**Measurements of area summation for luminance-modulated stimuli are typically confounded by variations in sensitivity across the retina. Recently we conducted a detailed analysis of sensitivity across the visual field (Baldwin, Meese, & Baker, 2012) and found it to be well described by a bilinear “witch's hat” function: Sensitivity declines rapidly over the first eight cycles or so, but more gently thereafter. Here we multiplied luminance-modulated stimuli (4 cycles/degree gratings and “Swiss cheeses”) by the inverse of the witch's hat function to compensate for the inhomogeneity. This revealed summation functions that were straight lines (on double log axes) with a slope of −1/4 extending to ≥33 cycles, demonstrating fourth-root summation of contrast over a wider area than has previously been reported for the central retina. Fourth-root summation is typically attributed to probability summation, but recent studies have rejected that interpretation in favor of a noisy energy model that performs local square-law transduction of the signal, adds noise at each location of the target, and then sums over signal area. Modeling shows our results to be consistent with a wide field application of such a contrast integrator. We reject a probability summation model, a quadratic model, and a matched template model of our results under the assumptions of signal detection theory. We also reject the high threshold theory of contrast detection under the assumption of probability summation over area.**

^{2}. It was viewed from a distance of 1.19 m, having a resolution of 48 pixels per degree of visual angle (12 pixels/cycle for the 4 cycles/degree stimuli used here).

*ϕ*= 90°) and anticosine (

*ϕ*= 270°) phases (Figure 1c and e). The centers of the stimuli were at the patterns' maximum and minimum contrasts for cosine and anticosine phase modulations, respectively. Eight sizes were used for the gratings (1.3 to 33.0 cycles in diameter), the larger five of which were also used for the Swiss cheeses. We express stimulus contrast in dB (re 1%), given by 20 × log

_{10}(

*c*), where

*c*is Michelson contrast in percent. For convenience, in the graphical presentation of our results we also express stimulus area as 20 times the log

_{10}of the nominal stimulus diameter squared, relative to the smallest stimulus area. Note also that our error measures were derived by calculating the root-mean-square (RMS) errors between empirical thresholds and model predictions, where each is expressed in dB.

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

*β*). Consistent with other studies (Mayer & Tyler, 1986; Meese & Summers, 2012; Wallis et al., 2013), psychometric slope did not vary systematically over different stimulus sizes (not shown). Combining data across the three observers gave the slopes for each condition shown in Figure 4, with an overall median

*β*of 2.8 (the individual observer values were 1.8, 3.2, and 3.0 for ASB, DHB, and TSM, respectively). Previous work by Robson and Graham (1981), Mayer and Tyler (1986), and Meese and Summers (2012) found average slopes of 3.5, 3.5, and 3.6, which are a little steeper than the slopes from the study here. Our values agree with those measured by Wallis et al. (2013), who also report slopes of 2.8. Previous work has shown that the psychometric function is essentially stationary for practiced observers and that slightly better estimates of the slope are achieved when the results are collapsed across multiple sessions before curve fitting, as we did here (Wallis et al., 2013).

**Figure 4**

**Figure 4**

*β*) equals 4 (Robson & Graham, 1981), though more generally, probability summation under HTT gives a summation slope of 1/

*β*. Note that HTT underlies the common conception of probability summation, where it is understood that one calculates overall sensitivity by combining the probabilities of detecting the individual components using the standard statistical procedure for combining probabilities. This model predicts an asymptotic summation slope of −1/

*β*for compensated stimuli.

*ϕ*= 90° and 270°) are shown with dashed and solid curves, respectively. The single parameter fit of the noisy energy model (purple curve in Figure 2a, b) is very good. Note how well it captures the initial steepness of the data (owing to within-filter summation) and then levels off to horizontal when there was no compensation (Figure 2a) and to a slope of −1/4 when the witch's hat stimulus compensation was in place (Figure 2b). As mentioned in the Introduction, this fourth-root behavior in the model is due to the cascading quadratic effects of square-law transduction and integration of internal noise with the signal (i.e., the internal noise at the decision variable increases with stimulus diameter). This is not seen in the uncompensated case (Figure 2a) because of the loss of sensitivity to the signal with eccentricity. The predictions (no free parameters) for the Swiss cheese stimuli (purple curves in Figure 3c, d) are also very good. Of the three models plotted in Figures 2 and 3, the RMS errors from the noisy energy model are by far the best.

*β*). The noisy energy model and the quadratic model both involve square-law (

*p*= 2) transduction of signal contrast (

*c*). In the absence of uncertainty (Pelli, 1985), this predicts

^{p}*β*= 1.3 × 2 = 2.6 (Pelli, 1987; Tyler & Chen, 2000; May & Solomon, 2013), very close to the average of

*β*= 2.8 here, and in agreement with the slopes from some of our individual conditions (see Figure 4). It seems likely that the small deviation from the

*β*= 2.6 prediction arises from uncertainty (see Meese & Summers, 2009), which appears to be greatest in the uncompensated case, particularly when

*ϕ*= 270° (i.e., when there was no signal contrast in the center of the visual field). Thus, it is clear that our preferred model from above (the noisy energy model) is consistent with the slopes of empirical psychometric functions.

*β*= 1.3 (Pelli, 1987), at odds with our estimates in this study (Figure 4). In principle, this shortcoming might be overcome by supposing a fairly high level of intrinsic uncertainty across all stimulus conditions. However, other experiments in which we have assessed intrinsic uncertainty by manipulating extrinsic uncertainty suggest that high levels of uncertainty are unlikely for grating stimuli (Meese & Summers, 2012). As mentioned earlier, one interpretation of the fourth-root summation model involves a contrast transducer of

*p*= 4; however, this predicts

*β*= 1.3 × 4 = 5.2, which is much higher than what we found empirically (see Figure 4).

*β*= 4 is implied by the fourth-root summation curve for detection thresholds. However, our psychometric slopes were not consistent with this prediction (Figure 4). Alternatively one might take the slope of the psychometric function to predict the slope of the summation function, but that predicts a summation function that is too steep (a slope of −1/2.8, on average) compared to the human results (Figure 2b; a slope of −1/2.8 lies between the upper two pale dashed lines which have slopes of −1/4 and −1/2). This mismatch between two empirical measures (slopes of psychometric functions and summation functions) serves as further evidence to reject HTT under the widely held assumption that area summation is achieved by probability summation under that model.

- The correct conclusions about the extent of contrast integration are drawn in our current work, with previous work being compromised by the loss of sensitivity with retinal eccentricity. For example, Baker and Meese (2011) built witch's hat compensation into their modeling, but not their stimuli (in which they manipulated carrier and modulator spatial frequencies, not diameter). A loss of experimental effect in the results (such as that in Figures 2a and 3a here) limits what the analysis can be expected to reveal. Indeed, Baker and Meese (2011) found it difficult to put a precise figure on the range of contrast integration, and aspects of their analysis hinted at a range of >20 cycles for two of their three observers. Baker and Meese (2014) made no allowance for eccentricity effects in their reverse correlation study. The contrast jitter applied to their target elements ensured they were above threshold, and so the effects of contrast constancy should come into play (Georgeson, 1991); however, we cannot rule out the possibilities that either (a) the contrast constancy process was incomplete or (b) internal noise effects not evident at detection threshold (e.g., signal dependent noise) compromised the conclusions.
- The correct conclusions about the extent of contrast integration come from our previous work. Our current work points to lawful fourth-root summation, but not necessarily signal integration across the full range. On this account, signal integration takes place up to a diameter of about 12 cycles and a different fourth-root summation processes take place beyond that point. For example, from our results here we cannot rule out the following possibility: Beyond an eccentricity of ∼1.5° the transducer becomes linear and overall sensitivity improves by probability summation (Tyler & Chen, 2000), but uncertainty (Pelli, 1985; Meese & Summers, 2012) for more peripheral targets causes the slope of the psychometric function to remain steeper than
*β*= 1.3 (May & Solomon, 2013).

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*slope*, then we can set the internal noise

*σ*to a fixed arbitrary value (

*σ*= 1).

*n*) is directly proportional to the square of the nominal stimulus diameter (the full width at half window height).

*d*′) for the general form of the noisy energy model is where

*c*is the stimulus contrast,

*s*is the amplitude of the

_{i}*i*

^{th}of

*p*pixels in the filtered “witch's hat attenuated” image (note that

*p*is constant for all images; it is the number of pixels in the display and does not depend on stimulus diameter),

*t*is the amplitude of the template at that pixel (in the noisy energy model presented here the template is matched to the outer boundary of the stimulus and does not match the Swiss cheese modulations), and

_{i}*σ*is its internal noise standard deviation (Figure A1). Although each pixel has two coordinates (

_{i}*x*and

*y*), we collapse these to a single dimension (

*i*) for ease of presentation. Solving Equation A1 for

*d*′ = 1 gives

**Figure 1**

**Figure 1**

*t = s*), meaning that the template for the nonsignal pixels is zero. In this case we consider the summation that occurs over the

*n*signal pixels determined by the area derived from nominal stimulus diameter, where the output of the filtering (sum of the sine and cosine phase filters) is uniform (thereby ignoring the minor effects of the blurred boundary to the stimuli).

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

**Figure 4**

**Figure 4**

*β*; Quick, 1974): and for a constant value of

*s*we have: for compensated gratings.

*β*= 4. Lower exponents are justified under particular conditions of uncertainty (

*β*< 4; see Tyler & Chen, 2000 for details), and higher exponents when the model includes an accelerating contrast transducer (equal to 4 times the transducer exponent; Meese & Summers, 2012).

**Figure 5**

**Figure 5**