To assess the effects of spatial frequency and phase alignment of mask components in pattern masking, target threshold vs. mask contrast (TvC) functions for a sine-wave grating (S) target were measured for five types of mask: a sine-wave grating (S), a square-wave grating (Q), a missing fundamental square-wave grating (M), harmonic complexes consisting of phase-scrambled harmonics of a square wave (Qp), and harmonic complexes consisting of phase-scrambled harmonics of a missing fundamental square wave (Mp). Target and masks had the same fundamental frequency (0.46 cpd) and the target was added in phase with the fundamental frequency component of the mask. Under monocular viewing conditions, the strength of masking depends on phase relationships among mask spatial frequencies far removed from that of the target, at least 3 times the target frequency, only when there are common target and mask spatial frequencies. Under dichoptic viewing conditions, S and Q masks produced similar masking to each other and the phase-scrambled masks (Qp and Mp) produced less masking. The results suggest that pattern masking is spatial frequency broadband in nature and sensitive to the phase alignments of spatial components.

*f*spectrum, reflecting the properties of natural images. We used broadband periodic masks consisting of spatial square wave and missing fundamental waveforms in which the harmonics were either phase aligned or phase scrambled. Stimuli were presented under monocular or dichoptic viewing conditions. Using these types of stimuli and manipulations can help us address the following issues: (1) the effect of within-scale masking and cross-scale masking and their interactions; the “within-scale” term refers to a stimulus consisting of a mask and target that have common spatial scale components, such as the sine-wave target and the square-wave mask. A “cross-scale” stimulus refers to a stimulus where the target and mask do not have any common scale components, such as the sine-wave target and the missing fundamental mask. The square-wave mask includes “within-scale” and “cross-scale” masking effects; (2) whether or not masking remains constant if the mask's contrast energy spectrum remains constant but relative phase relationships among higher harmonics change; and (3) the dependence of monocular/dichoptic viewing on the above two issues. Our results suggest that masking is spatial frequency broadband in nature and sensitive to the phase alignments of spatial components.

^{2}.

*L*

_{0}is mean luminance,

*f*is spatial frequency,

*x*

_{0}is a random central location that shifts the phase of the sine-wave grating, and

*c*is the Michelson contrast defined as

*f*is its fundamental frequency,

*c*is the Michelson contrast of the fundamental spatial frequency of the square wave, and

*θ*

_{ j }is the phase of each component,

*j*. For a square wave (Q), the phase of each component is set to zero. For a Qp mask,

*θ*

_{1}is set to zero but the rest are randomized and a uniform probability distribution was used to assign the phases. The missing fundamental pattern (M) is the square wave minus its fundamental frequency component:

*θ*values are set to zero; for Mp configuration, the

*θ*values are randomized and

*c*is the Michelson contrast of the fundamental spatial frequency of the corresponding square wave.

*x*-axis represent the contrast of the missing fundamental frequency needed to form the Q mask.

*independently*and

*linearly*from the different stimulus component contrasts within an orientation channel: The inhibitory signal within an orientation channel is a linear function of the component contrasts, raised to the power of

*q*. Since our Q (or Qp) mask is the sum of S and M (or Mp) masks, Foley's model predicts that the inhibitory signal due to the Q and Qp masks should be predictable from those of S and M, and Mp. We tested this prediction using the modeling procedure described in 1; this was done for the monocular viewing conditions only, because there is no agreed model at present for binocular combination that can accommodate different stimulus phases and spatial frequencies. Figure A2 shows performance of the model fitted to the S, M, and Mp conditions simultaneously and the predictions for the Q and Qp conditions using the fitted parameters (parameter values and model performance are given in Table A1). Figures A3 and A4 show performance of the model fitted separately to the phase-aligned (S and M) and phase-scrambled (S and Mp) conditions, respectively (parameter values and model performance are given in Table A2). The model fitted to S, M, and Mp (Figure A2) or just S and Mp (Figure A4) is able to predict the Qp data fairly well, but the model does not provide a satisfactory prediction of the Q data, even when the model is fitted just to the S and M conditions (Figure A3): The thresholds are higher than predicted from the independence assumption of Foley's model. The additional masking that occurs for the square wave shows that phase alignment between the stimulus components can increase the inhibitory signal for monocular masking, so that the masking effect is greater than we would expect if the inhibitory signals from different components of a stimulus were generated independently.

*q*. The model assumes that there are filters matched to the target component and to the other stimulus components. Because the first stage of Foley's model involves linear summation of stimulus components within an orientation channel (see Foley's Eq. 7), we can conceptually collapse the set of filters down to just two: one narrowband filter that is sensitive to the target (the “detection filter”) and another that is sensitive to all the other stimulus components; the latter would be constructed from the filters that respond to all the non-target components but can be considered to be a single, broadband, filter. We assume that detection performance is mediated by the narrowband “detection filter.” It might be considered more appropriate to use Foley and Chen's (1999) extension of Foley's model because only Foley and Chen's model includes filters selective for phase. However, because our mask components are always either in phase with the target or outside of the spatial frequency passband of the detection filter (or both), Foley and Chen's model is formally equivalent to Foley's model for our phase-aligned conditions and almost equivalent to it for the phase-randomized conditions. We assume that the detection filter is only sensitive to the S stimulus and the S component of Q. The excitatory responses,

*E*

_{S},

*E*

_{M}, and

*E*

_{Q}, of the detection filter in the S, M, and Q conditions, respectively, are given by

*Se*

_{S}is the detection filter's excitatory sensitivity to the S component,

*c*is the mask contrast, and Δ

*c*is zero for non-target stimuli and equal to the target contrast for target stimuli.

*I*

_{S}and

*I*

_{M}, for the S and M conditions, respectively, are given by

_{S}and Si

_{M}are the inhibitory sensitivities of the narrowband detection filter and the broadband filter, respectively. Because of the linear summation of inhibitory components across spatial frequency within an orientation channel in Foley's (1994) model, the inhibitory input,

*I*

_{Q}, for the Q mask (which is the sum of S and M masks) should be the linear sum of inhibitory inputs for the other two conditions:

*σ*is an additive constant, and

*E*is

*E*

_{S},

*E*

_{M}, or

*E*

_{Q}depending on the masking condition and similarly

*I*is

*I*

_{S},

*I*

_{M}, or

*I*

_{Q}. Suppose the threshold is determined by the difference between the response to the mask alone (

*R*

_{m}, when Δ

*c*= 0) and the mask plus target (

*R*

_{m+t}, when Δ

*c*is the target contrast). The threshold, Δ

*c,*is reached when the expected difference in response

_{M}and Si

_{Mp}for the inhibitory sensitivities in the M and Mp conditions, respectively. Then, the fitted parameters were used to predict the performance for Q and Qp masks based on the assumption that the broadband filter response to the Q (or Qp) mask is a linear function of the responses to the S and M (or Mp) masks (Equation A6). We constrained our parameters—

*p, q,*and

*σ*—to be the same for all conditions. Thus, the model has 7 parameters (

*Se*

_{S}, Si

_{S}, Si

_{M}, Si

_{Mp},

*p*,

*q*, and

*σ*), with

*Se*

_{S}being fixed at 100 and 6 parameters free to vary. The fitted results are shown in Figure A2 and the parameters and the

*x*

^{2}goodness-of-fit test are shown in Table A1. The divisive inhibition model fits the S, M, and Mp mask results quite well. Although the estimated parameters predict the performance of Qp mask fairly well, the derived parameters underestimated the masking effect found with the Q mask.

PCH | TD | |
---|---|---|

Se _{S}* | 100 | 100 |

Si_{S} | 57.03 | 69.19 |

Si_{M} | 9.02 | 9.48 |

Si_{Mp} | 9.79 | 11.57 |

p | 2.36 | 2.51 |

q | 1.89 | 2.04 |

σ | 7.89 | 7.49 |

Num_{obs} | 19 | 19 |

x ^{2} | 6.919 (p = 0.863) | 3.418 (p = 0.992) |

x _{Q} ^{2} | 24.402 (p < 0.01) | 46.837 (p < 0.01) |

x _{Qp} ^{2} | 9.123 (p = 0.167) | 2.236 (p = 0.897) |

*Notes*: *Fixed parameter. Num_{obs}: Number of data points used to fit the data. *x* ^{2}: The reduced chi-squared statistics were calculated according to the following formula, *x* _{reduced} ^{2} = 1 v ∑ ( J o b s − J p r e d ) 2 σ 2 , where *v* is the number of degrees of freedom, given by Num_{obs} − *n* − 1, with *n* being the number of fitted parameters, and *σ* is the known variance of the observation (here, we used averaged variance). *x* _{Q/Qp,Mp} ^{2}: The same equation as previously mentioned. The degrees of freedom are 7–1.

*Se*

_{S}was fixed at 100, and 5 parameters were free to vary—see Table A2). The fitted results for the phase-aligned configuration are shown in Figure A3 and the parameters and the

*x*

^{2}goodness-of-fit test are shown in Table A2. The divisive inhibition model fit the S and M mask data quite well. However, the derived parameters underestimated the masking effect found with the Q mask even though the model was fitted just to the phase-aligned data. The fitted results for the phase-scrambled configuration are shown in Figure A4 and the parameters and the

*x*

^{2}goodness-of-fit test are shown in Table A2. The divisive inhibition model fit the S/Mp mask data quite well and the estimated parameters predict the performance of Qp mask better than in the phase-aligned case. We also tested a non-linear summation model, where the inhibitory denominator involves non-linear summation, namely, the output of the

*I*

_{S}and

*I*

_{M}were summed together after each of them were raised by power

*q*. However, the goodness of fit was worse (

*x*

_{Q}

^{2}are 26.788, 52.834, and 22.904 for PCH, TD, and GM, respectively, for phase-aligned configuration;

*x*

_{Qp}

^{2}are 15.194 and 2.233 for PCH and TD, respectively, for phase-scrambled configuration). The modeling results suggest that phase alignment of mask components must play an important role in pattern masking.

Phase-aligned | Phase-scrambled | |||||
---|---|---|---|---|---|---|

PCH | TD | GM | PCH | TD | ||

Se _{S}* | 100 | 100 | 100 | Se _{S}* | 100 | 100 |

Si_{Sa} | 63.82 | 73.70 | 66.27 | Si_{Sp} | 48.35 | 70.80 |

Si_{M} | 11.74 | 8.26 | 20.25 | Si_{Mp} | 11.29 | 11.06 |

p _{a} | 2.60 | 2.72 | 2.19 | p _{p} | 2.13 | 2.60 |

q _{a} | 2.10 | 2.23 | 1.71 | q _{p} | 1.69 | 2.12 |

σ _{a} | 7.42 | 10.44 | 15.68 | σ _{p} | 5.51 | 8.47 |

Num_{obs} | 13 | 13 | 13 | Num_{obs} | 13 | 13 |

x _{S,M} ^{2} | 6.931 (p = 0.436) | 3.318 (p = 0.854) | 1.429 (p = 0.985) | x _{S,Mp} ^{2} | 2.973 (p = 0.887) | 2.256 (p = 0.944) |

x _{Q} ^{2} | 22.867 (p < 0.01) | 45.690 (p < 0.01) | 12.332 (p = 0.055) | x _{Qp} ^{2} | 9.060 (p = 0.170) | 2.157 (p = 0.905) |

*Notes*: *Fixed parameter. Num_{obs}: Number of data points used to fit the data. *x* _{S,M/Mp} ^{2}: The reduced chi-squared statistics were calculated as in Table A1. *x* _{Q/Qp} ^{2}: The same equation as previously mentioned. The degree of freedom is 7–1.