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In previous studies, quadratic modelling of natural images has resulted in cell models that react strongly to edges and bars. Here we apply quadratic Independent Component Analysis to natural image patches, and show that up to a small approximation error, the estimated components are computing conjunctions of two linear features. These conjunctive features appear to represent not only edges and bars, but also inherently two-dimensional stimuli, such as corners. In addition, we show that for many of the components, the underlying linear features have essentially V1 simple cell receptive field characteristics. Our results indicate that the development of the V2 cells preferring angles and corners may be partly explainable by the principle of unsupervised sparse coding of natural images.