**A Glass pattern consists of randomly distributed dot pairs, or dipoles, whose orientation is determined by a geometric transform that defines the global percept for this pattern. The perception of Glass patterns involves a local process to associate paired dots into dipoles and a global process to group the dipoles into a global structure. We used a variant of Glass patterns consisting of tripoles instead of dipoles to estimate the effect of luminance contrast on the global form percept. In each tripole, an anchor dot and two context dots formed the vertices of an equilateral triangle with the anchor dot pointing toward the center of the display. Grouping the anchor dot with one context dot would result in a global percept of a clockwise (CW) spiral and grouping with the other dot a counterclockwise (CCW) spiral. We manipulated the contrast of the context dots and measured the probability of a participant judging the patterns as a CW spiral. The CW spiral judging probability first increased then decreased with the contrast of the CW context dots, resulting in an inverted U shape. The peak also shifted to the right as the contrast of the competing CCW context dots increased. Our result cannot be explained by the existing models for Glass pattern perception. Instead, the data was well fit by a divisive inhibition model, in which the response of a global pattern is the excitation raised by a power and divided by the inhibition from all global patterns plus an additive constant.**

^{2}. The viewing distance between the eye position of the observers and the monitor was set at 75 cm so that each pixel occupied 2 min of one visual angle.

*|ΔL*|, and background luminance level,

*L*

_{0}, i.e., c = |

*ΔL*|

*/L*

_{0}. The Weber contrast was then converted to dB units by the operation 20 × log

_{10}(c). For each polarity, there were seven contrast levels for each context dot (CW and CCW dots) in the range between −30 dB (3%) and −1 dB (90%), and the anchor dot contrast remained −20 dB (10%).

*x*-axis indicates the contrast level of the CW dots; the

*y*-axis indicates the probability of the observer judging the Glass pattern as a CW spiral. Different icons denote the data collected at different CCW contrasts. The smooth curves are the fits of our model discussed below in the Model section. Each data point is the average of four measurements. The error bar represents the standard error of measurements in each condition.

*t*(3) ranged from 3.59 to 30.09,

*p*< 0.05, except the difference between −15 dB and −1 dB of the negative anchor polarity condition of observer YCJ,

*t*(3) = 2.34,

*p*= 0.051. This effect was less apparent as the CCW dot contrast increased.

*t*(13) ranged from 0.0803 to 0.8671,

*p*= 0.46 to 0.20.

*f*(

_{j}*x*,

*y*) is the receptive field of the

*j*th filter, which orientation conforms to the

*j*th global form in one Glass pattern;

*I*

_{1}(

*x*,

*y*) and

*I*

_{2}(

*x*,

*y*) are the spatial profiles; and

*C*

_{1}and

*C*

_{2}are the contrast of the two dots in the dipole, respectively. For simplicity, we assume that the receptive field centers are halfway between the two dots, and thus, the effects from both dots are symmetrical. Also, because we used identical square dots, the spatial profiles

*I*

_{1}and

*I*

_{2}are identical. Hence, Equation 1 can be simplified as where

*Cd*is the pooled contrast for the

_{j}*j*th dipole. Here we use constant

*A*to replace all factors not affected by contrast manipulation. In our tGP, there are three dots and thus three possible dipoles—the CW, CCW, and irrelevant (for decision making) dipoles (shown in Figure 5, left panel)—in one tripole, which globally produce three global forms: a CW spiral, a CCW spiral, and an irrelevant pattern. (The third pattern is deemed irrelevant because observers were not asked to respond to this global form.) Therefore,

_{j}*j*can refer to any one of these three forms. All other arrangements would either produce a nonoptimal response or unspecific orientation information.

*n*dipoles contributing to a global template: The excitation of

*j*th form, which is the sum of the excitation of all relevant dipoles, is We define

*nA*as the excitation sensitivity to the

_{j}*j*th global form and replace it with parameter

*Se*.

_{j}*j*th global form,

*I*, is defined in a similar fashion as where

_{j}*q*is the power parameter and

*Si*the inhibition sensitivity of the

_{k}*k*th template among all three aforementioned global forms. The response of the

*j*th global form is excitation,

*E*, raised by a power,

_{j}*p*, divided by the sum of the divisive inhibition,

*I*, and an additive constant,

_{j}*z*, as

*E*=

_{cw}*Se*×

*Cd*and

_{cw}*E*=

_{ccw}*Se*×

*Cd*, and inhibition (Equation 4) becomes

_{ccw}*I*= (

_{cw}*Si*×

_{t}*Cd*)

_{cw}*+ (*

^{q}*Si*

_{1}×

*Cd*)

_{ccw}*+ (*

^{q}*Si*

_{2}×

*Cd*)

_{irre}*, and*

^{q}*I*= (

_{ccw}*Si*×

_{t}*Cd*)

_{ccw}*+ (*

^{q}*Si*

_{1}×

*Cd*)

_{cw}*+ (*

^{q}*Si*

_{2}×

*Cd*)

_{irre}*is the inhibition from the corresponding template, and*

^{q}. Si_{t}*Si*

_{1}and

*Si*

_{2}are the inhibition sensitivities from the other two competing global form templates. Because there was no CW judging bias in the data, we used the same set of sensitivity parameters,

*Se*,

*Si*, and

_{t}*Si*

_{1}, in our model for CW and CCW responses.

*Cd*,

_{cw}*Cd*, and

_{ccw}*Cd*correspond to the pooled contrast of the CW dipole, CCW dipole, and irrelevant dipole, respectively. We empirically found that removing the irrelevant global form template did not affect the goodness of fit; thus, the parameter

_{irre}*Si*

_{2}was set as zero in the final fitting results.

*R*

^{2}) for all observers. Except for

*Si*,

_{t}*Si*,

_{1}*q*, and

*z*, all other parameters were fixed for

*R*

^{2}and did not differ empirically whether we set them free or not. The value of parameter

*p*was set to be 1 because we empirically found that fixing it at 1.00 did not affect the goodness of fit. Overall, our model explained 87% to 97% of the variance in the data with rooted mean square errors ranging from 0.0458 to 0.1034 and mean standard error from 0.0346 to 0.0583 across all observers.

*E*)/

_{CW}^{p}*z*, which is a monotonic increasing function of CW dipole contrast. Thus, it becomes a version of the energy model. We actually implemented the energy model, as shown in Figure 4a, by fixing parameter

*q*as 0 and

*p*as 2 in our model.

*E*)/

_{CW}^{p}*z*, which is a monotonic increasing function of CW contrast. This explains the rising part of the inverted U function. As CW dot contrast increases, the inhibition term becomes larger. Because the exponent parameter for inhibition,

*q*, is larger than that for excitation (Table 1), the inhibition quickly catches up with the excitation and drives the response down. J. A. Wilson et al. (2004) proposed a similar theory to explain why it is easier for an observer to integrate dots of the same contrast into a dipole. Such self-inhibition, with inverted U prediction, provides a way to implement the similarity model.

*R*=

_{cw}*E*/(

_{cw}^{p}*I*+

_{cw}*z*),

*I*= (

_{cw}*Si*×

_{t}*Cd*)

_{cw}*, but just affect the decision stage (Equation 6), in which the CCW dots only change the subtractive constant and thus have the same effect for all CW dot contrasts. That is, different CCW dot contrasts would only shift the inverted U function up and down rather than causing a lateral shift. This effect can be implemented by fixing parameter*

^{q}*Si*

_{1}as 0 and is visualized in Figure 4b.

*n*from Equation 4 and then multiplying it by the response function in Equation 7. Either operation can be readily absorbed by a change in the values of other parameters. In the context of our current study, the two models are mathematically equivalent given the assumptions we made for the current experiment. We chose the model in the current study simply to be consistent with Chen (2009). A future experiment may be able to distinguish these two models. For instance, suppose one repeated our experiment while manipulating the number of tripoles in the patterns. For the late summation model, the effect of the number of tripoles on the decision variable (Equation 6) would be linear, and for the early summation model, its effect would be exaggerated by the nonlinearity.

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