Fourth-root summation of contrast over area: No end in sight when spatially inhomogeneous sensitivity is compensated by a witch's hat

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.