Perceived location of bars and edges in one-dimensional images: computational models and human vision

Observers used a cursor to mark the location and polarity of all the bar and edge features seen in compound (f + 3f) gratings of moderate frequency and contrast. They almost always reported six bars and six edges per cycle of the fundamental frequency (f = 0.4 c/deg, contrast 32%), for all phases of the third harmonic (3f = 1.2 c/deg, contrast 10.7%). This general pattern of features was predicted by the positions of peaks and troughs in the outputs of even and odd filters applied to the stimulus waveform, but not by peaks of "local energy" since there were only two energy peaks per cycle. We considered a family of filters whose amplitude spectrum has slope p on a log-log plot. The best-fitting filter slope was determined for bars (even filter) and edges (odd filter) in conjunction with a classification rule in which all peaks and troughs in the response profile are counted as features. If bars were seen at luminance peaks, and edges seen at gradient peaks (zero-crossings in the second derivative) we should have found p = 0 for bars and p = 1 for edges. In fact, for both bars and edges the best-fitting slope was about p = 0.5. For edges, this is consistent with the use of a smoothed (Gaussian) derivative operator. The filters form a quadrature pair, as in the energy model, but features are not constrained to lie at energy peaks. A compressive transducer preceding the filters improved the goodness-of-fit for predicted edge locations, but did not affect the estimate of filter slopes, nor the goodness-of-fit for bar locations. In an experiment with single blurred edges we confirmed that the perceived location of edges is shifted towards the darker side of the edge in direct proportion to the contrast of the edge. This was well predicted by adding a compressive transducer to the filter model.