A Glass pattern consists of randomly distributed dot pairs (dipoles) whose orientations are determined by a geometric transform. To understand how an observer perceives the global structure, we investigated how the threshold for detecting a concentric or a radial Glass pattern (target) can be affected by the presence of another Glass pattern (masker). The Glass patterns had either concentric, radial, vertical, plaid, or spiral global forms. We used a 2AFC paradigm in which a mask was presented in both intervals while a target was randomly presented in one interval and a random dot pattern in the other. The target dot density thresholds were measured at 75% accuracy. For all masker types, the target threshold was constant at low masker densities and then increased with masker density. For concentric targets, concentric and spiral maskers had the best masking effect. For radial targets, a low-curvature spiral mask produced the best masking. The target threshold versus masker density functions were fit with a divisive inhibition model, in which the response of a global mechanism is the excitation of a linear template to the input image raised by a power and divided by the sum of an inhibition input and a constant.

*m*

_{1}signal dipoles and

*m*

_{2}noise dipoles while the masker contains

*n*

_{1}signal dipoles and

*n*

_{2}noise dipoles. That is, the target has a coherence

*m*

_{1}/ (

*m*

_{1}+

*m*

_{2}). When superimposed, the signal-plus-masker pattern contains

*m*

_{2}+

*n*

_{2}noise dipoles. This pattern can be taken as either a target of 1.0 coherence on a masker of

*n*

_{2}/ (

*m*

_{2}+

*n*

_{2}) coherence or a target of

*n*

_{1}/ (

*m*

_{2}+

*n*

_{2}) coherence on a masker of 1.0 coherence, or anything in between. There are other measurements of visibility in the literature, such as the maximum distance between dots in a dipole (Dakin, 1997; Kurki et al., 2003) or the maximum jitter of the orientation of the signal dipoles (Dakin, 1997). The former, however, may confound local and global processing, while the latter may create a new global structure different from that of the target (for instance, an orientation jitter of signal dipoles in a concentric pattern may result in a sum of two spiral patterns: one clockwise and the other counterclockwise). The second issue is that when a target is superimposed on a masker, the total number of dipoles is greater than that of the masker alone. Hence, an observer may simply use this difference in image statistics rather than perceived global structure to make responses.

^{2}.

*d*(the product of the density parameter) and half the number of pixels. The position of half of the dots, that is, the first dot in each dipole, was generated with a random number generator and distributed evenly in space. The position of the second dot in a dipole was determined by the position of the first dot and the desired global structure.

*k*was a constant depending on the position of the first dot,

*r*= (

*x*

^{2}+

*y*

^{2}),

*θ*= tan

^{−1}(

*y*/

*x*). The second dot in a dipole was placed such that the orientation of the dipoles was tangent to the contour, and the distance between the two dots in a dipole was always 5′. The parameter

*b*controlled the curvature of the spiral. We used spiral patterns of two curvatures: the high-curvature pattern, called Spiral 1, had

*b*= 0.16 and the low curvature one, Spiral 2,

*b*= 1.6, when the unit of

*θ*was radiance of arc length while the unit of

*h*be the response of each local dipole filter to a dipole. Hence, the contribution of

*k*th dipole to a

*j*th linear template in the

*i*th image would be

*a*

_{ i,jk}×

*h,*where

*a*

_{ i,jk}is a weighting constant. The value of the weighting constant

*a*

_{ i,jk}will be greater if the orientation of the

*k*th dipole is consistent with the sensitivity profile of the

*j*th linear template and smaller if not. Integrating all local filters, the excitation of the

*j*th linear template to

*i*th pattern,

*E*

_{ ij}, is

*n*is the number of dipoles in the image. That is, the excitation of the global linear template is a linear combination of the contribution of individual dipoles. The magnitude of excitation depends on both the number of dipoles in the image (

*n*) and the correspondence between the image and the template that determines the value and the distribution of

*a*

_{ i,jk}. Thus, the excitation can be rewritten as

_{ k}

_{=1}

^{ n}(

*a*

_{ i,jk}×

*h*) /

*n*.

*d*. Hence, Equation 3′ becomes

_{ ij}is Se′

_{ ij}times the number of possible dots in the image. Se

_{ ij}thus controls the contribution of the

*i*th pattern to the

*j*th template in our model. We constrained Se

_{ ij}to be positive for all conditions.

*p*and divide by the sum of a divisive inhibition term,

*I,*and an additive constant,

*z*. That is,

_{ j}= Σ

_{ m}(

*w*

_{ m}Se

_{ j}

^{ q}) is the sensitivity of the

*j*th mechanism to the divisive inhibition input.

*j*is the mechanism that gives the greatest response difference, and

*b*+

*t*denotes the pattern in the target-plus-masker interval, while

*b*+

*n*denotes the pattern in the noise-plus-masker interval. The target reaches the threshold when

*D*reaches a certain amount (Legge & Foley, 1980), designated as 1 in our model fitting.

Parameters | Observers | ||
---|---|---|---|

CC | CW | WL | |

p | 1.55 | 1.30 | 1.06 |

q | 1.09 | 1.00 ^{a} | 1.00 ^{a} |

z | 0.11 | 0.19 | 0.07 |

Se | |||

Concentric | 100 ^{a} | 100 ^{a} | 100 ^{a} |

Spiral 1 | 74.84 | 97.63 | 130.68 |

Spiral 2 | 100.92 | 109.01 | 55.21 |

Vertical | 29.05 | 16.40 | 1.96 |

Plaid | 28.33 | 34.45 | 65.88 |

Radial | 25.38 | 19.09 | 3.04 |

Noise | 27.95 | 36.70 | 57.69 |

Si | |||

Concentric | 32.30 | 17.08 | 3.71 |

Spiral 1 | 14.16 | 14.23 | 5.07 |

Spiral 2 | 23.03 | 23.17 | 14.78 |

Vertical | 26.19 | 16.66 | 6.87 |

Plaid | 28.21 | 3.48 | 3.33 |

Radial | 26.07 | 17.75 | 6.80 |

Noise | 22.54 | 6.81 | 2.27 |

*F*(2,53) = 1.03,

*p*= 0.36 for CW and

*F*(2,53) = 097,

*p*= 0.39 for WL) by fixing

*q*= 1. The parameter values in Tables 1 and 2 and fitted curves in Figures 2, 3, 5, and 6 reflect this constraint.

Parameters | Observers | ||
---|---|---|---|

CC | CW | WL | |

p | 1.23 | 0.98 | 1.71 |

q | 0.97 | 1.00 ^{a} | 1.00 ^{a} |

z | 0.12 | 0.07 | 0.09 |

Se | |||

Concentric | 81.88 | 250.46 | 0.38 |

Spiral 1 | 69.64 | 56.71 | 370.21 |

Spiral 2 | 225.49 | 1589.89 | 126.82 |

Vertical | 48.41 | 35.34 | 54.37 |

Plaid | 92.65 | 35.88 | 57.22 |

Radial | 100 ^{a} | 100 ^{a} | 100 ^{a} |

Noise | 63.07 | 83.43 | 82.35 |

Si | |||

Concentric | 15.40 | 20.31 | 3.71 |

Spiral 1 | 25.26 | 16.70 | 36.64 |

Spiral 2 | 18.66 | 54.75 | 10.57 |

Vertical | 23.58 | 9.30 | 113.90 |

Plaid | 7.30 | 9.30 | 151.82 |

Radial | 9.69 | 9.30 | 9.66 |

Noise | 6.34 | 10.51 | 7.05 |

*d*′) increased with the number of dipoles in the target and decreased with the number of dots in the noise masker and that

*d*′ could be explained by a function of the signal to noise ratio in the image. Their signal-to-noise ratio analysis implies that the threshold (i.e., the number of signal dots at a certain

*d*′ level) is proportional to the number of noise dots. This implies that the TvD function should have a fixed slope. This result, however, cannot be generalized to the various masking conditions we measured. The slope of TvD functions depends on both the target and the masker. As shown in Figure 9, the slope varied for different maskers. For the same concentric target, the slope of the TvD function for the noise masker and the concentric masker can differ more than twofold. Hence, the generality of this simple signal-to-noise ratio analysis is limited. A model that takes the interaction between global form detectors into account is necessary to explain our result.