Perception of second- and third-order orientation signals and their interactions

Abstract

Orientation signals, which are crucial to many aspects of visual function, are more complex and varied in the natural world than in the stimuli typically used for laboratory investigation. Gratings and lines have a single orientation, but in natural stimuli, local features have multiple orientations, and multiple orientations can occur even at the same location. Moreover, orientation cues can arise not only from pairwise spatial correlations, but from higher-order ones as well. To investigate these orientation cues and how they interact, we examined segmentation performance for visual textures in which the strengths of different kinds of orientation cues were varied independently, while controlling potential confounds such as differences in luminance statistics. Second-order cues (the kind present in gratings) at different orientations are largely processed independently: There is no cancellation of positive and negative signals at orientations that differ by 45°. Third-order orientation cues are readily detected and interact only minimally with second-order cues. However, they combine across orientations in a different way: Positive and negative signals largely cancel if the orientations differ by 90°. Two additional elements are superimposed on this picture. First, corners play a special role. When second-order orientation cues combine to produce corners, they provide a stronger signal for texture segregation than can be accounted for by their individual effects. Second, while the object versus background distinction does not influence processing of second-order orientation cues, this distinction influences the processing of third-order orientation cues.

Figure 1

(A) The experimental paradigm. Trials were subject-initiated and began with a fixation spot (300 ms), followed by a stimulus (120 ms), followed by a random mask (500 ms). (B) Example stimuli. Stimuli consisted of a 64 × 64 array of checks, in which a 16 × 64 target region is positioned eight checks from one of the four edges of the array. Stimuli were of two types: left, the background is random, and the target is structured (here, θ_⌋ = 1), or right, the background is structured and the target is random. The red rectangles indicate the location of the target and were not visible during the trials.

Figure 2

Psychometric functions for individual image statistics that carry orientation information. (A) Second-order correlations in cardinal (β_{_}, β_|) and oblique directions (β_\, β_/) directions. For each statistic, psychometric functions are shown for negative excursions (left element of each pair) and positive excursions (right element of each pair). Chance performance is 0.25. The patches above the psychometric functions show typical 32 × 32 samples of images defined by the corresponding second-order statistic, constructed with β = ±0.4. For second-order statistics, positive values correspond to correlation in one direction; negative values correspond to anticorrelation in that direction. (B) Third-order correlations in each of three directions (θ_⌋, θ_⌊, θ_⌈). Psychometric functions as in (A). The image samples are constructed with θ = ±0.72. For third-order statistics, positive values correspond to an excess of white triangular patches in a particular orientation; negative values correspond to an excess of dark triangular patches in the same orientation. Data from two representative subjects, MC and DT, out of N = 6.

Figure 3

Isodiscrimination contours for pairs of second-order statistics. The image at the left of each row shows the gamut of images that can be constructed with each pair of statistics. The random texture (the origin) is at the center of each gamut; image statistics increase with as distance from the origin grows and the ends of the axes correspond to values of ±1. The polar plots are isodiscrimination contours, i.e., the location in the gamut at which criterion performance (fraction correct = 0.625) was obtained, in each of eight directions (along each axis and along diagonals). Each colored trace corresponds to a separate subject; the black trace is the harmonic mean across subjects. The first column of polar plots was determined from all trials; the second column was determined from the half of the trials in which the background was structured according to the image statistics and the target was a random patch; the third column was obtained from the other half of the trials, in which the target was structured according to the image statistics and the background was a random patch. The isodiscrimination contours are generally circular or elliptical, but there is a consistent flattening for the (β_\, β_/)-pair, when both statistics are negative (blue arrow, lower row). This corresponds to the region of the gamut in which corners are present (rounded square, gamut). For all pairs of second-order statistics, there is little dependence on whether the background or the target was structured. N = 6.

Figure 4

Isodiscrimination contours for pairs of third-order statistics. As in Figure 3, the image at the left of each row shows the gamut of images for each pair of image statistics, and the polar plots are isodiscrimination contours, i.e., the location in the gamut at which criterion performance was reached. For other plotting conventions, see Figure 3. Note that for the pair (θ_⌈, θ_⌋), there is a strong dependence on whether the structured component of the stimulus was the background (near-circular contours, second column) versus the target (elongated contours, third column). N = 6.

Figure 5

Isodiscrimination contours for mixtures of second- and third-order statistics. As in Figure 3, the image at the left of each row shows the gamut of images for each pair of image statistics, and the polar plots are isodiscrimination contours, i.e., the location in the gamut at which criterion performance was reached. For other plotting conventions, see Figure 3. Isodiscrimination contours are approximately elliptical, with the short axis along the β-axis, and the long axis along the θ-axis, corresponding to the lower thresholds for β than for θ (see Figure 2). There is little dependence on whether the background or the target was structured. N = 4.

Figure 6

Pooling indices. The pooling index (Equation 8) describes how a pair of image statistics interact: It is the ratio of the thresholds when they have opposite signs, to when they have the same sign; values larger than one indicate cancellation in the opposite-sign condition. For individual subjects, pooling index is calculated from all trials. For the group means, the pooling index is calculated from all trials (solid bar), and for the subsets of trials separated according to whether the background was structured or the target was structured. N = 6 for pairings within order (second- or third-order), N = 4 for pairings between orders.

Figure 7

Thresholds for four-component mixtures of second-order statistics (A) and third-order statistics (B), and comparison with model predictions. Measured thresholds (solid symbols) are shown for each kind of mixture; these are compared with predictions based on independent processing of each statistic (upward triangle), complete pooling (downward triangle) and the ellipse model (bars). Predicted thresholds exceeding 1 (which is outside of the gamut) are plotted at 1. Since subject DC did not participate in experiments in which thresholds for pairs of image statistics were measured, the interaction parameters were determined either from subject MC (left open bar) or DT (right open bar); see text for further details. Example images, shown above each set of measured thresholds, have 32 × 32 checks, and are constructed with all image statistics set to ±0.225.