**Previous studies investigating signal integration in circular Glass patterns have concluded that the information in these patterns is linearly summed across the entire display for detection. Here we test whether an alternative form of summation, probability summation (PS), modeled under the assumptions of Signal Detection Theory (SDT), can be rejected as a model of Glass pattern detection. PS under SDT alone predicts that the exponent β of the Quick- (or Weibull-) fitted psychometric function should decrease with increasing signal area. We measured spatial integration in circular, radial, spiral, and parallel Glass patterns, as well as comparable patterns composed of Gabors instead of dot pairs. We measured the signal-to-noise ratio required for detection as a function of the size of the area containing signal, with the remaining area containing dot-pair or Gabor-orientation noise. Contrary to some previous studies, we found that the strength of summation never reached values close to linear summation for any stimuli. More importantly, the exponent β systematically decreased with signal area, as predicted by PS under SDT. We applied a model for PS under SDT and found that it gave a good account of the data. We conclude that probability summation is the most likely basis for the detection of circular, radial, spiral, and parallel orientation-defined textures.**

*β*in the Quick-fitted psychometric functions. The exponent

*β*, which is estimated from the fit to the data when the units of stimulus intensity are “raw” i.e., linear units, is related to the slope, or “steepness” of the psychometric function, in that

*β*characterizes the slope when the units of stimulus intensity are plotted logarithmically spaced, as shown later (see Discussion for an analysis of the relationship between

*β*and other measures of psychometric function slope; Strasburger, 2001). Historically

*β*has been, and continues to be widely employed as a psychometric slope parameter for modeling PS under both HTT and SDT (e.g., Bell & Badcock, 2008; Loffler et al., 2003; Meese & Summers, 2012; Meese & Williams, 2000; Morrone et al., 1995; Schmidtmann et al., 2012; Tan, Bowden, Dickinson, & Badcock, 2015; Tan et al., 2013; Tyler & Chen, 2000; Watson, 1979). To avoid any confusion however with psychometric function “steepness,” we will refer to

*β*as the “exponent” of the psychometric function.

*β*as a function of signal area under conditions in which subjects were unaware of the position of the signal sector, such that extrinsic uncertainty decreased with signal area. Assuming that intrinsic uncertainty is sufficiently low that our manipulation of extrinsic uncertainty has a meaningful impact on overall uncertainty, PS under SDT predicts a decrease in

*β*with increasing signal area because of the resulting decrease in overall uncertainty (Kingdom, Baldwin, & Schmidtmann, 2015; Meese & Summers, 2012; Pelli, 1985; Tyler & Chen, 2000), whereas PS under HTT predicts no change in

*β*(Mayer & Tyler, 1986). A failure to find such decreases in

*β*would therefore constitute a valid basis for rejecting PS under SDT as a model for the detection of textures. This would leave open the possibility that either additive (including linear) summation, or probability summation under HTT is the correct model of detection.

^{2}), under the control of an Apple Mac Pro (3.33 GHz). Observers viewed the stimuli at a distance of 120 cm. At this distance one pixel subtends 0.018° of visual angle. Experiments were performed in a dimly-illuminated room. Routines from the Psychophysics Toolbox were employed to present the stimuli (Brainard, 1997; Pelli, 1997).

**Figure 1**

**Figure 1**

*fminsearch*function. The Quick function is: where

*x*is the proportion of signal elements within each signal sector

*, α*is the proportion of signal elements (Gabors or dot-pairs) within each sector required for 75% correct detection, and

*β*is the exponent that is related to the slope of the function.

*α*and exponents

*β*were estimated from the psychometric functions of proportion correct as a function of proportion of signal elements. Thresholds were measured for circular, radial, spiral, and parallel textures. The results are presented in Figure 2, where the left most graph shows detection thresholds

*α*(averaged across all observers) for each texture type and the right most graph shows the values of

*β*(averaged across observers and texture types), both as a function of the signal area size (percentage of total stimulus area).

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

*SEM*± 0.024). The strength of summation is about half as strong as reported by some investigators (Wilson et al., 1997; Wilson & Wilkinson, 1998), but not others (Dakin & Bex, 2002; Kurki et al., 2003; see Discussion). A two-way repeated-measures ANOVA with factors texture type and signal area showed a significant decrease in threshold with increasing signal area,

*F*(9, 36) = 71.87,

*p*< 0.001, but no difference between the different texture types,

*F*(3, 12) = 3.612,

*p*= 0.108. The interaction term was not significant,

*F*(27, 108) = 1.016,

*p*= 0.455.

*F*(3, 12) = 8.237,

*p*= 0.003. Subsequent post hoc test (Bonferroni corrected, here and throughout) showed a significant difference only between spiral and parallel textures (

*p*= 0.025). Performance is similar for circular, radial and spiral texture and tends to be best for parallel texture. This is contrary to previous results showing much better performance for circular Glass patterns (Wilson et al., 1997; Wilson & Wilkinson, 1998; cf. Dakin & Bex, 2002). The same studies also showed the poorest performance for parallel textures.

*β*values as a function of signal area.

*β*values range from 2.97 (

*SEM*± 0.21) for 10% signal area to 1.25 (

*SEM*± 0.097) for 100% signal area. This translates to a decrease in

*β*with signal area with a slope on a log–log plot of −0.39 (

*SEM*± 0.021). The statistical treatment of the

*β*values will be described later in the modeling section.

*SEM*± 0.035). The summation slopes are even shallower than in the Low-density experiment and range between −0.27 for parallel textures to −0.43 for radial textures. A two-way repeated-measures ANOVA (within-subjects) with factors of texture type and signal area revealed a significant increase in performance with increasing signal area,

*F*(3, 12) = 81.97,

*p*< 0.001, but no significant difference between the four different texture types,

*F*(3, 12) = 1.711,

*p*= 0.218. No significant interactions were found,

*F*(9, 36) = 1.912,

*p*= 0.082.

*F*(1, 19) = 81.881,

*p*< 0.001, and area,

*F*(3, 57) = 208.664,

*p*< 0.001. Subsequent post hoc tests showed significant differences between the two densities and for all signal areas. Increasing the density leads to a decrease in detection thresholds. The analysis revealed a significant interaction between density and area,

*F*(3, 57) = 31.055,

*p*< 0.001, which reflects an initial steeper decrease in thresholds with increasing area.

*F*(3, 12) = 1.449,

*p*= 0.277. The average

*β*values range between 2.29 (

*SEM*± 0.23) for 10% signal area and 1.352 (

*SEM*± 0.083) for 100% signal area. Similar to the Low-density Experiment,

*β*decreases with increasing signal area [slope = −0.21 (

*SEM*± 0.024)], as predicted by PS under SDT (see Discussion and Figure 4).

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

**Figure 6**

**Figure 6**

*SEM*± 0.027), consistent with the summation slopes measured in Experiments 1 and 2, but still only half the magnitude of the summation slopes reported previously (Wilson et al., 1997; Wilson & Wilkinson, 1998; cf. Dakin & Bex, 2002; Kurki et al., 2003). Similar to the Low-density and High-density stimuli, a two-way repeated-measures ANOVA revealed significant effect of signal area, F(2, 8) = 34.918,

*p*< 0.001, but no significant difference in performance is as a function of texture type,

*F*(3, 12) = 1.02,

*p*= 0.42. No significant interaction was found. In a separate statistical analysis (one-way ANOVA) we compared the thresholds for the different texture types at 100% signal area. There is a tendency for a superior performance for circular textures, but the statistical analysis reveals no significant differences between the different textures,

*F*(3, 12) = 1.461,

*p*= 0.274.

*β*values range from 2.016 (

*SEM*± 0.182) to 1.352 (

*SEM*± 0.083). Similar to the results in Experiments 1 and 2,

*β*decreases with increasing signal area, with a log–log slope of −0.37 (

*SEM*± 0.12).

*SEM*± 0.067), similarly shallow the Gabor textures in Experiment 3 (−0.37). A two-way ANOVA with texture type and area as factors revealed significant effects for both texture,

*F*(3, 12) = 11.00,

*p*= 0.001, and area,

*F*(2, 8) = 18.652,

*p*= 0.001. No significant interactions were found. Subsequent post hoc tests showed that detection thresholds for spiral Glass patterns were significantly higher compared to circular (

*p*= 0.001) and radial Glass patterns (

*p*= 0.046). In a separate one-way ANOVA we compared the detection thresholds for the different texture types for 100% signal area. The analysis showed a significant main effect,

*F*(3, 9) = 5.78,

*p*= 0.017. Additional post hoc tests showed that detection thresholds for circular Glass patterns were significantly lower than spiral ones (

*p*= 0.003). Analyzing detection thresholds for Gabor textures at 100% signal area shows that performance is similar for circular, radial, and parallel Glass patterns. There is a bias towards a better performance for circular and radial Glass patterns, but these effects are not statistically significant. Furthermore, we compared detection thresholds for each texture type for the Gabor textures in Experiment 3 with the detection thresholds for Glass patterns with separate one-way ANOVAS. These revealed significant differences only between the spiral textures, F(1, 3) = 16.18,

*p*= 0.003, showing that spiral Glass patterns are significantly harder to detect than spiral Gabor textures. Interestingly, there are no statistically significant differences between all the other Glass patterns and the Gabor textures, which suggests that the orientation information can be equally sufficiently extracted from dot pairs compared to oriented Gabors.

*N*= 5) with a presentation time of 160 ms, which falls within the range of earlier studies (Dakin & Bex, 2002: 147 ms; Wilson et al., 1997: 167 ms; Wilson & Wilkinson, 1998: 167 ms). The results are presented in green dashed line and neither thresholds nor slopes are significantly different compared with the 500 ms data.

*β*decreases with increasing signal area, ranging from 2.24 (

*SEM*± 0.22) for 33% signal area to 1.53 (

*SEM*± 0.16) for 100% signal area. The average log–log slope of

*β*as a function of signal area is −0.35 (

*SEM*± 0.061).

*β*as a function of signal area. If the positions/orientations of the sectors containing the signals are randomized on each trial, as in all our experiments, PS under SDT predicts that

*β*will decrease with signal area (Meese & Summers, 2012; Tyler & Chen, 2000). This was observed in all experiments. This is a novel result that has significant consequences for how we interpret increases in performance with signal area in textured stimuli.

*β*values, and, in particular, how they vary with signal area.

*β*(Graham, 1989; Quick, 1974; Robson & Graham, 1981; Watson, 1979). A complete derivation from first principles of this model prediction can be found in Kingdom and Prins (2016). Most importantly, HTT predicts that

*β*is constant across signal area, as also predicted by additive (including linear) summation models.

*β*on stimulus uncertainty when the observer attends to all parts of the stimulus in which the signal might occur (Meese & Summers, 2012; Tyler & Chen, 2000). Our results consistently show that

*β*decreases with signal area, i.e., as uncertainty decreases, as predicted by SDT. Although at a qualitative level our results are therefore consistent with PS under SDT, and not consistent with either PS under HTT, or additive (including linear) summation, we wanted to determine how quantitatively our data could be modeled by PS under SDT.

*M-*AFC (

*M*-IFC) tasks (Green & Swets, 1988; Kingdom & Prins, 2016), and generalizes the PS model of Shimozaki et al. (2003) to include the full gamut of relevant parameters. As is typical in SDT models, the noise-alone (“noise”) and stimulus-plus-noise (“stimulus”) intervals/locations are modeled as normal distributions with equal variance, with signal intensity given in

*Z*, i.e., standard deviation units with respect to the center of the noise distribution. The separation of the two distributions is symbolized by

*d'*, which is thus a measure of the internal strength of the stimulus (Figure 3). The aim of the observer in a 2-IFC task is to identify on each trial the interval containing the target stimulus. The assumed strategy is the optimal decision rule: select the interval with the biggest signal. This is termed the MAX-rule.

*t*is the intensity of a

*sample*signal these are

*ϕ*(

*t*) and

*ϕ*(

*t-d*′), which are the heights, or relative likelihoods of

*t*in the noise and stimulus distributions, and Φ(

*t*) and Φ(

*t-d′*), which are the areas of the noise and stimulus distributions below

*t*. Kingdom et al. (2015) showed that when multiple stimuli are combined by PS, proportion correct (

*Pc*) detection is given by the following: with where

*M*is the number of task intervals/alternatives (here 2),

*Q*the number of monitored channels,

*n*the number of those channels containing stimulus (signal),

*τ*the exponent of the transducer,

*s*the intensity, or amplitude of the stimulus, and

*g*a stimulus intensity scaling factor. Equation 2, which uses numerical integration, has been verified by Monte Carlo simulation, and is implemented by the routine PAL_SDT_PS_SLtoPC in the Palamedes Toolbox (Prins & Kingdom, 2009). The routine calculates proportion correct for any

*s*,

*g*,

*τ*,

*M*,

*Q,*and

*n*. Equation 2 can be thought of as an equation for a psychometric function, since it describes

*Pc*as a function of stimulus strength

*s*, given the parameters

*g*,

*τ*,

*Q,*and

*n*. It may be used therefore to

*simulate*psychometric functions of

*Pc*against

*s*, which can then be fitted with a conventional psychometric function such as a Weibull or Quick function (Kingdom et al., 2015). That is how we have used it here. We employ Equation 2 to simulate psychometric functions, which are then fitted with Quick functions. This enables us to determine how Quick function parameters such as threshold

*α*and exponent

*β*vary with the parameters

*n*,

*τ,*and

*Q*(see again Kingdom et al., 2015).

*α*and

*β*in the Quick function would be expected to vary as a function of both

*Q*and

*n,*using simulated psychometric functions generated from the model as described above. To do this we set the scaling factor

*g*to unity,

*M*to 2 and the transducer exponent

*τ*to 1.125, which we found to be the average value that best fitted the data, as described in the following material. Then, for each combination of

*Q*and

*n*, we used 30 equally-spaced values of

*s*and generated a psychometric function that spanned the range 50% to 99.9% correct (less than 100% to avoid

*d'*becoming infinity). Each function was then fitted by a Quick function and the threshold

*α*and exponent

*β*estimated.

*α*and

*β*are predicted to decline with

*n*for various values of

*Q*. Table 1 presents the slopes of the decline when a straight line is fitted to portions of the log—log model data. As Table 1 shows, the modelled decline in

*α*and

*β*varies to some extent with

*Q*as well as the range of

*n*selected, with least variation across

*Q*when only the upper 33% to 100% of

*n*is selected.

**Table 1**

*β*values because for

*β*there are only two free model parameters, the exponent on the transducer

*τ*and the number of monitored channels

*Q*(for

*α*there are three free parameters,

*g*,

*τ,*and

*Q*, where

*g*is the stimulus intensity scaling factor). In the context of our stimuli and model,

*Q*can be thought of as the number of independently-monitored regions of the stimulus. The model proposed here may be considered as a probability summation version of the model previously suggested for concentric Glass patterns (Wilson et al., 1997; Wilson & Wilkinson, 1998; Wilson & Wilkinson, 2015). This model consists of three processing stages: (1) First-stage local orientation filters, e.g., V1 simple cells, followed by a full-wave rectification, (2) pooling by larger second-stage filters in V2, and (3) global pooling across the whole stimulus by neurons in V4 (Wilson et al., 1997; Wilson & Wilkinson, 1998). Wilson and colleagues' second-stage filters, which act like end-stopped cells, are designed to extract local curvature (V2; see also Dobbins, Zucker, & Cynader, 1987; Wilson, 1999). Their responses are then linearly pooled by the third stage (V4). However, Wilson et al.'s (1997) model is conceivable without this intermediate stage, whereas in our model the stage is necessary to encode the local geometric arrangement of the elements (e.g., circular, radial, etc.) prior to the final probability summation stage. It is important to emphasize that the size of the putative second-stage filters, or the number and size of their first-stage inputs, is not specified in Wilson's (1997) model and, to our knowledge, there is no consensus from either psychophysics or physiology as to what their values might be. Hence, we can only speculate about the size of the second-stage filters and their degree of spatial overlap, and hence only speculate as to the precise value of

*Q,*the number of monitored channels/filters. As in Wilson et al.'s model, we assume that the second-stage filters and their first-stage inputs are oriented to match the local parts of the stimulus. In other words, the visual system, even though it probability summates the outputs of those second-stage filters via the MAX-rule, monitors the totality of filters that are matched to the particular stimulus arrangement (circular, radial, etc.). We initially explored various combinations of

*τ*and

*Q*to determine which combinations minimized the difference between model and data, but for nearly all conditions we were unable to find an optimum value of

*Q*. The reason for this is that the observed

*β*-versus-

*n*slopes were in nearly all cases slightly steeper than the model

*β*-versus-

*n*slopes, whichever value of

*Q*we tried (up to 5,000). It should be born in mind, however, that

*Q*(unlike

*τ*) has only a small effect on the absolute model values of

*β*and a negligible effect on the

*β*-versus-

*n*slopes, as can be seen in Table 1. Despite the nondependency of the modeled results on

*Q*, we have chosen to use different values of

*Q*for each experiment. The values are chosen according to the number of Gabor elements or dot-pairs used in each stimulus. Specifically, Experiment 1 Low-density: Number of elements (NoE) = 150,

*Q*= 75; Experiments 2 and 3 High-density and Windmill: NoE = 2,000,

*Q*= 1,000; Experiment 4 Glass patterns: NoE = 3,000,

*Q*= 1,500. Having set the value of

*Q*, we then found the optimal value of

*τ*for the

*β*data, and used this value to predict the

*α*-versus-

*n*slope

*s*(for which the stimulus scaling factor

*g*is arbitrary). Note, however, that the fundamental difference between Wilson et al.'s (1997, 1998) Glass pattern model and our model is the final pooling stage of the second-level filters, which are pooled linearly in Wilson et al.'s model, but in our model summed by probability summation.

*α*values have been scaled vertically to fit the data—however, this does not affect the model's critical

*α*-versus-

*n*slopes. No scaling is applied to the

*β*values. Figure 5 shows the model fits to the data as green lines, and Table 2 provides a comparison between model and data

*α*-versus-

*n*as well as

*β*-versus-

*n*slope values. Except for Experiment 2, the model slightly underestimates the decline of

*β*with signal area but otherwise captures the decline in both

*α*and

*β*with increasing signal area very well. The goodness of fit between the model and the data was evaluated by calculating the coefficient of determination

*R*

^{2}for each experiment, which is given for each graph in Figure 5. In order to validate the model predictions statistically, one-tailed significance probability values of the correlation coefficient

*R*between data and model predictions for both

*α*and

*β*were calculated and

*p*-values are stated in each graph of Figure 6. Note, that for four out of the eight conditions the

*p*< 0.05 significance level is not reached owing to the fact that, in spite of the high correlation, the number of data points is only three (Figure 5E and F and Figure 5G and H, respectively). The small sample size is due to the reuse of the windmill stimulus design used in previous studies (Wilson et al., 1997; Wilson & Wilkinson, 1998), in order to enable consistent comparisons between studies.

**Table 2**

*Q*that we chose: Whatever value we had chosen, the decline in

*β*with signal area would have been predicted, though slightly less well had we chosen smaller values of

*Q*.

*β*, we have also analyzed model predictions with respect to the derivative of the psychometric function (see Strasburger, 2001, for an analysis of the relationship between

*β*and other measures of psychometric function slope). The derivative of the Quick function is:

*β*and the derivatives of the psychometric functions. Note that for this observer

*β*drastically decreases with increasing signal area (

*β*ranges from 4.49–1.83). However across observers, the average decline in

*β*(see Figure 5 and graphs in left column of Figure 2) is more subtle (Experiment 1

*β*; range: 2.97–1.25; Experiment 2: 2.29–1.352; Experiment 3: 2.016–1.55; Experiment 4: 2.24–1.53). The extreme subject's decline in

*β*leads to the decrease in slopes as measured by the derivative of the psychometric functions, despite the fact that our model predicts an increase in the derivatives. Figure 6 shows plots of the derivatives of the extreme subject case (6B), a more typical subject case (6D), and the average of the derivatives across subjects (

*N*= 5; 6F) for this condition, with each plot also showing the AS and PS predictions of our model.

*β*.

*β*declines with signal area that does not preclude the possibility that the maximum, i.e., whole-area signal condition is detected by a global linear integrator. He suggests that the visual system might employ linear filters matched in shape to the various signal-shape conditions. Thus for the single pie-wedge and windmill conditions these would be pie-wedge and windmill-shaped filters matched to the signal area, culminating in full-circle global linear integrators for the 100% signal area conditions.

*β*with increasing signal area reflects the reduction in uncertainty as the matched filter grows to fill-up the stimulus area. Remember that with the pie-wedge shaped and windmill-shaped signals, their orientations were randomized on each stimulus presentation. This means that if we consider the matched filter as capturing all the signals in each signal-area condition, the number of nonsignal locations that the filter would likely sample, would decline with increasing signal area. From the point of view of the MAX-rule it means that the visual system would examine independently each possible signal location with the matched filter and select the interval containing the maximum filter response. In terms of the SDT PS formulation described above, the relationships between

*Q*(number of monitored channels or stimulus locations),

*n*(number of signals), and signal area

*A*are therefore different for the mixed compared to the pure PS model advanced above. In the pure PS model,

*Q*is fixed and as signal area

*A*is increases and so is

*n*. In the matched filter model it works the other way round: as signal area

*A*increases

*n*remains fixed and

*Q*declines.

**Figure 7**

**Figure 7**

*n*of 1, that is, we consider the area containing the signal to stimulate a single matched filter, i.e., a single stimulus/channel. We define

*A*, the area of the signal (and matched filter) as the percentage of total stimulus area. We then assume that each matched filter integrates

*N*local samples from the stimulus, with

*N*being equal to the area of the matched filter, i.e., ranging from 10 to 100. By “local samples” we mean local input signals with their own independent noise sources (see Figure 7). Finally, we assume that

*Q*, which embodies the degree of uncertainty in the model, is equal to 100/

*A*, thus ranging from 10 to 1 as signal area increases from 10% to 100%. In other words, as the signal area and with it matched filter area increases, the number of nonsignal locations sampled on each trial by the matched filter,

*Q*-1, declines proportionately.

*α*-versus-

*A*and

*β*-versus-

*A*slope predictions we use the PS formula in Equation 2, with

*n*set to 1 and

*Q*ranging from 10 to 1. Note that in the limiting case when

*n*=

*Q*= 1 Equation 2 reduces to the standard formula for calculating proportion correct for a single signal in an M-AFC task according to the MAX-rule under SDT (Kingdom & Prins, 2016). The linear summation component of the mixed model is achieved by computing for each condition

*d'*= (

*gs*)

*√*

^{τ}*N*with

*g*(stimulus gain) and

*τ*(transducer exponent) set to unity. A

*τ*of 1 is the model value needed to produce the average

*β*of 1.3 observed in the 100% signal area condition. The term √

*N*captures the fact that for a linear filter as

*N*increases the amount of internal noise increases in proportion to the square root of the number of noise samples (because

*N*/√

*N*= √

*N*). Psychometric functions of proportion correct as a function of stimulus strength

*s*were then calculated for the 10 values of

*Q*and fitted by Quick functions as described previously. The resulting

*α*-versus-

*A*and

*β*-versus-

*A*plots were then fitted as before by a straight line applied to the log–log data. The resulting slopes of the mixed model are −0.79 for

*α*-versus-

*A*and −0.184 for

*β*-versus-

*A*. These values are what one would expect from a mixed linear and PS model. The pure PS model predictions were −0.42 for the average

*α*-versus-

*A*slopes and −0.25 for average

*β*-versus-

*A*slopes. A pure linear summation model (under the fixed attention window scenario) would predict an

*α*-versus-

*A*slope of −1.0 and a

*β*-versus-

*A*slope of 0. The mixed model slopes therefore fall in between their corresponding pure probability and linear summation model values.

*α*-versus-

*A*slope of −0.79 is too steep and the

*β*-versus-

*A*slope of −0.184 too shallow compared to the data, whose average values are −0.44 and −0.30, respectively. If we make

*N*proportional to

*A*and set

^{j}*j*to 0.25 instead of 1.0, we can achieve an

*α*-versus-

*A*slope of −0.44. This would mean that the density of local samples integrated by the matched filter declines with the fourth root of the size of the filter, which to us seems implausible. Regarding the

*β*-versus-

*A*slope, its steepness is determined by the rate at which

*Q*declines as a function of signal area, so it is clear that the mixed model rate of decline is insufficient to predict the data. However, it is not obvious to us what plausible mechanism would increase the rate of decline of

*Q*with signal area. If rather than discretely positioning the matched filter on the available nonsignal location in order to tile the stimulus, the visual system performed a continuous convolution of the matched filter with the stimulus, i.e., “around the clock,” this would presumably not increase the rate of

*Q*decline, or if anything reduce it. Thus while the matched filter model is an important theoretical possibility, it does not appear to account for the data as well as the pure PS model advanced above, at least for the type of stimuli considered here.

*β*of the psychometric functions should decrease as signal area is increased. Our results showed this characteristic, both in Glass patterns and Gabor textures. In addition, we found that the strength of summation across signal area was approximately half as strong as some previous reports. Moreover, we found no evidence for specialized detectors for circular textures; detection sensitivity was independent of texture type. Taken together, our findings show that probability

*not*linear summation is the most likely basis for the detection of circular orientation-defined textures, and provide no support for the idea of specialized detectors for circular textures in vision.

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