We investigate how well the direction of the expansion gradient and the direction of a spatial step in expansion can be extracted from a complex optical flow field. The expansion gradient is an invariant of the second-order structure under rotations of the image plane. Our stimuli consist of sparse random dot patterns in a circular window with a diameter of 20 . We find that translational velocity masks the direction of both the expansion gradient and the step in expansion. Thresholds for both the expansion gradient and the step in expansion are found to be similar for a wide range of expansions. We introduce two simple models that use the thresholds for the step in expansion to predict thresholds for the expansion gradient. Results of these models are comparable with the results of our subjects, showing that we do not need a hypothetical separate mechanism for the detection of second-order structure to explain the data. # 1997 Optical Society of America [S0740-3232(97)01504-4]
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