Spatial summation in simple (Fourier) and complex (non-Fourier) texture channels

Complex (non-Fourier, second-order) channels have been proposed to explain aspects of texture-based region segregation and related perceptual tasks. Complex channels contain two stages of linear filtering with an intermediate pointwise nonlinearity. The intermediate nonlinearity is crucial. Without it, a complex channel is equivalent to a single linear filter (a simple channel). Here we asked whether the intermediate nonlinearity is piecewise-linear (an ordinary rectifier), or compressive, or expansive. We measured the perceptual segregation between element-arrangement textures where the contrast and area of the individual elements were systematically varied. For solid-square elements, the tradeoff between contrast and area was approximately linear, consistent with simple linear channels. For Gabor-patch elements, however, the tradeoff was highly nonlinear, consistent with complex channels in which the intermediate nonlinearity is expansive (with an exponent somewhat higher than 2). Also, substantial individual differences in certain details were explainable by differential intrusion from "off-frequency" complex channels. Lastly, the results reported here (in conjunction with those of other studies) suggest that the strongly compressive intensive nonlinearity previously known to act in texture segregation cannot be attributed to a compressive nonlinearity acting locally and relatively early (before the spatial-frequency and orientation-selective channels) but could result from inhibition among the channels (as in a normalization network).