1. Introduction

Several color difference metrics have been developed for split fields [1]. These color difference metrics allow to predict at any color center the visibility threshold between two uniform fields. However, the visibility threshold of color differences also depends on the spatial parameters of a stimulus. Spatial sinusoidal patterns are used to investigate and characterize the spatial dependencies of the detection threshold of achromatic and chromatic contrast variations, e.g. References [2–4]. This method is important for a variety of applications, e.g. displays [5,6], image compression [7], medical [8] and lighting [9], but also for the understanding of the underlying color vision mechanisms [10]. To predict at any color center the sensitivity to detect color changes, a color difference metric which includes the spatial parameters of the sinusoidal gratings is necessary. Euclidean spatial color difference metrics have been proposed [11–14]. However, a Riemannian color difference metric is required to quantify adequately the distance between two similar colors [15]. In this paper we describe a new Riemannian spatial color difference metric, which is based on our earlier development of a Riemannian color difference metric for split fields [16].

The organization of this paper is as follows: The next section gives an outline of the Riemannian color difference metric for split-fields. In section 3 the definition of the contrast of spatial sinusoidal stimuli and the contrast sensitivity is presented. An overview of contrast sensitivity measurements and models, and Euclidean spatial color difference metrics is presented in section 4. In sections 5, we introduce the achromatic and chromatic contrast sensitivity functions. In section 6, the formulation of the new Riemannian spatial color difference metric is presented. In section 7, the predictions of the new model are compared with experimental results available in literature. This section includes also comparisons with the Euclidean spatial metric of Mantiuk et al. [14]. The model parameters of the new Riemannian spatial color difference metric for eight data sets are compared and discussed in section 8.

2. Riemannian color difference metric for split fields

We introduced in Ref. [16] a new line element with a Riemannian metric (denoted as 'split field LE') for the prediction of color difference thresholds of split fields with uniform color of the sub-fields. In this paper we describe an extension of the split field LE for test fields with sinusoidal color variations. The following is a brief outline of the split field LE, which is the basis for the extension to sinusoidal color variations. We use the spectral sensitivities of the foveal cone photoreceptors as determined by Smith-Pokorny [17] and the MacLeod-Boynton chromaticities [18,19], viz. $l=\frac {L}{Y}$, $m=\frac {M}{Y}$ and $s=\frac {S}{Y}$, with $L$, $M$ and $S$ the cone responses of respectively the long-wavelength, middle-wavelength and short-wavelength sensitive cone photoreceptors, and $Y=L+M$ the retinal illuminance [td] (troland). The split field LE is defined in a 3-dimensional space with coordinates $(\frac {dY}{Y}, \frac {dl}{l}, \frac {ds}{s})$, which we call the MacLeod-Boynton contrast space. This space is a linear transformation of the cone contrast space with coordinates $(\frac {dL}{L},\frac {dM}{M},\frac {dS}{S})$ [cf. Appendix A]. The axes of the MacLeod-Boynton contrast space represent the directions of stimuli which isolate respectively the achromatic, red-green (RG) and blue-yellow (BY) detection mechanisms. These directions are called ’cardinal directions’ [20] (pp.11.31-11.33). The 3-dimensional space defined by the responses of the detection mechanisms, known as the Derrington-Krauskopf-Lennie (DKL) space, is a linear transformation of the cone contrast space [21–23] [cf. Appendix A.]. From the available research literature about experimentally determined discrimination thresholds along the three cardinal directions, we derived the threshold expressions $\psi _A$, $\psi _T$ and $\psi _D$ in respectively the achromatic, RG and BY cardinal direction. For a just-noticeable difference, in an arbitrary direction of the color space, it is required that the contrast vector $[\frac {dY}{Y} \; \frac {dl}{l} \; \frac {ds}{s}]$ causes just enough ’energy’ to ’trigger a hypothetical detector in the brain’. This is expressed as a positive definite differential quadratic form equated to a constant:

Mathematically the line element $d\sigma$ is an infinitesimal quantity, whereas the distance in color space corresponding with a just-noticeable difference is a finite quantity. However, the distance between just-noticeable color differences is small in the sense that the metric tensor elements do not change appreciably over a distance corresponding to just-noticeable differences. The Riemannian metric tensor $\mathsf {G_{MLB}}$ is scaled in such a way that a just-noticeable difference for the human eye corresponds to a distance $d\sigma =1$. However, a just-noticeable difference and the actual magnitude of the threshold distance $d\sigma$ depend on the specific threshold criteria and the measurement conditions [16,24]. For $d\sigma =1$, the diagonal elements $g_{ii}$ are given by:

When the adapting background color (luminance and chromaticities) is equal to the reference color of the test field, then the threshold expressions for a standard (binocular) observer are given by:

3. Contrast and contrast sensitivity

In this study we consider static sinusoidal hard edge gratings and Gabor gratings, oriented vertically or horizontally, and viewed foveally. Hard edge gratings have typically a square or circular aperture with a sharp boundary. Gabor gratings are limited by a Gaussian envelope [25]. Illustrations of such stimuli can be found in Ref. [26] (Fig. 3.6) (achromatic hard edge grating), Ref. [27] (Fig. 1 C1) (achromatic Gabor grating) and Ref. [28] (Fig. 1) (isoluminous red-green and blue-yellow Gabor grating).

The contrast of an achromatic grating is calculated from the grating’s luminance distribution. The Michelson contrast [23,25,29], denoted as $C_{\text {M}}$, is used as the contrast measure for sinusoidal achromatic gratings, and is calculated as the ratio of the amplitude $\delta Y=\frac {1}{2}(Y_{\text {max}}-Y_{\text {min}})$ of the luminance variation over the average luminance $Y=\frac {1}{2}(Y_{\text {max}}+Y_{\text {min}})$:

Notice that $0 \leq C_\text {M} \leq 1$. Typically, the luminance of the adapting background is equal to the average luminance $Y$. The contrast sensitivity of the achromatic mechanism is the reciprocal of the Michelson contrast at which the pattern is just detectable against the uniform background [23]. Usually the contrast sensitivity is expressed as a function of the spatial frequency of the sinusoidal pattern, and is called the 'contrast sensitivity function' (CSF). Stimuli in the RG and BY cardinal direction are isoluminous, only the chromaticity changes with spatial location. Therefore another contrast definition than Eq. (7) is required for isoluminous chromatic stimuli. We define the contrast of stimuli in an arbitrary color direction as the Euclidean distance in the MacLeod-Boynton contrast space, given by:

4. Previous experiments and models

The results of many psychophysical measurements of achromatic CSFs for foveal vision and static sinusoidal gratings have been reported, e.g. References [2,3,31–39]. Foveal vison means that the center of the grating is imaged on the center of the fovea, but the image of the grating may be substantially larger than the fovea [40] (p. 65). Models for the achromatic CSF were developed by Barten [40] (pp. 36-37) [41,42], Daly [43] and a research team around Rovamo [34–37,44,45]. Barten’s model has been adopted for display applications [5,6,46,47]. For a good fit between the Barten’s model and the experimental data, some model parameters are adapted case by case. The CSF models of Barten and Rovamo are very similar. According to these models the CSF depends on the capability of the human visual system to discriminate a sinusoidal luminance pattern from a noisy background. Several psychophysical experiments measured the isoluminuous chromatic CSF in two to six directions of the color space at one or more color centers. [4,9,27,48–62]. A noise-based model for isoluminuous chromatic RG and BY CSFs was developed by the research team around Rovamo [45,53,55]. Some achromatic and isoluminuous chromatic CSF measurements used hard edge gratings, whereas other measurements used Gabor gratings to avoid edge effects [52,63]. Zhang and Wandell (1997) [11] proposed a spatial extension of the CIELAB color difference metric [64]. In a similar manner, Choudhury et al. (2020) [12] proposed a spatial extension of the ICtCp color difference metric [65]. Mantiuk et al. (2020) [14] devised a color difference metric for Gabor gratings. This metric is defined in a space of three specific color detection mechanisms, which are different from the color detection mechanisms of the DKL space [21]. In this model, it is assumed that the cone responses are encoded to contrast signals (normalization) prior to the transformation to mechanism responses. In the cone contrast space the metric tensor depends on the luminance, but is independent of the chromaticities of the color center. Therefore, this metric is Euclidean for isoluminuous stimuli. The model was fitted on contrast detection thresholds from five different studies. The complete model as described in Ref. [14] requires 32 fitting parameters

5. Contrast sensitivity functions

In this section we introduce the new CSF models for the isolated achromatic, RG and BY detection mechanism. Based on the findings of Poirson and Wandell (1996) [66], it is assumed that these mechanisms have their own independent spatial filtering, and that the chromaticity processing is separate from the spatial processing. Notice that these findings were implicitly assumed in the studies mentioned in section 4. These studies also show the following properties of the CSFs for cardinal stimuli: in the achromatic cardinal direction, the CSF has a bandpass shape, and is independent of the chromaticity of the grating [2], but its shape and peak sensitivity depend on the retinal illuminance of the color center; the CSFs in the RG and BY cardinal direction have both a lowpass shape and the maxima depend on the luminance and the chromaticities of the color center [4,50], though a bandpass shape was found in some studies [62]. In the next paragraph we formulate the achromatic CSF. This is primarily based on the CSF models developed by Barten [40] (pp.25-64) [41,42] and by the research team around Rovamo [34–36,44].

5.1 Achromatic contrast sensitivity function

The model for the achromatic CSF comprises two distinct internal noise sources: photon noise and late neural noise. The number of photons absorbed by the retinal photoreceptors fluctuates in accordance with Poisson statistics about a mean number of absorptions and a variance proportional with the luminance of the stimulus. These fluctuations are called photon noise. The late neural noise is caused by spontaneous neural activity in the brain [40] (pp. 31-32). The spontaneous activation of the photopigments, referred to as ’dark light’ or early noise, is not included in the model because it is only relevant at very low light levels ($Y$< 1 td) [38,67,68]. The noise present in the sinusoidal luminance pattern is not included in the model, because it is assumed that the external noise is negligible compared to the photon noise and the late neural noise. The complicated neural processing in the human visual system is approximated by a simple ’image processor’ [36]. The model’s sequence of processing stages and noise sources is as follows [40] (pp. 27) [44]. The visual stimuli are first filtered by the lowpass filter of the eye optics. The photon noise is then added, because the statistical fluctuations of the number of photons occur at the event of photon absorptions by the photoreceptors. Thereafter, there is a highpass filtering caused by the neuron’s spatial receptive fields [69] (pp. 135-137), referred to as the lateral inhibition process [40] (pp. 32-35), and the addition of the late neural noise. Finally the processed visual stimuli and noise are the input of a hypothetical detector, which takes into account the limited spatial and temporal integration capabilities of the human visual system. The equation for the resulting achromatic CSF for binocular viewing proposed by Barten [40] (p. 36) is then given by:

$\mathcal {D}(u,d_\text {p},555)$ represents the MTF of an incoherent optical system limited only by diffraction at a wavelength of 555 nm and with a circular aperture with diameter $d_\text {p}$ mm [74] (pp. 137-144) [73]. Inter-observer optical MTF differences [cf. Ref. [75]] can be modeled by adapting the exponent $\gamma$ [38]. $\Phi _{\text {ph}}(Y)$ represents the spectral density of the photon noise and is given by:

For low luminance levels $\Phi _{\text {ph}}(Y)>\Phi _0/M_{\text {lat}}^2(u)$, the achromatic CSF is proportional with $\sqrt {Y}$, which is known as the de Vries-Rose law. For high luminance levels $\Phi _{\text {ph}}(Y)<\Phi _0/M_{\text {lat}}^2(u)$, the achromatic CSF is independent of the luminance level, which is known as Weber’s law. The critical retinal illuminance $Y_{cA}(u)$, which marks the transition from the de Vries-Rose zone to the Weber zone, increases with an increase of the spatial frequency [31,35] and is given by:

Temporal and spatial integration are important mechanisms to average out the photon and neural noise and are therefore important to discriminate the grating from the noise. Only static images are considered. The integration time of the eye, represented by $\mathcal {T}$, is assumed to be constant under all conditions. The approximate cone integration time is 100 ms [40] (p.19). $\mathcal {A}_{\text {e}A}(u,\mathcal {A})$ represents the effective angular area over which the eye can integrate. In Ref. [36] the effective integration area is approximated by:

Achromatic contrast sensitivity measurements by Van Nes and Bouman (1967) [2] showed that the achromatic CSF does not change with the chromaticity of the color center. Monocular observation has higher achromatic visibility thresholds compared to binocular observation. Campbell and Green (1965) [77] found for achromatic gratings in the spatial frequency range of 1.85 to 46 cpd that the visibility threshold is about a factor $\sqrt {2}$ higher for monocular viewing. Kim et al. [30] found for achromatic gratings, in the spatial frequency range of 0.24 to 9.57 cpd, that the ratio of the visibility thresholds for binocular viewing over the visibility threshold for monocular viewing increases as the spatial frequency increases. However, in this study we assume a constant ratio of $\sqrt {2}$.

Taking into account the above considerations, we can express the achromatic CSF as:

In this paper we use Eq. (17) as the model for the achromatic CSF. In the next paragraph we construct the CSFs of the isolated RG and BY detection mechanisms. This is mainly based on the models developed by the research team around Rovamo [45,53–55] and the color difference metric for split fields presented in Ref. [16].

5.2 Isoluminous chromatic contrast sensitivity functions

There are similarities and important differences between the achromatic CSF and the isoluminous chromatic CSFs. In this paragraph, we construct the model for the isoluminous RG and BY CSFs by adapting Eq. (17) taking into account the results of various experiments reported in literature. Rovamo et al. (2001) [55] found that the critical retinal illuminance of isoluminous RG and BY gratings, which marks the transition point from the Rose-deVries zone to the Weber zone, is independent of the spatial frequency. They found a value for the critical retinal illuminance, denoted as $Y_c$, of about 165 td. Therefore, in Eq. (17) we replace $Y_{cA}(u)$ by $Y_c$. Notice that in the experiments of Ref. [55] a white background was used with CIE1931 chromaticities $(x,y)=(0.297,0.310)$. In various experiments [4,50,52,56] it was found that isoluminous chromatic CSFs have a lowpass shape whereas the achromatic CSF has a bandpass shape. The lowpass shape is due to a lack of the lateral inhibition process for equiluminous chromatic gratings [45]. To express this in the isoluminous chromatic CSFs, we replace the highpass filter $M_{\text {lat}}(u)$ in Eq. (17) by a constant. In various experiments [52–54] it was found that isoluminous chromatic gratings have a similar spatial integration function as achromatic gratings [Eq. (15)]. Therefore, the effective integration area is approximated by:

Notice that: (1) in Eqs. (21) and (22) the spatial processing and the chromaticity processing are separated, which is in accordance with the findings in Ref. [66]; (2) Eqs. (21) and (22) reduce to Eq. (20) for the EEW color center supposed $Y_T=Y_D=Y_c$. This is approximately the case for the white color center of Ref. [55]. In this paper we use Eqs. (21) and (22) with $Y_T=Y_D=Y_c$ as the model for the isoluminous RG and BY CSFs, respectively.

6. Line element for spatial gratings

At threshold, the modulation in the achromatic cardinal direction $(\delta l=0 \text { and } \delta s=0)$ excites only the achromatic detection mechanism. At threshold, the modulation in the isoluminous RG $(\delta Y=0 \text { and } \delta s=0)$ and BY cardinal direction $(\delta Y=0 \text { and } \delta l=0)$ excites only the RG and the BY detection mechanism, respectively. Similarly as for the split field LE [Eq. (1)], we propose, for modulations in the three cardinal directions simultaneously, the line element given by:

From Eqs. (17), (21), (22), (23) and $d\sigma =1$ it follows that:

Similar as for the split field LE [Eqs. (6)], the off-diagonal elements are given by:

We denote the line element given by Eqs. (23), (24) and (25) as the ’grating LE’. Notice that the foveal small field (2$^\circ$ field of view) color matching obeys Grassmann’s proportionality law [79] (p. 118) in the range of retinal illuminance levels from about 1 td to 8000 td [80] (pp. 116-117). This means that outside this range the colorimetric representation of colors does not hold because the shape of the color matching functions change.

7. Results

In this section we validate the contrast sensitivity functions given by Eqs. (17) (21) and (22) against the measurements by Carlson (1982) [3], Mullen (1985) [4], McKeefry et al. (2001) [56] and Kim et al. (2013) [27]. Furthermore, we validate the grating LE [Eqs. (23) (24) and (25)] against the measurements by Kitanovski et al. (2019) [81] (vertically oriented grating) and Xu et al. (2020) [62]. The predictions of the model devised by Mantiuk et al. [14] are included for the experiments by Kim et al. (2013) and Xu et al. (2020). For the achromatic CSF measurements by Carlson (1982) [3] (abbreviated as CA82) the calculation results are included in paragraph 7.4 and Table 1. A detailed description of this experiment is not included, because it has been discussed in detail by Barten [40] (pp. 51-52). The calculation results for the measurements by Kitanovski et al. (2019) [81] (abbreviated KI19) are shown in Table 1. A detailed description of these measurements is not included, because the set-up is similar to the one used by Xu et al. (2020) [62], and the measurements are limited to a white color center. For the other experiments considered a comprehensive description of the measurements and the modeling is given in the next paragraphs. In Supplement 1 an application example of the line element for spatial gratings is explained.

Table 1. Main experimental parameters for the eight data sets considered and the corresponding values of the parameters $K_\alpha$, $u_\alpha$, $\gamma$ and $\mathcal {A}_\alpha$. The bottom rows show the root-mean-square errors $E_{\text {rms} \alpha }$ of the CSFs and the measure of dissimilarity $d_{\text {rms}}$ for the two threshold ellipse measurements. Symbols and abbreviations: $\dagger$ Luminance of the isoluminous RG gratings for the CSF measurements or the luminance of the isoluminous gratings in six color directions for the threshold ellipse measurements; $\ddagger$ the field angle is the angular length of the side for a square grating ($\square$) or the angular diameter for a circular grating ($\bigcirc$); HE/GA: hard edge or Gabor grating; m/b: monocular or binocular viewing; n/a: natural pupil or artificial pupil; $\S$ psychophysical method: method of adjustment (adj), staircase method (sc), four-alternative forced-choice (4AFC), two-alternative forced-choice (2AFC), adaptive staircase procedure for estimating thresholds QUEST [89] (Q); $\gamma$ is the exponent of the optical MTF [Eq. (10)].

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7.1 Model parameters and constants

The CSFs given by Eqs. (17), (21) and (22) use a fair number of model parameters and constants. For some model parameters we use the typical values as given by Barten [40] (pp. 37,61) : $u_0=7$ cpd and $(\eta \, p \, \Phi _0)=\frac {1}{896}~$td. We use the values $Y_T=Y_D=165$ td as reported in Ref. [55]. The values of $K_{\alpha }$, $\mathcal {A}_{\alpha }$ and $u_{\alpha }$, with $\alpha \in \{A,T,D \}$, depend on the measurement conditions and the threshold criteria. The average value of the MTF exponent $\gamma$ is equal to 0.62, as reported in Ref. [73], but adaptations are necessary for dealing with inter-observer differences and different grating sizes. $B$ is given by Eq. (18). The MacLeod-Boynton chromaticities for EEW are $l_E=0.6654$ and $s_E=0.01608$. We use $k_{23}=5$ for threshold differences as reported in Ref. [16]. For each of the measurements mentioned in the preamble of this section, the values of the free parameters $K_{\alpha }$, $\mathcal {A}_{\alpha }$, $u_{\alpha }$ and $\gamma$, as determined in this study, are summarized in Table 1.

7.2 Measurements by Mullen

Mullen (1985) [4] (abbreviated as MU85) measured CSFs using sinusoidal monochromatic isoluminous RG (526 and 602 nm) and BY (470 and 577 nm) gratings, and sinusoidal monochromatic green (526 nm) and yellow (577 nm) luminance gratings. The gratings were produced by optical superposition of pairs of sinusoidal monochromatic gratings. For the isoluminous RG grating the 526 and 602 nm gratings were presented 180$^{\circ }$ out of phase and combined optically. The Michelson luminance contrast of the two monochromatic gratings was always equal. At threshold the inverse of the Michelson luminance contrast of one of the monochromatic gratings is defined as the contrast sensitivity. Several preventive measures were taken to avoid the visibility of luminance variations in the chromatic patterns. The same was done for the isoluminous BY (470 and 577 nm) gratings. For the measurement of the achromatic CSF the monochromatic gratings were presented in-phase and combined optically. The wavelength pairs of the gratings were chosen based on the chromatic response functions of Hurvich and Jameson (1955) [79] (pp. 642-643) [82]. The grating patch was circular and probably hard-edged as in Ref. [51]. The surround was uniform with the same mean color and a reduced mean luminance. The grating pattern was horizontally oriented. The spatial frequency range was from about 0.06 to 6 cpd for the chromatic gratings and from about 0.06 to 20 cpd for the achromatic gratings. The field diameter varied stepwise in the range from 2.2 to 23.5 deg to make sure that the contrast sensitivity was independent of the number of cycles. The stimulus was shown continuously and was phase reversed sinusoidally at 0.4 Hz. Viewing was monocular with a natural pupil. For the RG gratings the mean luminance was 15 cd m$^{-2}$ and the pupil diameter 4 mm, and for the BY grating respectively 2.1 cd m$^{-2}$ and 6 mm. A single staircase procedure was used to determine the contrast thresholds. The procedure started at a randomly selected contrast above or below threshold. The measurement results of two observers were reported. As mentioned above, for these measurements the reciprocal of the threshold Michelson contrast of one of the monochromatic gratings is defined as the contrast sensitivity. The Michelson contrast at threshold is denoted as $C_\text {M}$. The MacLeod-Boynton chromaticities $(l,s)$ of the color centers are $(0.688,0.0005)$ and $(0.566,0.098)$ for the RG and the BY grating, respectively. In the RG modulation direction we have $\frac {ds}{dl}\approx -0.4{\%}$. Therefore, in this direction we neglect the change of the S-cone response. In the BY modulation direction there is a substantial change of the L-cone response. In Appendix B we estimate that the contrast sensitivity in the measured BY direction is about $\mathcal {E}=7{\%}$ lower compared with the contrast sensitivity in the BY cardinal direction. In Appendix C the relation is explained between the luminance Michelson contrast at threshold of one of the chromatic components and the cone contrast. From Eqs. (24) and (37) it follows that:

 

Fig. 1. (a) Observer RMC: data points of the measured isoluminous RG CSF (red circles) and the achromatic CSF (green, 526 nm) (green circles). Calculated curves of $\text {CSF}_T(u)$ [Eq. (21)] (red line). Calculated curves of $\text {CSF}_A(u)$ [Eq. (17)] (green line). (b) Observer KT: data points of the measured isoluminous RG CSF (red circles), isoluminous BY CSF (blue circles) and achromatic CSF (yellow, 577 nm) (brown circles). Calculated curves of $\text {CSF}_T(u)$ [Eq. (21)] (red line). Calculated curves of $\text {CSF}_D(u)$ [Eq. (22)] (blue line). Calculated curves of $\text {CSF}_A(u)$ [Eq. (17)] (brown line).

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7.3 Measurements by McKeefry et al

McKeefry et al. (2001) [56] (abbreviated as MK01) measured CSFs using isoluminous RG (543 and 599 nm) and BY (439 and 568 nm) gratings. The experimental set up was similar as in MU85 [4]. The wavelength pair of the RG grating corresponds to the RG cardinal direction, whereas the wavelength pair of the BY grating departs slightly from the BY cardinal direction. The detection thresholds were determined by the method of adjustment. Viewing was monocular through an artificial pupil with 5 mm diameter. For both gratings the mean retinal illuminance was 60 td. The size of the grating at the various spatial frequencies is not mentioned, but a minimum of 3 cycles was visible in all the experiments. The measurement results of two observers were reported. The spatial frequency range was from 0.4 to 7 cpd for observer JJK and from 0.4 to 4 cpd for observer DMcK. The gratings were presented in pattern reversal mode at a rate of 1 Hz (the spatial phase was reversed each 500 ms, square wave time modulation). At threshold, the reciprocal of the Michelson luminance contrast of one of the monochromatic gratings is defined as the chromatic contrast sensitivity. The Michelson contrast at threshold is denoted as $C_\text {M}$. The MacLeod-Boynton chromaticities $(l,s)$ of the color centers are $(0.697,0.0001)$ and $(0.585,0.352)$ for the RG and the BY grating, respectively. In the RG modulation direction we have $\frac {ds}{dl}\approx -0.12{\%}$. Therefore, for this direction we neglect the change of the S-cone response. In order to realize in the complete frequency range a sufficient range of contrast levels, the BY modulation direction deviated from the BY cardinal direction. However, in MK01 [56] this direction was considered to be sufficiently close to the BY cardinal direction. This was verified with measurements in the BY cardinal direction at low spatial frequencies ($u<2$ cpd). In this case the same relation between $C_{\text {M},\alpha }^{-1}(u)$ and $\text {CSF}_\alpha (u)$ as given by Eq. (26) $(\text {with }\mathcal {E}=0)$ holds, with $l_1=0.596$ and $l_2=0.798$ the $l$-chromaticities of respectively the monochromatic red and green light, and $s_1=0.704$ and $s_2=4.2 \times 10^{-5}$ the $s$-chromaticities of respectively the monochromatic blue and yellow light. The data points of the observers JJK and DMcK, scaled according to Eqs. (26) with $\mathcal {E}=0$, $w_T=0.145$ and $w_D=1.0$, and the curves of $\text {CSF}_T(u)$ and $\text {CSF}_D(u)$ according to Eqs. (21) and (22), respectively, are shown in Figs. 2(b) and 2(a). The fitting parameters $K_\alpha$ and $u_\alpha$ of the functions $\text {CSF}_\alpha (u)$ are determined as in section 7.2. We assumed a constant circular grating with diameter of 7.5 deg. The results are mentioned in Table 1, together with the chosen values of $\gamma$ and $\mathcal {A}_{\alpha }$. The other parameters are mentioned in section 7.1.

 

Fig. 2. (a) Observer DMcK: data points of the measured isoluminous RG CSF (red circles) and BY CSF (blue circles). Calculated curves of $\text {CSF}_T(u)$ [Eq. (21)] (red line). Calculated curves of $\text {CSF}_D(u)$ [Eq. (22)] (blue line). (b) Observer JJK: data points of the measured isoluminous RG CSF (red circles) and BY CSF (blue circles). Calculated curves of $\text {CSF}_T(u)$ [Eq. (21)] (red line). Calculated curves of $\text {CSF}_D(u)$ [Eq. (22)] (blue line).

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7.4 Measurements by Kim et al

Kim et al. (2013) [27] (abbreviated as KM13) measured achromatic and chromatic CSFs at various luminance levels and on a D65 background. Isoluminous red to green (RG) and violet to yellow-green (BY) gratings were used for the chromatic CSFs. We consider the data from 0.2 to 150 cd m$^{-2}$ for the achromatic gratings and 0.2 to 200 cd m$^{-2}$ for the chromatic gratings. The spatial frequency range varied from 0.125 to 16 cpd for the achromatic gratings and from 0.25 to 8 cpd for the chromatic gratings. The stimuli were shown on a 24 inch LCD screen with RGB LED backlight. The chromaticity pairs of the RG and the BY gratings correspond to the RG and the BY cardinal directions, respectively. The stimuli consisted of horizontally oriented sinusoidal gratings multiplied by a 2D Gaussian envelope (Gabor grating). At the two lowest spatial frequencies (0.125 and 0.25 cpd) the standard deviation (SD) of the Gaussian envelope was equal to 3 deg, and for all other spatial frequencies (0.5 to 8 cpd) the SD was equal to 1.5 deg. The area of the screen outside the grating was set to the mean luminance level. The presentation time was not limited, and the stimuli were not modulated over time. The detection of the grating was under natural viewing conditions i.e. natural pupil, without corrections for chromatic aberrations of the eye and binocular. The detection thresholds were determined with the four-alternative forced-choice (4AFC) method. More than four observers participated in the measurements. In KM13 [27] the contrast is defined as:

 

Fig. 3. (a) Data points of the measured isoluminous RG CSF at mean luminance levels from 0.2 to 200 cd m$^{-2}$ (open circles). Calculated curves of $\text {CSF}_T(u)$ [Eq. (21)] (solid lines). Calculated curves of $(\frac {\delta L}{L})^{-1}(u)$ according to MT20 [14] (dotted lines). (b) Data points of the measured isoluminous BY CSF at mean luminance levels from 0.2 to 200 cd m$^{-2}$ (open circles). Calculated curves of $\text {CSF}_D(u)$ [Eq. (22)] (solid lines). Calculated curves of $(\frac {\delta S}{S})^{-1}(u)$ according to MT20 [14] (dotted lines).

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Fig. 4. (a) Data points of the measured achromatic CSFs in KM13 [27] at mean luminance levels from 0.2 to 150 cd m$^{-2}$ (open circles). Calculated curves of $\text {CSF}_A(u)$ [Eq. (17)] (solid lines). Calculated curves of $(\frac {\delta Y}{Y})^{-1}(u)$ according to MT20 [14] (dotted lines). (b) Data points of the measured achromatic CSFs in CA82 [3] for field sizes of $0.5^\circ$ to $60^\circ$ (length of the side of a square grating), the mean luminance was constant at 108 cd m$^{-2}$ (open circles). Calculated curves of $\text {CSF}_A(u)$ [Eq. (17)] (solid lines).

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7.5 Measurements by Xu et al

Xu et al. (2020) [62] (abbreviated as XU20) measured the visibility thresholds in six different directions of the UCS $(u',v')$ plane [86] (pp. 72–73) with isoluminuous chromatic gratings at various spatial frequencies. This was carried out at five color centers: white (W) $(u',v',Y[\text { cd m}^{-2}])=(0.1979, 0.4695, 72)$, red (R) $(0.3155, 0.5016, 14)$, green (G) $(0.1449, 0.4758, 24)$, yellow (Y) $(0.2109, 0.5234, 50)$ and blue (B) $(0.1700, 0.3772, 9)$. The gratings were shown on a 27 inch LCD screen with 2560$\times$1440 pixels. The area of the screen outside the grating was set to the same color center as the grating. Viewing was binocular with natural pupil at a distance of 500 mm from the screen, the total field of view was $61.5^\circ \times 37.6^\circ$. The stimuli consisted of vertical or horizontal gratings multiplied by a Gaussian envelope with a standard deviation (SD) of $37.6^{\circ }/4$ (Gabor gratings). The detection thresholds were determined with the two-alternative forced-choice (2AFC) method. A staircase procedure was followed. Horizontally or vertically oriented gratings were shown sequentially. The observers had to indicate the orientation of the grating. The thresholds were measured at 0.06, 0.12, 0.24, 0.48, 0.96, 1.92 and 3.84 cpd. In total 89 observers participated in the experiment. For each color center the visual data was collected for 20 observers. The parameters of the 35 measured threshold ellipses are not available. However, we derived the parameters of the measured threshold ellipses in the $(u',v')$ plane from the data points in Fig. 6(a) of XU20 [62], using the WebPlotDigitizer [87]. In this figure the values of the reciprocal of the Euclidean distance $\sqrt {du'^2+dv'^2}$ at threshold are shown for the five color centers at the seven spatial frequencies in six directions. From this data we calculated the parameters of the threshold ellipses in the $(du',dv')$ plane, using the MATLAB code of Ref. [88], and transformed these ellipses to the $(dl,ds)$ plane [cf. Appendix A.]. In a next step we calculate at each spatial frequency and color center the threshold ellipses according to the grating LE [Eqs. (23), (24) and (25)]. The parameters $K_\alpha$ and $u_\alpha$, with $\alpha \in \{T,D\}$, are calculated for an optimized fit between the 35 measured and calculated threshold ellipses in the $(dl,ds)$ plane. Similarly as in Ref. [16], using MATLAB solver fmincon, we minimize the root-mean-square (RMS) of the dissimilarities, denoted as $d_{\text {rms}}$, between the measured and calculated threshold ellipses of the complete data set. For the calculation of the contrast sensitivity functions $\text {CSF}_\alpha (u)$, $\alpha \in \{T,D\}$ we used for the Gabor gratings a grating area $\mathcal {A}=\pi \, \text {SD}^2$ as in Refs. [14,60]. The results are mentioned in Table 1, together with the chosen values of $\gamma$ and $\mathcal {A}_{\alpha }$. The other parameters are mentioned in section 7.1. For the optimized fit with the 35 measured ellipses the dissimilarity $d_{\text {rms}}=0.393$. The RMS of the dissimilarities per spatial frequency for the five color centers W$\dots$B are shown in Fig. 5(a). The RMS of the dissimilarities per color for the seven spatial frequencies $u=0.06 \dots 3.84$ cpd are shown in Fig. 5(b).

 

Fig. 5. (a) The RMS of the dissimilarities per spatial frequency for the five color centers W$\dots$B, according to the grating LE (green squares) and MT20 [14] (for $E=0.520$) (red squares). (b) The RMS of the dissimilarities per color for the seven spatial frequencies 0.06$\dots$3.84 cpd, according the grating LE (green squares) and MT20 [14] (for $E=0.520$) (red squares). (a) and (b) The $d_{\text {rms}}$ value (complete data set) equals 0.393 for the grating LE (green dashed line) and 0.502 for MT20 [14] (red dashed line).

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Fig. 6. Measured and calculated threshold ellipses of the XU20 [62] data set. Each panel shows the threshold ellipses of the XU20 [62] data set, at the five color centers and at one of the spatial frequencies (thick solid lines), and the calculated threshold ellipses according to the grating LE (dotted lines) and according to the model of MT20 [14] (thin solid lines).

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We also calculate the threshold ellipses according to the color difference metric of MT20 [14]. The threshold distance in MT20 [14] is denoted as $E$. We optimized the value of $E$ for minimal dissimilarity $d_{\text {rms}}$ for the 35 threshold ellipses of the XU20 [62] data set. Using the fitting parameters given in Ref. [85] we find for $E=0.520$ the lowest dissimilarity with value $d_{\text {rms}}=0.502$. The RMS of the dissimilarities per spatial frequency for the five colors centers W$\dots$B are shown in Fig. 5(a). The RMS of the dissimilarities per color for the seven spatial frequencies $u=0.06 \dots 3.84$ cpd are shown in Fig. 5(b).

The experimentally determined threshold ellipses of the XU20 [62] data set and the calculated threshold ellipses according to the grating LE and the model of MT20 [14] are shown in Fig. 6.

8. Discussion

In this study we have applied the CSF models [Eqs. (17), (21) and (22)] to the CA82 [3], MU85 [4], MK01 [56] and KM13 [27] CSF measurements, and the complete model [Eqs. (23), (24) and (25)] is applied to the KI19 [81] and the XU20 [62] threshold ellipse experiments. Each CSF has a fair number of parameters. However, for each CSF we use only two fitting parameters to optimize the fit with the various data sets, viz. $K_\alpha$ and $u_\alpha$ with $\alpha \in \{A,T,D\}$. The parameters $\gamma$ and $\mathcal {A}_\alpha$ are adapted to the individual measurements, because these parameters depend on the observer(s) and the measurement conditions. For the other parameters we use the values as mentioned in paragraph 7.1. The goodness of fit with the CSF data sets was evaluated with Eq. (27), and for the threshold ellipse data sets of KI19 [81] and XU20 [62] we use the dissimilarity measure $d_{\text {rms}}$ [16]. The results are given in Table 1. In the next part we discuss how well the model fits the data sets and the potential causes of the variances between the parameter values across the different experiments.

A preliminary remark should be made on the size of the grating field in relation to the inhomogeneous distribution of the L, M and S cones in the retina [90] (p.244), and the decline of the contrast sensitivity away from the fovea [10]. The achromatic CSF given by Eq. (17) is mainly based on Barten’s contrast sensitivity model [40] (pp. 25-64). As stated by Barten, this model is valid in situations for which the center of the grating is imaged on the center of the fovea, but for which the image of the grating may be substantially larger than the fovea [40] (p. 65). The RG and BY isoluminous CSFs are partially based on the experiments by Rovamo et al. [55]. For these experiments the field of the grating was about $16^{\circ }\times 16^{\circ }$ at the lowest spatial frequency. Mullen and Kingdom (2002) [10] showed that the RG cone contrast sensitivity declines steeply away from the retina, whereas the BY cone contrast sensitivity and the achromatic cone contrast sensitivity have a much shallower decline.

The achromatic detection mechanism has an effective spatial integration area modeled according to Eq. (15), with as parameters the maximum extent of the spatial integration $\mathcal {A}_A$ and the critical spatial frequency $u_A$. In Barten’s study [40] (p.20) the typical value $\mathcal {A}_A=144$ deg$^2$ was proposed, which was derived from the CA82 [3] measurements. In the same study, the maximum number of cycles given by $n_A=u_A\, \sqrt {\mathcal {A}_A}$ was used, and a large spread of this parameter (from 5 to 25) across different studies was reported in Ref. [40] (p.21). According to Barten, this large spread is probably caused by the different measurement conditions. As typical value, Barten proposed $n_A=15$ cycles, which corresponds with $u_A=1.25$ cpd for $\mathcal {A}_A=144$ deg$^2$. We found a good matching between the data points of CA82 [3] and the curves of the calculated $\text {CSF}_A$ given by Eq. (17) with $\mathcal {A}_A=144$ deg$^2$ and $u_A=0.97$ cpd [cf. Table 1 and Fig. 4(b)]. Notice that the value of $\mathcal {A}_A$ might vary considerably across different experiments. In a study by Rovamo et al. (1993) [36] the parameter values $\mathcal {A}_A=269$ deg$^2$ and $u_A=0.650$ cpd were reported. In another study by Rovamo et al. (1994) [44], the parameter values $\mathcal {A}_A=320$ deg$^2$ and $u_A=0.465$ cpd were reported. In the MU85 [4] measurements the size of the grating field was changed at some specific spatial frequencies, and with $\mathcal {A}_A=144$ deg$^2$, as typical value proposed by Barten, the curve of the calculated $\text {CSF}_A(u)$ shows substantial jumps at these frequencies. However, the achromatic CSF data in MU85 [4] doesn’t show jumps when the grating field size changes. In our analysis of the MU85 [4] achromatic CSFs, we found that a value of $\mathcal {A}_A\approx 15$ deg$^2$ is necessary to match the measurement data with the curve of the calculated $\text {CSF}_A(u)$ [Eq. (17)]. Similarly, in the KM13 [27] measurements the size of the grating field is increased at the two lowest frequencies. In this case, a good matching with the measurement data is also possible with $\mathcal {A}_A\approx 15$ deg$^2$. For the isoluminous RG and BY gratings, we use $\mathcal {A}_\alpha =15$ deg$^{2}$ with $\alpha \in \{T,D\}$ for the same reason.

As shown in Table 1, across the different data sets we found different values for the parameter $K_\alpha$ with $\alpha \in \{A,T,D\}$. The value of $K_\alpha$ across the different data sets depends on e.g. the detection threshold criteria, the experimental conditions, and inter-observer differences. The detection thresholds of the experiments in Table 1 were estimated using different psychophysical methods [91]. Each of these methods might have a different psychometric function and the threshold criteria might be different. The measurements in CA82 [3], MU85 [4] and KM13 [27] were carried out with respectively a square window hard edge grating, a circular window hard edge grating and a Gabor grating. As shown by Kelly (1970) [63] the particular phase of a hard edge sinusoidal grating may change the achromatic contrast sensitivity. The gratings in MU85 [4] were phase reversed sinusoidally at 0.4 Hz and therefore not strictly static. On the other hand, there was no time-modulation for the experiments of CA82 [3] and KM13 [27]. Intraocular scattering is another factor that has impact on $K_\alpha$. Intraocular light scattering causes a veil of light on the retina and reduces the contrast of the retinal image. The amount of straylight varies per observer, may vary per eye of an observer, and increases with the age of the observer [92,93]. We estimate this variability with Eq. (6) in Ref. [73]: for an observer with 20 years of age and a mean blue Caucasian eye, the contrast sensitivity is about 12% lower compared to an observer of the same age with a mean dark-brown Caucasian eye; for an observer with 60 years of age and a blue Caucasian eye, the contrast sensitivity is about 8% lower compared to an observer at the age of 20 and the same eye pigment. The error of the estimated $K_\alpha$ depends also on the difference between the estimated pupil diameter and the actual pupil diameter, because e.g. the retinal illuminance depends on the pupil diameter. Notice that there are large inter-observer differences of the pupil diameter [79] (p.105) [94] and the optical MTF of the eye optics [75]. Nevertheless, for most of the data sets of Tabel 1, a good fit is achieved with the typical value of the exponent $\gamma =0.62$ of the MTF model of the eye optics [Eq. (10)]. However, for the CA82 [3] and the KI19 [81] data set other values of $\gamma$ are necessary. In the CA82 [3] case, $\gamma =1.2$ was necessary for a faster fall-off of the optical MTF as a function of the spatial frequency. This faster fall-off may be attributed to the large field angle of a number of data points of the CA82 [3] data set [95,96] in combination with the high spatial frequencies. Notice that Barten also needed a faster fall-off of the optical MTF for a good matching with the CA82 [3] data set [40](p. 52). In the case of KI19 [81], a slower fall-off of the optical MTF was necessary with $\gamma =0.30$. This slower fall-off may be attributed to the small field angle of the grating used for the KI19 [81] data set [95,96] in combination with the high spatial frequencies.

The $E_{{\text {rms}\alpha }}$ values of Table 1 indicate an adequate fit between the CSF measurements and the CSF models. For the CA82 [3] and the KM13 [27] data sets, the $E_{{\text {rms}\alpha }}$ values are larger compared to the MU85 [4] and MK01 [56] data sets. The reason for this are the large ranges of the grating areas and the luminance levels for the CA82 [3] and KM13 [27] data sets, respectively.

The isoluminous threshold measurement of KI19 [81] were carried out at intermediate and high spatial frequencies (2.4 to 19.1 cpd) and were limited to a white (D50) color center and one luminance level of 32.7 cd m$^{-2}$. The root-mean-square dissimilarity $d_{\text {rms}}$ is equal to 0.270 for the complete data set. This indicates an adequate fit with the measured threshold ellipses.

The XU20 [62] isoluminous threshold measurements were carried out at five color centers and relative low spatial frequencies. The lowest spatial frequency was 0.06 cpd, which is equal to the lowest spatial frequency of the MU85 [4] measurements. The reciprocal of the area of the measured threshold ellipses as a function of the spatial frequency for the five color centers is shown in Fig. 7(a). The reciprocal of a threshold ellipse area is an aggregated measure for the contrast detection sensitivity at a specific color center and spatial frequency. Unlike the lowpass shape of the isoluminous CSFs of the MU85 [4] measurements, the curves in Fig. 7(a) show a bandpass shape with a maximum at 0.24 cpd. The decreasing sensitivity for the spatial frequencies $u<0.24$ cpd might have several causes. In XU20 [62] it is mentioned that the cause might be the low number of grating cycles. There are in this case, at the lowest spatial frequency of 0.06 cpd, about 1.12 cycles in an angular width of $2\times$SD of the Gaussian window. This is close to the 1.41 cycles of the hard edge grating of the MU85 [4] measurements at 0.06 cpd, and the MU85 [4] isoluminous chromatic CSFs have definitely a lowpass shape. Therefore, we assume that for the XU20 [62] measurements the response of the achromatic detection mechanism is not sufficiently silenced, which is caused by perceivable residual luminance variations in the grating’s retinal image. It is known that the production of chromatic gratings that have an equiluminous retinal image is a challenge [97,98], because equiluminance is affected by the longitudinal and transverse chromatic aberration of the eye optics [99,100]. Moreover, optimal focussing by the accommodation system is less effective due to chromatic aberration [97,101]. In addition, neural factors such as inter-observer spectral sensitivity differences [102] and retinal inhomogeneity [97,103], might cause visible achromatic contrast variations. Because of the aforementioned factors, silencing the achromatic detection mechanism depends on the grating’s spectrum, spatial frequency and field size, and is also observer dependent. The reciprocal of the area of the calculated threshold ellipses as a function of the spatial frequency for the five color centers and the conditions of the XU20 [62] measurements is shown in Fig. 7(b). Contrary to the XU20 [62] measurements, the model shows a lowpass shape for the aggregated sensitivity as a function of the spatial frequency.

 

Fig. 7. (a) Reciprocal of the threshold ellipse area according to the XU20 [62] measurements. (b) Reciprocal of the threshold ellipse area according to the grating LE [Eqs. (23), (24) and (25)].

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The XU20 [62] threshold ellipses and the threshold ellipses according to the model [Eqs. (23), (24) and (25)] are shown in Fig. 6. This figure gives the qualitative impression of an adequate correlation between the measured and calculated threshold ellipses. This is confirmed by the low measure of dissimilarity $d_{\text {rms}}$ equal to 0.393 for the 35 threshold ellipses. The RMS of the dissimilarities per spatial frequency for the five color centers is shown in Fig. 5(a). With the exception of the lowest spatial frequency of 0.06 cpd, there are small variations of the RMS dissimilarities around the $d_{\text {rms}}=0.393$ of the complete data set. The larger RMS dissimilarity for 0.06 cpd is due to the bandpass and lowpass shape of the sensitivity for the measurements and the model, respectively, as shown in Figs. 7(a) and 7(b). The RMS of the dissimilarities per color for the seven spatial frequencies is shown in Fig. 5(b). This figure shows small variations of the RMS dissimilarities around the $d_{\text {rms}}=0.393$ of the complete data set.

The threshold ellipses according to the MT20 [14] model are also shown in Fig. 6. For this model we find the dissimilarity measure $d_{\text {rms}}$ equal to 0.502 for the 35 ellipses. With exception of the blue color center, Fig. 6 shows adequate agreement between the measured and calculated threshold ellipses. This deviation for the blue color center is also shown in Fig. 5(b). The metric tensor in the cone contrast space of the MT20 [14] model is independent of the color center chromaticities, i.e. the metric is Euclidean for isoluminuous stimuli. This is the root cause of this deviation for the blue color center. As shown in Fig. 5(a) there are small deviations of the RMS dissimilarities around the $d_{\text {rms}}=0.502$, also at the lowest spatial frequency. This is because the two isoluminous chromatic CSFs of the MT20 [14] model have a bandpass shape.

The mean value and the coefficient of variation (CV) [104] of $K_\alpha$, $K_A/K_T$, $K_T/K_D$ and $u_\alpha$ across the measurements of Table 1 are shown in Table 2. The parameters $K_A$ and $K_T$ have a CV equal to 0.33 and 0.36, respectively. As comparison, across the 17 achromatic CSF measurements of Barten the CV of the parameter $k$ is equal to 0.19 [40] (p. 57). The ratio $K_A/K_T$ has a mean equal to 2.6 and a CV of 0.07, while the ratio $K_T/K_D$ has a mean equal to 16 and a CV of 0.38. Also the critical frequencies $u_\alpha$ have a high CV.

Table 2. Mean values and the coefficient of variation (CV) of the fitting parameters $K_\alpha$ and $u_\alpha$, and the ratios $K_A/K_T$ and $K_T/K_D$ across the data sets.

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9. Conclusions

A Riemannian color difference metric is necessary for the adequate prediction of threshold ellipsoids in color space. We developed a new Riemannian color difference metric for spatial sinusoidal color variations. With this metric one can predict, in any direction of the color space, the detection threshold of sinusoidal color variation as a function of the grating’s luminance, chromaticity, spatial frequency and size. This is important for various applications. The metric is based on a Riemannian color difference metric for split fields and models for the contrast sensitivity function of the isolated achromatic, red-green and blue-yellow detection mechanisms. Since these contrast sensitivity functions are new key-constituents of the metric, we validated the models against various contrast sensitivity measurements in the cardinal directions, available in literature. We also validated the complete Riemannian color difference metric against the threshold ellipse measurements of isoluminous chromatic Gabor gratings at various spatial frequencies and various color centers. Adjustments of a limited set of model parameters are necessary to take into account various differences in the measurement set-ups and inter-observer differences. We found adequate agreement with the measurements considered.

Appendix A: Transformations

The 3-dimensional space defined by the mechanism responses $dD_A$, $dD_T$ and $dD_D$ is a linear transformation of the cone contrast space [21,23] and the MacLeod-Boynton contrast space [16]:

(30)$$\begin{bmatrix} dD_A \\ dD_T \\ dD_D \end{bmatrix} = \begin{bmatrix} L & M & 0 \\ L & -L & 0 \\ -L & -M & L+M \end{bmatrix} \begin{bmatrix} \frac{dL}{L} \\ \frac{dM}{M} \\ \frac{dS}{S} \end{bmatrix}= Y \begin{bmatrix}1 & 0 & 0 \\ 0 & \frac{L}{M} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{dY}{Y} \\ \frac{dl}{l} \\ \frac{ds}{s} \end{bmatrix}$$

Consider the Uniform Chromaticity Scale (UCS) plane with chromaticity coordinates $(u',v')$ [86] (pp.72-73). The transformation from the $(du',dv')$ plane to the $(dl,ds)$ plane is given by:

(31)$$\begin{bmatrix} dl \\ ds \end{bmatrix}= \begin{bmatrix} \frac{a_{11}-a_{13}}{y} & \frac{(a_{13}-a_{11})x-a_{13}}{y^2} \\ \frac{-a_{33}}{y} & \frac{a_{33}(x-1)}{y^2} \end{bmatrix} \begin{bmatrix} \frac{48y+12}{V} & \frac{-48x}{V} \\ \frac{18y}{V} & \frac{-18x+27}{V} \end{bmatrix}^{{-}1} \begin{bmatrix} du' \\ dv' \end{bmatrix}\; ,$$

with: $a_{11}=0.15514$, $a_{13}=-0.03286$ and $a_{33}=0.01608$, $x=\frac {9u'}{6u'-16v'+12}$, $y=\frac {4v'}{6u'-16v'+12}$ and $V=(-2x+12y+3)^2$.

Appendix B: Cone contrast thresholds in isoluminuous blue-yellow modulation directions

The blue-yellow modulation direction of the MU85 [4] contrast sensitivity measurements was not in a direction that isolates the blue-yellow mechanism. In this section we estimate the impact of this deviation on the contrast sensitivity function. Consider a color center with cone responses $(L_0,M_0,S_0)$. For small isoluminuous color modulations around the color center, the cone contrasts are given by: $\frac {dL}{L_0}=\frac {dl}{l_0}$, $\frac {dM}{M_0}=-q \, \frac {dl}{l_0}$ and $\frac {dS}{S_0}=\frac {ds}{s_0}$ with $q=\frac {l_0}{m_0}$. Assume that the direction of the isoluminuous color modulation is determined by the coordinates $(l_1,s_1)$ and $(l_2,s_2)$, then we have:

(32)$$\frac{dl}{l_0}=\rho \, \frac{ds}{s_0}\; , \qquad \rho=\frac{(l_1-l_2)(s_1+s_2)}{(s_1-s_2)(l_1+l_2)}\, , \quad l_0=\frac{l_1+l_2}{2} \quad \text{and} \quad s_0=\frac{s_1+s_2}{2}\, .$$

For the MU85 [4] measurements, the isoluminuous chromatic blue-yellow grating is composed of two monochromatic gratings with wavelengths $\lambda _1=470$ nm and $\lambda _2=577$ nm, and corresponding chromaticities $(l_1,s_1)=(0.439,0.197)$ and $(l_2,s_2)=(0.694,3.52\,10^{-5})$, respectively. For these chromaticities we have $\rho =-0.225$. In a second step we calculate, with the LE, the cone contrast threshold in the isolating direction $[\begin {smallmatrix} \frac {dl}{l_0} & \frac {ds}{s_0} \end {smallmatrix}]^T=[\begin {smallmatrix} 0 & 1 \end {smallmatrix}]^T$ and in the direction of the actual measurements $[\begin {smallmatrix} \frac {dl}{l_0} & \frac {ds}{s_0} \end {smallmatrix}]^T=[\begin {smallmatrix} \rho & 1 \end {smallmatrix}]^T$. From Eqs. (23), (24) and (25) it follows that the reciprocal of the cone contrast threshold in the isolating direction and in the direction of the actual measurements are respectively given by:

(33)$$S_D(u) =\sqrt{h_{33}} \qquad \text{and} \qquad S_m(\rho,u)=(1+\rho^2)^{{-}0.5} \; \left\{ \left[\begin{smallmatrix} \rho & 1 \end{smallmatrix}\right] \; \; \mathsf{h} \; \; \left[\begin{smallmatrix} \rho \\ 1 \end{smallmatrix}\right] \right\}^{0.5} \, ,$$

with $\mathsf {h}=\left [\begin {smallmatrix} h_{22} & h_{23} \\ h_{23} & h_{33} \end {smallmatrix}\right ]$. The relative error $\mathcal {E}(\rho,u)=\frac {S_m(\rho,u)-S_D(u)}{S_D(u)}\times 100$ [%] is a measure for the deviation of the contrast sensitivity caused by the different modulation directions. For the MU85 [4] measurements we calculate $\mathcal {E}(\rho,u)$ at the spatial frequencies 0.1, 1, 3 and 5 cpd for $\rho =-0.225$. The results are shown in Table 3. The contrast sensitivity is for the spatial frequencies considered about 7% lower in the measuring direction compared with the isolating direction.

Table 3. Relative error $\mathcal {E}(\rho,u)$ at the spatial frequencies 0.1, 1, 3 and 5 cpd for the MU85 [4] measurements.

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Appendix C: Optical superposition of two achromatic gratings

The Michelson contrast of an achromatic grating is given by:

(34)$$C_{\text{M}} = \frac{Y_{\text{max}}-Y_{\text{min}}}{Y_{\text{max}}+Y_{\text{min}}}=\frac{\delta Y}{Y_{\text{avg}}}$$

with $\delta Y=\frac {1}{2}(Y_{\text {max}}-Y_{\text {min}})$ the amplitude of the retinal illuminance variation and $Y_{\text {avg}}=\frac {1}{2}(Y_{\text {max}}+Y_{\text {min}})$ the average retinal illuminance. Consider the superposition of two sinusoidal achromatic gratings oriented along the $x$-axis, with color centers $(L_1,M_1,S_1)$ and $(L_2,M_2,S_2)$, opposite phases, spatial frequency $u$, Michelson contrast $C_{\text {M}}$ and average retinal illuminance $Y_0=L_1+M_1=L_2+M_2$:

(35)$$\begin{bmatrix} L(x) \\ M(x) \\ S(x) \end{bmatrix}= \begin{bmatrix} L_1 \\ M_1 \\ S_1 \end{bmatrix} \{1+C_{\text{M}} \, \sin(2 \pi \, u \, x)\} + \begin{bmatrix} L_2 \\ M_2 \\ S_2 \end{bmatrix} \{ 1-C_{\text{M}}\, \sin(2 \pi \, u \, x) \} \, .$$

The average cone responses and the amplitude of the cone response differences of the superposition are respectively given by:

(36)$$\begin{bmatrix} L_0 \\ M_0 \\ S_0 \end{bmatrix}= \begin{bmatrix} l_1 + l_2\\ m_1 + m_2 \\ s_1 + s2 \end{bmatrix} Y_0 \qquad \text{and} \qquad \begin{bmatrix} \delta L \\ \delta M \\ \delta S \end{bmatrix}= \begin{bmatrix}| l_1 - l_2 | \\| m_1 - m_2|\\ |s_1 - s_2 | \end{bmatrix} \; C_{\text{M}} \; Y_0 \, .$$

It follows that the cone contrast values are given by:

(37)$$\frac{\delta L}{L_0}=\frac{\delta l}{l_0}=\frac{|l_1-l_2|}{l_1+l_2} \, C_{\text{M}} \, , \qquad \frac{\delta M}{M_0}=\frac{\delta m}{m_0}=\frac{|m_1-m_2|}{m_1+m_2} \, C_{\text{M}} \, , \qquad \frac{\delta S}{S_0}=\frac{\delta s}{s_0}=\frac{|s_1-s_2|}{s_1+s_2} \, C_{\text{M}} \, .$$

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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