We examined the perception of structure in a family of visual textures whose second-order correlation structure is flat. These textures were generated by two-dimensional recursion rules, in a manner which extends the construction of Julesz, Gilbert and Victor (1978; Biological Cybernetics, 31, 137-140). Textures generated by some recursion rules elicited a visually salient percept of structure, while textures generated by other recursion rules did not. Textures whose statistical structure was visually salient produced evoked responses which differed from the response evoked by completely random textures. The size of this VEP difference correlated well with psychophysical measures. Since the textures were constructed to have identical global spatial frequency spectra, models for the extraction of visual structure must be essentially nonlinear. Models based on symmetry, information content, or simple spatial extent (but not pattern) of correlation fail to explain the observed results. Models based on the cooperative interaction of pairs of nonlinear subunits provide a reasonable qualitative account of the findings. The critical model features are (i) the presence of multiple nonlinear subunits, and (ii) a second nonlinearity, such as a threshold, at the stage of combination of subunit signals.