**Narrowly tuned, selective noise masking of chromatic detection has been taken as evidence for the existence of a large number of color mechanisms (i.e., higher order color mechanisms). Here we replicate earlier observations of selective masking of tests in the (L,M) plane of cone space when the noise is placed near the corners of the detection contour. We used unipolar Gaussian blob tests with three different noise color directions, and we show that there are substantial asymmetries in the detection contours—asymmetries that would have been missed with bipolar tests such as Gabor patches. We develop a new chromatic detection model, which is based on probability summation of linear cone combinations, and incorporates a linear contrast energy versus noise power relationship that predicts how the sensitivity of these mechanisms changes with noise contrast and chromaticity. With only six unipolar color mechanisms (the same number as the cardinal model), the new model accounts for the threshold contours across the different noise conditions, including the asymmetries and the selective effects of the noises. The key for producing selective noise masking in the (L,M) plane is having more than two mechanisms with opposed L- and M-cone inputs, in which case selective masking can be produced without large numbers of color mechanisms.**

**Figure 1**

**Figure 1**

*σ*= 1°) presented against a gray background field with a rapid-start “sawtooth” profile of 333-ms total duration (Figure 2). The sawtooth profile was chosen to maximize the likelihood of separating On and Off responses. Fixation was guided by four black diagonal lines pointing at the center of the screen (ending 1.5° from the center), which were present throughout the experiment.

**Figure 2**

**Figure 2**

**n**| (Appendix A) was 0.498, 0.414, and 0.267 for the 42°/222°, 48°/228°, and 64°/244° noises, respectively.

**t|**, is defined as the Euclidean distance from the origin to the (ΔL/L, ΔM/M) point representing the peak of the stimulus: Reported contrasts have been halved (from the nominal, peak value) to facilitate comparison with studies not using half-toned stimuli.

**Figure 4**

**Figure 4**

*W*and

_{L}*W*for the L- and M-cones, respectively), with that sum defined to be 1.0 at threshold. The mechanisms are half-wave rectified so that each mechanism responds only in one half of the chromatic space (Figure 1; Eskew, 2009). The mechanisms are combined by an approximation to probability summation, with a Minkowski exponent of 4.0. These features of the model are identical or nearly identical to those used previously by several authors (Cole, Hine, & McIlhagga, 1993; Eskew, McLellan, & Giulianini, 1999; Eskew, Newton, & Giulianini, 2001; Giulianini & Eskew, 1998; Newton & Eskew, 2003; Sankeralli & Mullen, 1996).

_{M}*τ*, in the presence of the noise of contrast power |

**n**|

^{2}at angle

*ν*, is given by (derived in Appendix A). The half brackets (⌊ ⌋) represent half-wave rectification (i.e., values less than zero are set to zero). The vector of cone contrast weights (the “mechanism vector”; Eskew et al., 1999), is

**f**, which takes polar angle

*α*in the (L,M) plane (

*α*= 0 is the L-cone increment direction). The subscripted Qs are calculated constants that represent the energy and power in test (

*t*) and noise (

*n*), respectively, and the fitted parameter

*b*represents the sensitivity of the mechanism to the spatiotemporal characteristics of the noise. The first term in the radical represents the baseline condition in which no noise was added, and the second term raises the threshold due to the noise.

**f**(or, equivalently, the magnitude |

**f**| and its angle

*α*) and the noise sensitivity parameter

*b*. Below, we report the polar angle of the mechanism vector and vector length in the (ΔL/L, ΔM/M) plane (see Table 1), with asymptotic standard error estimates from the fits (and after applying the appropriate propagation of error formulae in converting from the cone weights to angles and vector lengths). The degrees of freedom for fitting the model is the number of thresholds across all noise conditions (e.g., 92 for observer TGS) minus three times the number of mechanisms (e.g., 74 degrees of freedom for observer TGS's six-mechanism model).

**Table 1**

**f**. Even more important, for every test angle, the masking effect of the noise on the mechanism must be proportional to cos

^{2}(

*α*−

*ν*), with the constant of proportionality

*bQ*the same for all test and noise angles detected by that mechanism. These model features are derived from theory and are consistent with prior results in both luminance and chromatic detection (Gegenfurtner & Kiper, 1992; Giulianini & Eskew, 1998; Legge, Kersten, & Burgess, 1987; Pelli, 1981; Wang et al., 2014), and collectively they make model comparisons much more powerful. For example, comparing models with different numbers of these mechanisms is not based on curve fitting but rather is a comparison of theoretical accounts of the data.

_{n}/Q_{t}*SE*(based on between-sessions variability only); in many cases the error bars are smaller than the symbols. Colored lines on the plots represent mechanism thresholds (discussed below), and the smooth closed contour is the probability sum of those mechanisms. The line color is a rough indication of the hue of stimuli that lie on that line according to informal observations of the observers. For example, stimuli on or near the orange line appear “orangey red,” and stimuli on or near the blue line appear “bluish.”

**Figure 5**

**Figure 5**

**Figure 6**

**Figure 6**

**Figure 7**

**Figure 7**

**Figure 8**

**Figure 8**

*R*

^{2}≥ 0.98), but there are a few areas along some of the contours where the fit is poor (see Figures 5 through 7). For observer TGS, the model slightly overestimates the thresholds along the flanks and slightly underestimates thresholds in the corners of the 48°/228° condition. For observer CLM, the model overestimates thresholds in QI in the 64°/244° condition. There is also a slight overestimation along the greenish flank (QII) in the 42°/222° noise condition. These small discrepancies result from the constraint that the slopes of each mechanism line must be the same in across all the conditions, for each observer, because the model was fit to all of the noise conditions simultaneously (Equation A1 and Appendix A).

**Table 2**

**Figure 9**

**Figure 9**

*R*

^{2}> 0.97. However, the model fits with these symmetry and sensitivity constraints were significantly worse than the original six-mechanism model where both the mechanism's sensitivity and cone weights were free to vary, even after taking into account the reduced number of free parameters produced by the symmetry constraint. These conclusions hold whether or not the noise sensitivity parameter

*b*was constrained to be the same for the two members of a pair. Details of these analyses are given in the Supplementary Material. Allowing slight asymmetries in the model is necessary to satisfactorily fit the data across multiple noise conditions.

^{2}

*(*

^{γ}*α*−

*ν*), with

*γ*a free parameter—for all or some of the mechanisms. This type of nonlinearity, which has been used in several previous studies (D'Zmura & Knoblauch, 1998; Goda & Fujii, 2001; Hansen & Gegenfurtner, 2006; McKeefry, McGraw, Vakrou, & Whitaker, 2004), forces a symmetric tuning curve for the masking effect of the noise, and it failed here.

*only*S-cone input, for which there is no evidence). Because some of the mechanisms of Figure 10 must get S-cone input, we have depicted it in the figure for completeness. Although the cardinal model (Figure 1) has the S-cone input opposed by a sum of L and M, substantial prior evidence indicates that S-cone detection mechanisms have long-wavelength inputs of opposite sign, as shown here (De Valois & De Valois, 1993; McLellan & Eskew, 2000; Wang et al., 2014; Wisowaty, 1983).

**Figure 10**

**Figure 10**

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*E*represents the test contrast energy (proportional to squared contrast),

*N*is the applied noise contrast power (also proportional to squared contrast), and

*N*is a constant representing the level of intrinsic noise in the detection mechanism. For a rectified but otherwise linear chromatic detection mechanism, Equation A1 may be written as (compare equation 9 of Wang et al., 2014). In Equation A2,

_{0}**f**is the vector of cone contrast weights of the mechanism, (

*W*,

_{L}*W*). The vector

_{M}**t**represents the cone contrasts (ΔL/L, ΔM/M) of the test, and

**n**is the corresponding vector representing the noise. The half brackets (⌊ ⌋) represent half-wave rectification (i.e., values less than zero are set to zero).

*Q*is the constant of proportionality between the noise contrast squared and the noise power. Its value was taken to be the unit contrast noise power at DC, which is 1.07 × 10

_{n}^{−3}deg·s, after accounting for half toning. This value was calculated in Wang et al. (2014) for radially symmetric noise. The value here is the same considering the vertical dimension of the pattern. The analogous constant for the contrast energy of the blob test, again considering only the vertical dimension, is in which

*h*(

*t*) gives the time course of the test presentation (Figure 2b) and

*w*(

*y*) is the half-toning function (which sets alternate 2-pixel lines to zero contrast). Explicitly including

*Q*and

_{t}*Q*in the model factors these stimulus-specific constants out of the mechanism vector and, in principle, makes the cone contrast weights independent of the spatiotemporal characteristics of the test and noise. Comparison with previously published cone contrast mechanism vector lengths from studies that did not explicitly take the Qs into account (e.g., the summary in Eskew et al., 1999) requires multiplying the vector lengths in Table 1 by .

_{n}*N*

_{0}= 1, effectively scaling in terms of the intrinsic noise. It is convenient to express the relationship of Equation A2 in polar coordinates: where

*α*,

*τ*, and

*ν*are the angles of the mechanism, test, and noise vectors, respectively, in the (L,M) plane. The vector length of the test at threshold is then

**f**|. The second term in the radical represents the effect of the noise (of contrast

**|n|**). Because the mechanism sensitivity |

**f**| affects the response to the noise and the test equally, the mechanism vector length does not appear in the second term, which is to say that the

*relative*degree of elevation by the noise does not depend on the sensitivity of the mechanism. Equation A4 was applied to each mechanism, simultaneously across all the noise conditions, to estimate values of the cone weights (

*W*,

_{L}*W*; determining |

_{M}**f**| and

*α*) and

*b*, with the responses of the mechanisms combined by probability summation (Minkowski exponent of 4.0; Eskew et al., 1999).

**t**that would produce a threshold response in the presence of chromatic noise, with the vector of cone weights

**f**fixed for each hypothetical mechanism. Because we were fitting measured thresholds, we varied the mechanism parameters instead of varying simulated tests. Stated simply, for a single mechanism and noise condition, we found the vector of cone weights

**f**that satisfied

**f**·

**t**= 1, with

**t**determined by experiment, whereas Hansen and Gegenfurtner (2006) found values of

**t**satisfying the same relationship, with

**f**determined by assumption.