**We measured and modeled visibility thresholds of spatial chromatic sine-wave gratings at isoluminance. In two experiments we manipulated the base color, direction of chromatic modulation, spatial frequency, the number of cycles in the grating, and grating orientation. In Experiment 1 (18 participants) we studied four chromatic modulation directions around three base colors, for spatial frequencies 0.15–5 cycles/deg. Results show that the location, size and orientation of fitted ellipses through the observer-averaged thresholds varied with spatial frequency and base color. As expected, visibility threshold decreased with decreasing spatial frequency, except for the lowest spatial frequency, for which the number of cycles was only three. In Experiment 2 (27 participants) we investigated the effect of the number of cycles at spatial frequencies down to 0.025 cycles/deg. This showed that the threshold elevation at 0.15 cycles/deg in Experiment 1 was at least partly explained by the small number of cycles. We developed two types of chromatic detection models and fitted these to the threshold data. Both models incorporate probability summation across spatially weighted chromatic contrast signals, but differ in the stage at which the contrast signal is calculated. In one, chromatic contrast is determined at the cone receptor level, the dominant procedure in literature. In the other model, it is determined at a postreceptoral level, that is, after cone signals have been transformed into chromatic-opponent channels. We applied Akaike's Information Criterion to compare the performance of the models and calculated their relative probabilities and evidence ratios. We found evidence in favor of the second model and conclude that postreceptoral contrast is the most accurate determinant for chromatic contrast sensitivity.**

^{2}. As mentioned above, we only report studies or parts of studies in the table that concern static sine-waves.

*X*and

*Y*represent coordinates in the range (−0.5, 0.5) relative to the display center. This attenuation profile is flat and equal to 1 at the central stimulus area and rapidly falls off to the value of 0 to the edge of the stimulus area (18.9°).

*SD*= 8.6).

*Y*= 108 cd/m

^{2}), oriented either horizontally or vertically. In Experiment 1 the orientation was randomly assigned per trial, in Experiment 2 all stimuli were shown in both orientations. The perceived orientation of the pattern was indicated by the participants by pressing the left or right arrow key for horizontal, and up or down arrow key for vertical orientation on a keyboard. Visibility thresholds were determined using a one-down/three-up weighted staircase method that converges to 75% correct responses (Kaernbach, 1991; Levitt, 1971). There was no reason to expect a response bias in the judgment of a horizontal or vertical grating. Research has shown inconclusive results on the effect of orientation on chromatic contrast sensitivity. Kelly (1975) reported no oblique effect on the contrast sensitivity to chromatic stimuli. Murasugi and Cavanagh (1988) showed an orientation effect for a 2

*°*red-green sinewave grating (drifting at 2 Hz). Three participants had higher sensitivity to horizontal orientation, but in the fourth participant the effect was inverted. Webster et al. (1990) showed an estimated 10% difference in contrast threshold (for both L-M grating and an S gratings) at 90

*°*orientation difference, but only for a spatial frequency of 2 cycles/deg.

*p*values 0.0041, 0.914, 0.0012, <0.001, <0.001, 0.20, respectively) at the 95% CI. Note that these ratios are calculated over all levels of the chromatic contrast, so including values below, around, and above threshold.

^{−6}). The ellipses are approximately centered on the base color and, at first impression, show an increasing size with increasing spatial frequency. So, chromatic contrast sensitivity (the inverse of threshold modulation depth) is lower for the higher spatial frequencies, which is the expected result for a low-pass sensitivity function usually associated with chromatic channels. Moreover, the ellipses also rotate in u′v′ space which is not easily understood and is certainly not confined to this particular choice of color space. For instance, in Figure 6 we show the same data plotted in MacLeod-Boynton chromaticities L/(L+M) versus S/(L+M) (CIE, 2015) . The vertical axes in this figure are on a different scale than the horizontal axes, so while they may look a bit less elliptical than the plots in u′v′ space, they are actually elongated along the vertical axes when the figure is drawn on equal axis. We will show in the modeling part of this paper that the changes in size and rotation are accounted for by a chromatic detection model of the visual system.

*base color*,

*spatial frequency,*and

*color direction*on the visibility thresholds. The threshold data were first log-transformed to obtain a normal distribution. Main effects for

*base color, F*(2, 1199) = 8.1,

*p*= 0.0003;

*spatial frequency, F*(5, 1199) = 592.18,

*p*< 0.001); and

*color direction, F*(3, 1199) = 7.56,

*p*= 0.0001, were found to be significant at the 95% CI. The first order interaction effect between

*spatial frequency*and

*base color*was not significant (

*p*= 0.0528); the other interactions were highly significant (

*p*< 0.001). One salient aspect of the thresholds shown in Figure 5 is that the ellipse size decreases with decreasing spatial frequency, but increases again for the lowest spatial frequency (0.15 cycles/deg, in bold red line). The mean threshold at spatial frequencies of 0.15 is significantly different from all others, meaning that the increase in threshold at 0.15 cycles/deg is significant with respect to the next higher spatial frequency (0.3 cycles/deg). We suspected that this was related to the diminishing number of sine-wave cycles that can be shown in a fixed stimulus area when spatial frequency is lowered. This initiated Experiment 2 in which we studied the effect of the number of cycles in the stimulus. Before presenting the results of that experiment, we first report on the variability of the thresholds between participants.

*threshold*as dependent variable and

*orientation*and

*spatial frequency*as independent variables. A significant main effect of

*spatial frequency*on

*threshold*was found,

*F*(4, 269) = 67.37,

*p*< 0.001. The main effect of

*orientation*was not significant,

*F*(1, 269) = 1.58,

*p*= 0.21. No significant interaction effect between the two independent variables was found,

*F*(4, 269) = 1.81,

*p*= 0.13.

*xy*color space, but the 1976 u′v′ color space was designed to minimize these variations. We here show strong additional changes in ellipse shape when spatial frequency at a fixed color point is varied within a constant stimulus size, underlining the limited applicability of the MacAdam ellipses to nonuniform stimuli. We should note that the MacAdam ellipses were measured for adaptation to the chromaticity of illuminant C, whereas in our experiments participants were adapted to base colors of correlated color temperature 2600, 3800, and 5700 K. A difference in adaptation point leads to different discrimination ellipses, as shown for example, by Opstelten and Rinzema (1987) and by Krauskopf and Gegenfurtner (1992). Moreover, MacAdam used a 2° bipartite test field having a luminance of about 48 cd/m

^{2}, twice that of the background, whereas we used isoluminant sine-wave patterns at 108 cd/m

^{2}. So, the two types of stimuli also have very different spatial frequency content.

^{2}and for a single participant. Also in the achromatic domain it has been reported that the number of cycles becomes dominant in contrast sensitivity measurements at the lowest spatial frequencies (Savoy & McCann, 1975). This is linked to the fact that the physical limitations in display size simply do not allow the presentation of more cycles. Decreased sensitivity in that case is not necessarily caused by the visual system, but may result from a methodological artefact. Rovamo et al. (1993) described contrast sensitivity in the luminance domain as a function of stimulus size, spatial frequency, and number of cycles. In the chromatic domain, however, the effects of the interplay of these parameters have not been unified into a single model yet.

*°*orientation difference, measured at 2 cycles/deg, differed about 10% from the value of 1, which corresponds well with our data measured at 1.5 and 3.0 cycles/deg (see Figure 3).

*XYZ*and LMS space, for our NEC display we also derived the following matrix relationship between CIE 1931 XYZ tristimulus values and

*LMS*cone excitations:

*Y*to

*XYZ*, and then applying Equation 2 to obtain the

*L*,

*M*, and

*S*cone excitations.The matrix elements in Equation 2 were calculated by minimizing the pooled model error for a set of 27 spectral measurements, where for example,

*L*is modeled as

*L*= m

_{11}×

*X*+ m

_{12}×

*Y*+ m

_{13}×

*Z*and m

_{1j}represent the matrix elements in the first row. The average absolute error was 0.0001, negligibly small compared to the cone excitation values of around 100.

*L*,

*M*, and

*S*cone excitations. In addition, for both models we study the impact of having two or three chromatic mechanisms on the explained data variance. We expect a model with two chromatic channels to be more appropriate for our experimental data, since it represents chromatic thresholds at isoluminance, that is, one of the color dimensions is silenced.

*L*/

*L*, Δ

*M*/

*M*, Δ

*S*/

*S*, where Δ

*L*/

*L*for instance is calculated as (

*L*

_{threshold}−

*L*

_{base color})/

*L*

_{base color}, that is, the threshold increment or decrement relative to the level of the base color, which is assumed to correspond to the adapting color. Using a 3 × 3 matrix, we linearly recombine these cone contrasts into chromatic channels, here arbitrarily labeled as A, B, C:

_{i}(i = A, B, C) that depend on spatial frequency

*f*, and we then apply probability summation with a Minkowski norm to compute a pooled detection signal:

*cc*stands for cone-contrast and

*p*represents the Minkowski norm coefficient. In the data fitting procedure (least squares) we simultaneously optimize the matrix elements in Equation 3, the sensitivities S

_{A}(f), S

_{B}(f), S

_{C}(f) at each spatial frequency, and the Minkowski coefficient in Equation 4 to minimize the pooled model error:

*(*

_{cc}*i*) would equal 1, which is an arbitrary choice but conveniently indicates the threshold level. In the case of the three-channel model we have 28 parameters to estimate (i.e., 9 matrix coefficients in Equation 3 for the color transformation, 3 × 6 = 18 for the spatial sensitivity, and the Minkowski coefficient). In the case of the two-channel model, we arbitrarily omit channel A and matrix coefficients a

_{11}, a

_{12}, a

_{13}in Equation 3, and we also leave out the corresponding contrast signal A in Equation 4. Optimization of the two-channel model thus requires the estimation of 19 parameters (i.e., 6 color matrix coefficients + 2 × 6 spatial sensitivities +1 Minkoswki coefficient).

*L*,

*M*, and

*S*cone excitations into chromatic channels

*A*/

*A*, Δ

*B*/

*B*and Δ

*C*/

*C*, where Δ

*A*/

*A*for instance is calculated as (

*A*

_{threshold}−

*A*

_{base color})/

*A*

_{base color}, that is, the threshold increment or decrement relative to the level of the base color. The contrast signals are weighted by a sensitivity function S that depends on spatial frequency (f), and we then apply probability summation with a Minkowski norm to compute a detection signal:

*prc*stands for postreceptoral contrast. As for the cone contrast model, we estimate 28 parameters for the three-channel model and 19 parameters for the two-channel model, by minimizing the pooled error

_{A,}S

_{B,}and S

_{C}) at the six spatial frequencies, next to these the values of the nine coefficients in the color matrix and the Minkowski norm parameter are shown. In general, the sensitivity functions show a low-pass behavior for spatial frequency, except for the data point at the lowest spatial frequency (0.15 cycles/deg). Also, the sensitivity curve for channel B in the three-channel cone contrast model (top left graph) is more of a band-pass type typically found for an achromatic channel. Indeed, the coefficients in the corresponding color matrix show that the channel is constructed by mainly adding L and M cone excitations (the achromatic channel is usually defined as L+M).

*B*/

*B*and Δ

*C*/

*C*, which are 0.0011 and 0.014, respectively.

*R*

^{2}only (shown in Figure 10), it would be concluded that the three-channel models outperform the two-channel models. However, the contrast signals in two channels of the three-channel models were highly correlated, indicating an overfit, whereas the contrast signals in the two-channel models were almost uncorrelated. In the next section we use the Akaike Information Criterion to compare the models and show that a model having two chromatic channels is more likely than one having three, and postreceptoral contrast calculation is much more likely than cone contrast.

*value is the preferred model in the sense that it has the lowest loss of Kullback-Leibler information when approximating a true distribution with the model distribution of data values (Burnham, Anderson, & Huyvaert, 2011). From Equation 9 it follows indeed that a model having a smaller residual sum of squares gives rise to a smaller AIC*

_{c}*value. In Table 3 the AIC performance measures of the model fits described above are presented. It shows two important things. Firstly, the two-channel models have lower AIC*

_{c}*values (in the order of ΔAIC*

_{c}*= 10) than the three-channel models. Secondly, the postreceptoral contrast models have lower AIC*

_{c}*values (in the order of ΔAIC*

_{c}*= 20) than the cone contrast models.*

_{c}_{c}values we calculated the so called Akaike weights (e.g., Wagenmakers & Farrell, 2004) defined as

*M*models. The ΔAIC values appearing in Equation 10 are obtained by subtracting the minimum AIC value of the pool of models from that of model i: Δ

_{i}AIC = AIC

_{c,i}− AIC

_{c,min}. The ratio of the Akaike weights of two models can be interpreted as the evidence ratio of one model to the other. The last column in Table 3 presents the evidence ratios for model three. So, the ratio of models three and four, having the lowest and second lowest absolute AIC

_{c}value, equals 0.997/0.00286 = 349, implying that model 3 (postreceptoral, two chromatic channels) is 349 times more likely to be the true model in comparison to the postreceptoral three-channel model. Likewise, the ratio of the Akaike weights of models three and one is 0.997/5.547E – 05 = 17,910, indicating that within the two-channel models, the postreceptoral contrast model is much more likely than the cone-contrast model. We will come back to this salient result in the discussion section.

^{2}) and its peak around a spatial frequency of about 3 cycles/deg (Kim et al., 2013), the influence of the achromatic component would have occurred for our stimuli at 3 and 5 cycles/deg. For the lower spatial frequencies we studied, achromatic contrast sensitivity is negligibly small relative to chromatic contrast sensitivity.

_{c}value of −215, much worse than the values reported in Table 3. So, when forcing the traditional chromatically opponent channels, model performance drops substantially. In general, since our best result was obtained by estimating all model parameters in a single run, any constraint put on the model parameters will lead to a decrease in the reported performance.

*n*,

_{L}*n*and

_{M,}*n*respectively). For the two-channel postreceptoral model this resulted in a slightly lower RSS of 0.785 (compared to the 0.807 reported in Table 3), but at the cost of three additional model parameters. The accompanying AIC

_{S},_{c}value was −251.9, not better than our best result.

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