m , ℓ } . (2)Here, the parameters β i represent the fraction of each type of cone in the retina of theobserver, and the curves q i ( λ ) are the cone fundamentals describing the wavelengthdependence of the absorption probability of each type. The space of all possible light2pectra is hence projected on the -dimensional space spanned by the vector k . Im-portantly, the projection is probabilistic, and in different trials, the same spectrum I ( λ ) may generate different k -vectors. The mean value of the number of absorbed photonsof each type is h k i = α = ( α s , α m , α ℓ ) t .Equation 2 not only provides an algorithm with which to calculate the mean andvariance of the distribution of Eq. 1, but also constitutes a linear projection that trans-forms a spectrum E ( λ ) into the triplet α . Thus interpreted, Eq. 2 provides the LM S color coordinates (Wyszecki and Fielder, 1971).Nowadays, neuroscientists studying vision know that the only signal that reaches thebrain carrying chromatic information is a function of the vector k . Consequently, thefiltering operation produced by cones is always present in our description of behavioralexperiments. Yet, long before photoreceptors were described, Hermann von Helmholtzexplored the phsychophysics of color vision (von Helmholtz, 1910), and arrived to theconclusion that any chromatic sensation can be perceptually equated with a combinationof three monochromatic beams of adjustable intensity, the so-called primary colors . Thetriplet of primaries is not unique, since many choices can be used, as long as the mixturetwo of the colors does not produce the chromatic sensation of the third.In the th century, Hermann Grassmann (Grassmann, 1853) introduced the lawsthat carry his name, and govern the rules of color matching: symmetry, transitivity,proportionality and additivity (Wyszecki and Stiles, 2000). In 1931, the Commissioninternationale de l’éclairage (CIE) reported the results for a collection of the exper-iments called color matching experiments (Commission Internationale de l’Eclairage,1932). Subjects were instructed to adjust the gains g , g , g of three monochromaticbeams of wavelengths λ , λ , λ and intensities I , I , I (the primaries) to match a tar-get spectral color of wavelength λ t . The experimenter showed a bipartite field on ascreen. One of the halves was illuminated with the target stimulus, of spectrum I t ( λ ) = I t δ ( λ − λ t ) , (3)3nd the other half displayed the matched color, of spectrum I m ( λ ) = g I δ ( λ − λ ) + g I δ ( λ − λ ) + g I δ ( λ − λ ) . (4)The values of g , g and g were collected from 18 subjects, for a set of target wave-lengths λ t ∈ [380 nm ,
780 nm] every nm . The population mean of each g i ( λ t ) wasdefined as the red, the green and the blue color matching functions (CMF) of the so-called “standard observer” of Fig. 1A (Wyszecki and Stiles, 2000).The CIE 1931 chose primaries of wavelength λ = 700 nm, λ = 546 . nm and λ = 435 . nm. For λ t between and nm no gains ( g , g , g ) could achieve aperceptual match. If, however, the red primary was added to the target spectrum witha specific intensity g , observers were able to find positive gains g and g to achievethe match. By convention, then, the CMF evaluated at the target wavelength λ t weredefined by the gains ( g , g , − g ) . The negative sign of the last component indicatesthat the beam of wavelength λ was added to the target field (as opposed to the matchedfield) with gain g .A match between the perceived target and constructed colors implies that the prob-ability distributions P ( k | α ) of both beams coincide. The only way of achieving thisequality is by inserting Eqs. 3 and 4 into Eq. 2, and obtaining exactly the same triplet ( α s , α m , α ℓ ) . Therefore, the gains g , g , g must be chosen so that (Brainard and Stockman,2010) X j ∈{ , , } g j I j q i ( λ j ) = I t q i ( λ t ) , ∀ i ∈ { s , m , ℓ } . (5)The parameters β i describing the composition of the retina of the observer (Eq. 2) arecancelled out, so they do not appear in Eq. 5. As a consequence, the gains g j chosen byobservers with different retinas coincide.The linear relation of Eq. 5 between the column vector g = ( g , g , g ) t of thegains and the column vector t = ( q s ( λ t ) , q m ( λ t ) , q ℓ ( λ t )) t of the target stimulus can beshortened by defining the matrix Q with entries Q ij = q i ( λ j ) , with i ∈ { s , m , ℓ } and j ∈ { , , } D with entries D jj = I j I t . The change of base matrix C = Q · D (6)then relates g and t : C g = t . This equation can be solved uniquely for g for all non-singular C matrices, yielding g = C − t . (7)The requirement of a non-singular C is met by all triplets of non-coinciding primaries,as long as D is invertible, that is, none of the beams is turned off. If two primaries, how-ever, are close to each other, the matrix C is close to singular, and one of its eigenvaluesis close to zero. Unrealistically large gains g j may then be required. If less than threeprimaries are used, then C is a rectangular matrix with more rows than columns, andcannot be inverted, implying that no match can be found. If, instead, more than threeprimaries are employed, C is a rectangular matrix with more columns than rows, andthe system has infinite solutions. One of the primaries can be obtained by combinationof the remaining three, so whichever gain is assigned to that primary, could also havebeen distributed among the other three.Since the vector t depends on the wavelength λ t of the target beam, Eq. 7 relates theCMFs g j ( λ t ) to the spectral selectivity of the photon absorption process (through Q ),and the properties of the three chosen primaries (through the wavelengths ( λ , λ , λ ) and the associated intensities ( I , I , I ) appearing in D . Hence, the three CMFs arelinear combinations of the cone fundamentals q i ( λ t ) , and the coefficients of the linearcombination, which depend on the three chosen primaries, define the change-of-basematrix C .Figure 1A displays the original CMF reported by the CIE 1931, with the prediction5
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Target wa elength (nm) T h e o r e t i c a l p o s i t i e C M F ( a . u . ) C Figure 1: A: Color matching functions reported by Guild (Guild, 1932), employedby the CIE 1931 to construct their RGB and XYZ color spaces ( λ = 435 . nm inblue dots, λ = 546 . nm in green dots and λ = 700 nm in red dots) normalized tounit Euclidean norm, and the corresponding normalized theoretical prediction in solidlines (Eq. 7 calculated with the cone fundamentals of Stockman and Sharpe (2000)). B:Predicted normalized CMF corresponding to primaries λ = 455 nm (blue), λ = 550 nm (green), λ = 625 nm (red), selected to minimize the scalar product between thecurves. C: Predicted normalized CMF corresponding to primaries λ = 380 nm (blue), λ = 510 nm (green), λ = 775 nm (red), selected to maximize the range of λ t valuesfor which the curves are positive.for g i ( λ t ) of Eq. 7, with i ∈ { , , } . The diagonal elements I i /I t of matrix D wereset to unity, which yields CMFs of unit Euclidean norm.If instead of combining three monochromatic primaries, matching experiments areperformed with light beams of arbitrary spectra e ( λ ) , e ( λ ) , e ( λ ) , the gains g j ( λ t ) arestill given by Eq. 7, but with a matrix Q with elements Q ij = h q i , e j i . The resultingCMF g j ( λ t ) can still be obtained, and they still represent the gains of the three beams.So far, the target beam was assumed to be monochromatic. If this restriction isrelaxed, the spectrum I ′ t ( λ ) can be an arbitrary (non-negative) function. The linearity ofGrassman’s laws implies that the three gains ( g ′ , g ′ , g ′ ) required to achieve the matchare linear combinations of the CMFs obtained for monochromatic targets, that is, g ′ j = Z g j ( λ ) I ′ t ( λ ) d λ, (8)where g j ( λ ) are given by Eq. 7. The values g ′ , g ′ , g ′ are the “tri-stimulus values” of thebeam I ′ t , and constitute one possible system of coordinates in which the chromaticityof I ′ t ( λ ) is represented. Different choices of primaries result in different coordinatesystems, since they yield different CMFs. 6hen the choice of primaries is only meant to produce CMFs that define a coordi-nate system (that is, whenever the actual execution of the color matching experimentis not required) unattainable primaries, often termed imaginary primaries, may be em-ployed. Imaginary primaries are defined by power spectra that contain negative values,and therefore, cannot be instantiated in reality. Such is the case, for example, of theprimaries that underlie the LM S , the
RGB and the
Y XY coordinate systems.Equation 8 implies that the tri-stimulus values are the projection of the target spec-trum I ′ t on the CMFs. Within this framework, the CMFs act as a base of the subspaceof spectra that trichromats perceive. The first goal of this paper is to reveal two tripletsof primary colors that produce CMFs that are particularly convenient.The choice of the first triplet is guided by the requirement of obtaining CMFs thatare as orthogonal as possible. Coordinate systems constructed with orthogonal basesare desirable, since correlations in the resulting tri-stimulus values reflect correlationsin the original spectra, as opposed to correlations in the chosen base.The color matching functions reported by the CIE 1931 were not far from orthog-onal. The scalar products of the normalized version of those curves were h g , g i =0 . , h g , g i = 0 . , and h g , g i = − . , where the sub-indices , , referto the primaries with wavelengths . , . and nm, respectively. The scalarproducts are small, but they can still be improved by diminishing h g , g i .The search for primaries that produce orthogonal CMFs has been undertaken be-fore (Thornton, 1999; Brill and Worthey, 2007; Worthey, 2012), by finding a lineartransformation of some set of previously reported CMFs. However, the resulting pri-maries were imaginary. To produce an (almost) orthogonal base that is connected toa realizable color-matching experiment, here we performed an exhaustive numericalsearch of all triplets of monochromatic primaries between and nm, in stepsof nm., calculated their CMFs through Eq. 7, and retained the triplet that minimizedthe function h g , g i + h g , g i + h g , g i . The optimal triplet was λ = 455 nm,7 = 550 nm, and λ = 625 nm. The main difference with the CIE 1931 primariesis that the wavelength of the red beam is diminished. The resulting CMFs are dis-played in Fig. 1B, and the most noticeable difference with the CMFs of CIE 1931 isthat g contains larger negative regions flanking both sides of its maximum, therebydiminishing the overlap with g . The inner products between the resulting CMFs are h g , g i = 0 . , h g , g i = 0 . , h g , g i = − . .In the second place, we search for primaries that produce CMFs with maximal do-main of positive values. Such primaries are the optimal choice when attempting toconstruct metamers of monochromatic beams with the largest possible range of targetwavelengths, since the negative portion of CMFs reflect a failure to construct the targetpercept. This request is relevant, for example, when choosing the LEDs of computerscreens.Again, we performed a numerical, exhaustive search of monochromatic primaries,and maximized the sum of the domains where the resulting CMFs were positive. Theoptimal triplet had wavelengths λ = 380 nm, λ = 510 nm, and λ = 775 nm. Inthis case, the wavelengths are more separated from one another than in the originalCIE 1931 primaries, and reached the minimal and maximal values employed in oursearch. Clearly, if no restriction is imposed on the amplitude of the gains, even moreseparated primaries would produce still more positive CMFs, since cones of differenttype would never be activated simultaneously. The normalized CMFs obtained with oursearch algorithm are displayed in Fig. 1C. Negative values could not be avoided for g and g , but the reached values were small ( − . and − . , respectively), so it maybe hypothesized that for those wavelengths, replacing a negative gain by zero wouldproduce a minimal perceptual shift.We now turn to the second goal of this paper, namely, to provide a principled deriva-tion of the trial-to-trial variability and correlation structure of the CMFs, capturing thedispersion and the structure of the observer’s responses. In our derivation, the source ofvariability is the stochastic nature of photon absorption (Eq. 1). We are aware of the ex-8stence of additional sources of variability. Still, here the aim is to assess how much ofthe experimental variability can be accounted for, taking only the stochasticity of pho-ton absorption of Eq. 1 into account. The advantage of describing photon absorptionalone, is that the probability distribution of Eq. 1 can be derived from first principles(da Fonseca and Samengo, 2016).We interpret the matching experiment as the observer’s attempt to estimate the tri-stimulus values of the target stimulus. In other words, if the coordinates of the targetmonochromatic stimulus obtained from Eq. 7 are ( g , g , g ) , the behavioral responseobtained in the color matching task in a single trial ˆ g = (ˆ g , ˆ g , ˆ g ) t can be interpretedas an estimator of the true g performed by the subject from the absorbed photons k =( k s , k m , k ℓ ) t . The trial-to-trial fluctuations of ˆ g are captured by the × mean quadraticerror matrix E of entries E ab ( g ) = h [ˆ g a ( k ) − h ˆ g a i ] [ˆ g b ( k ) − h ˆ g b i ] i , where the brackets represent an expectation value weighted with P ( k | g ) . The diagonalelements of E jj represent the variances of the measured g i values, and the off-diagonalelements E ab , the covariances.The Crámer-Rao bound (Rao, 1945; Cramér, 1946; Cover and Thomas, 2012) statesthat the mean quadratic error E of any unbiased estimator is bounded from below bythe inverse of the Fisher Information J ( g ) , a × matrix of entries J ab ( g ) = − (cid:28) ∂ ln P ( k | g ) ∂g a ∂g b (cid:29) . (9)The Fisher Information matrix is the metric tensor with which infinitesimal distancesin color space can be calculated, such that traversing a unit of distance in color spacemodifies the distribution of k vectors in a fixed amount (Amari and Nagaoka, 2000).The bound reads E · J ≥ , (10)and states that all the eigenvalues of the matrix product E · J must be larger or equalthan unity. It implies that inasmuch as J is associated to the notion of information, J −
9s associated to the notion of minimal mean quadratic estimation error. The larger theinformation, the smaller the error, and vice versa. The fact that Eq. 10 is expressed inmatrix form means that the bound is directional. In other words, the mean quadraticerror may take different values along different directions: Along the eigenvectors of E , the error is equal to the corresponding eigenvalues. Equation 10 is only valid forunbiased estimators, that is, those for which h ˆ g ( k ) i = g . A more complex formulais required in the biased case (Cover and Thomas, 2012). However, if the stimulus isnot surrounded by a chromatic background (and such is the case of the color matchingexperiment discussed here), behavioral errors have zero mean (Klauke and Wachtler,2015, 2016), so we work under the assumption that the nervous system is able to im-plement at least one unbiased estimator.Equation 10 is an inequality, so the Fisher Information can be employed to bound,but not to calculate, the mean quadratic error. Even so, in this paper we assume that theequality holds, and derive the mean quadratic error analytically as E ≈ J − , (11)since J can be obtained analytically from Eqs. 9 and 1. The assumption is only validif all subsequent processing stages, downstream from photon absorption, preserve theinformation encoded by the vector k . In 2016, we showed that the mean quadratic errorobtained by assuming that the equality holds captures 87% of the variance of behavioraldiscrimination experiments (da Fonseca and Samengo, 2016). Assuming the equality,hence, seems to be justified up to a reasonable degree. Continuing with this line ofthought, we here explore the consequences of this assumption in the mean quadraticerror of behavioral matching experiments.The Fisher Information matrix was obtained in da Fonseca and Samengo (2016), inthe LM S coordinate system, obtaining a diagonal matrix of entries J ( α ) ab = 1 α a δ ab , (12)where δ ab is the Kronecker delta symbol. To predict the trial-to-trial fluctuations inmatching experiments, this tensor must be transformed to the g coordinate system. To10hat end, we define the diagonal matrix B with entries B ij = β i δ ij , containing the fractions β i of each type of cones. The coordinate transformation be-tween α and g is α = B · C g . (13)Consequently, the transformation rule for the metric tensor is (da Fonseca and Samengo,2016) J ( α ) = ( B · C ) t · J ( g ) · ( B · C ) . (14)Inserting Eq. 12 into Eq. 14, using the expression 13, and solving for J ( g ) , the FisherInformation can be obtained analytically, J ( g ) ab = I a I b X i ∈ { s , m ,ℓ } β i q i ( λ a ) q i ( λ b ) P j =1 q i ( λ j ) I j g j . (15)If the color-matching experiment is performed with monochromatic target stimuli(Eq. 3), the Fisher Information matrix of Eq. 15 reduces to J ( g ) ab = I a I b I t X i ∈ { s , m ,ℓ } β i q i ( λ a ) q i ( λ b ) q i ( λ t ) . (16)The Fisher Information matrix bears an explicit dependence on ( β s , β m , β ℓ ) , implyingthat observers with different retinal composition respond with trial-to-trial fluctuationsof varying structure. Moreover, writing the intensities ( I , I , I ) in units of I t revealsthat the Fisher Information is linear in the target intensity I t . Therefore, the varianceof the responded g j is inversely proportional to the total light intensity employed in theexperiment.The Fisher Information matrix of Eq. 16 can be inverted to yield the mean quadraticerror under the assumption of Eq. 11. In Fig. 2A, the diagonal elements E aa are dis-played (the variances), for a retinal composition of β s = 0 . , β m = 0 . , βℓ = 0 . ,which is quite typical for human trichromats. We also verified that modifying thesevalues within the physiological range produced only minor changes in the derived vari-ances and covariances. The ratios I j /I t were set to unity, in order for the resulting11MFs to have unit norm. The global scaling factor I t was set to . , in order forthe maximum of the variance in g (peak of the red curve in Fig. 2A) to equate theexperimental height (Fig. 2C, see below).The off-diagonal elements E ab can be seen in Fig. 2B, showing that all correlationsare negative, and they tend to be particularly significant in those regions of the spectrumwhere the two corresponding CMFs overlap. Negative correlations imply that if, in oneparticular trial, the observer sets one of the gains above average, they are likely to setthe other two below average, at least, if there is an overlap between the correspondingCMFs.
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1e 3 D Figure 2: A: Normalized theoretical prediction for the variance of the CMFs of Fig. 1A,as a function of the target wavelength λ t . The three curves are the diagonal terms ofthe inverse of the matrix in Eq. 16. B: Correlations between the CMFs, obtained fromthe off-diagonal terms of the inverse of the matrix in Eq. 16. C and D: Variances andcorrelations obtained from multiple subjects performing the color-matching experiment(Stiles and Burch, 1959) 12ome previous studies have addressed the subject-to-subject variability of color-matching experiments (Stiles and Burch, 1959; Wyszecki and Fielder, 1971; Alfvin and Fairchild,1997; Fairchild and Heckaman, 2013, 2016; Asano et al., 2016a,b; Emery et al., 2017;Murdoch and Fairchild, 2019; Emery and Webster, 2019), and only a few have exploredthe trial-to-trial variability of the responses of a single subject (Wyszecki and Fielder,1971; Alfvin and Fairchild, 1997; Sarkar et al., 2010; Asano, 2015). The two types ofvariability derive from different sources. Subject-to-subject variability is mainly dueto individual differences in biophysical and physiological properties, and describes thedegree of agreement in the percept produced by a given stimulus in a population of ob-servers. Trial-to-trial variability, instead, reflects the inherent uncertainty with which agiven observer perceives a given stimulus, and stems from noisy processes both outsideand inside the visual system.The inter-subject variability, has been more exhaustively characterized, probably forcommercial purposes, and can be depicted as a function of wavelength (Fig. 2C and D).Theoretical studies (Fairchild and Heckaman, 2013; Asano, 2015; Asano et al., 2016a;Murdoch and Fairchild, 2019) on the inter-subject variability take into account individ-ual differences in lens and macular pigment density, retinal composition (matrix B ),and variations in the shape of the cone fundamentals q s ( λ ) , q m and q ℓ ( λ ) . Experimentaldata with the CMFs of a population of 49 subjects (Stiles and Burch, 1959) are availableonline (Fig. 2C and D).Unfortunately, we lack experimental data on the trial-to-trial fluctuations of a sin-gle observer, at least, beyond crude estimation performed with very few samples. Theresults of Wyszecki and Fielder (1971) are difficult to interpret, since the variability ofa single subject in different sessions (separated by several weeks or months) is consid-erably larger than one obtained in a single session of multiple trials, suggesting thatsome experimental conditions may have changed from one session to the next. Ex-periments estimating the intra-observer variability from matches performed by eachsubject were published in the PhD Dissertation of Yuta Asano Asano (2015). This trial-to-trial variability was approximately half the inter-observer variability, in accordance13ith earlier estimations (Alfvin and Fairchild, 1997; Sarkar et al., 2010). Importantly,the variability was estimated for a collection of non-monochromatic target colors, so itcannot be displayed as a function of the target wavelength.To our knowledge, the present study is the first analytical derivation of the trial-to-trial variability of the CMFs, deduced from one well identified source of noise: ThePoissonian nature of photon absorption. The result can be displayed as a function ofthe target wavelength (Fig.2A and B). Since our results cannot be reliably comparedwith experimental data recorded with multiple trials in a single observer, we comparethem with those obtained with a single trial of multiple observers, understanding thatdifferences are expected, due to the diverse sources of variability.The experimental result of the population variances exhibit a marked peak for thered primary (Fig. 2C), the position of which was fairly well reproduced by the theoreti-cal variance for a single observer (Fig. 2A). Therefore, part of the variance reported inthe experimental result could potentially stem from intra-subject variability. The exper-imental variances of the other two primaries (red and blue curves in Fig. 2A) are toonoisy to be useful. However, the relative size of the blue curve (compared to the red)in the theoretical result does not coincide with the experimental relation. Therefore,the population variability present in Fig. 2C and absent from Fig. 2A probably affectsdifferentially the two primaries. No conclusions can be drawn about the absolute mag-nitude of the theoretical and experimental curves, since the analytical result contains aglobal scale factor I t , which was fixated in Fig. 2A and B only to draw the variances.The experimental covariances of the gains are also noisy (Fig. 2D). Both the theo-retical and experimental covariances become significantly different from zero in thoseregions of the spectrum where the corresponding CMFs overlap. The theoretical resultcaptures the sign of the (negative) covariance for the green-red interaction (yellowishcurves) and the blue-green interaction (cyan), but not for the blue-red case (magenta).This discrepancy implies that, at least in the case of the blue-red interaction, either (a)subject-to-subject variability, or (b) additional stochasticity in downstream processing14tages of a single subject, play an important role in the co-variation of g and g .In summary, in this letter we presented a theoretical derivation of the variances andco-variances expected in color-matching experiments when the sole source of noise isthe stochasticity inherent to Poissonian photon absorption by cones. We were not able tofind experimental data on intra-subject variability of CMFs obtained for monochromatictarget stimuli, so we hope that the present study motivates psychophysical experiments.If the measured variances and covariances coincide with the analytical result obtainedhere, photoreceptor noise may be concluded to be a crucial ingredient in the perceptualvariability of chromatic vision. Instead, if experiments happen to reveal a differentbehavior, subsequent stages in color processing may be concluded to play the lead. Acknowledgements
This work was supported by Agencia Nacional de Investigaciones Científicas y Técni-cas, Consejo Nacional de Investigaciones Científicas y Técnicas, Comisión Nacional deEnergía Atómica and Universidad Nacional de Cuyo, all from Argentina.
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