Detection thresholds for spiral Glass patterns

Abstract

We measured thresholds for the detection of spiral Glass patterns in the presence of random noise. The patterns were constructed so that the orientation content did not vary as a function of spiral angle or signal level. We found that spiral patterns had higher thresholds than either radial or concentric Glass patterns. The results support the idea that the human visual system is specialized to detect radial and concentric patterns.

1. Introduction

The primary visual cortex extracts information about the local orientation and size of simple contours and edges from the retinal image. However, this is just one of the beginning stages in form vision. How the neural operations combine the local contour orientations to form the complete global structure implicit in the response of V1 neurons remains unsolved.

During psychophysical experiments using concentric, radial, parallel, and hyperbolic Glass patterns (Glass, 1969; Glass & Perez, 1973) Wilson and Wilkinson (1998) discovered that detection thresholds, measured by adding random noise, varied consistently across patterns. Concentric patterns had the lowest thresholds, parallel patterns had the highest and hyperbolic and radial patterns had intermediate threshold values. Psychophysical data (Wilson, Wilkinson, & Asaad, 1997) also demonstrate that orientation information in the concentric and radial random-dot Glass pattern is summed linearly to extract a global form percept.

We have further explored the issue of how global structure is extracted by measuring detection thresholds for spiral Glass patterns. Spiral Glass patterns are designed by randomly positioning dot pairs within a circular region, but orientating the pairs relative to lines extending from the center of the circle to the circumference (Prazdny, 1984). We refer to the orientation of the dot pairs relative to the radii as the spiral angle. Radial and concentric Glass patterns are special cases that have spiral angles of 0° and 90°, respectively. Spiral angles between 0° and 90° give rise to true spirals. One advantage of spirals is that one can construct sets of patterns that differ in spiral angle but not in the overall distribution of orientations. All patterns within a set of spirals have the same isotropic distribution of orientations so that differences in detection thresholds cannot be explained on the basis of orientation content.

2. Methods

Glass patterns consisted of a fixed number of dot pairs that were divided into signal pairs and noise pairs. Signal pairs were positioned randomly within a circular region, but placed such that the orientation of each pair was tangent to the imaginary contour lines that defined each pattern (Prazdny, 1984). Noise pairs were positioned and oriented randomly. Each individual dot was 2×2 pixels (0.06 deg²). The separation between the dots in a pair was 0.2° and was constant throughout the pattern. Approximately 500 of these dot pairs were spread over a circular region with a radius of 4.0°, yielding a density of 20 dots per deg² (the dots occupied 7% of the pixels within the aperture). Both the separation of the dots within a pair and the overall dot density were within the range used by Wilson and Wilkinson (1998). The spiral angle was measured as the orientation of the signal pairs relative to the radii of the circular aperture. Seven spiral angles were used in each experiment, ranging from 0° to ±90°, which progressed in 15° intervals. Positive and negative spiral angles were tested in separate experiments. Glass patterns with spiral angles of 0°, 45° and 90° are shown in Fig. 1A–C.

Fig. 1.

Glass patterns and Fourier spectra. (A–C) Patterns with spiral angles of 0°, 45° and 90°. All patterns have 100% signal. (D–F) Power spectra for patterns in A–C. The small black square in the center is the DC component, which was removed to enhance the contrast of the other frequencies. (G–I) Distribution of dot pair orientations for patterns in A–C. Orientation histograms are summed over ten randomly generated patterns of each type. Bin width=5°.

Six signal dot percentages were used, spanning the range 0–50% in 10% increments (for parallel patterns, 11 signal levels were used 0–100% in 10% increments). The signal dot percentages determined the signal-to-noise ratio, and the strength at which a Glass pattern could be detected. All patterns used in this study had nearly indistinguishable Fourier power spectra (Fig. 1D–F) except for possible differences in spatial frequency content. The orientation content did not vary as a function of signal level or spiral angle. Orientation histograms for all patterns were uniformly flat (Fig. 1G–I). Patterns with signal = 0 (i.e. noise) were identical to the limiting case shown in Fig. 3 of Glass and Switkes (1976).

Stimuli were produced using Matlab and the Psychophysics Toolbox software (Brainard, 1997) and were presented on a Sony 20-in. CRT display. Screen resolution was 1280 × 1024 pixels, which subtended 38.7 × 29.2° at a viewing distance of 57 cm. The frame rate was 75 Hz. Viewing was binocular with the subject’s head comfortably positioned in a chin rest. The patterns consisted of white squares on a dark background. The mean luminance was 5.0 cd/m². Outside the aperture, the screen was a uniform gray matched to the mean luminance of the pattern to avoid an afterimage. The rest of the testing room was dimly illuminated with overhead incandescent lights.

Thresholds for detecting global structure were measured using a two-interval forced-choice task. On each trial a signal level and spiral angle were selected at random. Subjects were told to fixate on a small white dot in the middle of the screen indicating where the patterns would appear. Subjects initiated each trial with a button press and were then presented successively with two patterns in random order. One pattern had signal in the range 0–50% and the other always had 0% signal. The subject’s task was to report the interval in which he/she detected any global pattern. The duration of each stimulus was 300 ms with a 500 ms ISI. Auditory tones provided feedback at the end of each trial. Two new patterns were computed between each trial based on unique random number seeds so that it was highly unlikely that any subject ever saw exactly the same configuration of dots more than once.

Subjects were given a brief training run of 150 trials in which one interval contained 50%, 75% or 100% signal patterns with spiral angles randomly chosen from the set used in the full experiment. The full experiment comprised 6 signal levels × 7 spiral angles × 25 repetitions for a total of 1050 trials that took approximately 1 h to complete. Subjects were allowed to pause between trials.

Nine subjects between the ages of 16 and 40 participated in this experiment. Of these, two were the authors (LS and VF), and seven were naïve to the purpose of the experiment. Subjects LS, VF, NM and JW performed at least six experiments each (at least three repetitions with both positive and negative spiral angles). The other five subjects performed one experiment each. All subjects had normal or corrected-to-normal vision. Upon completion of the experiment the percentage of correct responses was computed as a function of signal dot percentage. The resulting data were fit by a Quick (1974) function using a maximum likelihood procedure, and the threshold was taken to be the 81% correct point estimated from this fit.

3. Results

Detection thresholds for spiral Glass patterns were measured as a function of spiral angle. Patterns with intermediate spiral angles had threshold levels that were generally higher than thresholds for concentric and radial patterns. When thresholds were compared within experiments, spiral patterns had higher thresholds than concentric patterns in 132 out of 155 (85%) conditions (31 experiments × 5 intermediate spiral angles), and higher thresholds than radial patterns in 111/155 (72%) conditions. Mean threshold averaged over all subjects (Fig. 2A) showed a strong dependence on spiral angle with a maximum value of 36.7% when the spiral angle was 30°. The dependence of threshold on spiral angle was consistent across subjects (Fig. 2B and C), although there was substantial variability in the absolute threshold values. A one-way ANOVA on the combined data for all subjects (independent factor, spiral angle; dependent variable, threshold) showed the effect of spiral angle was statistically significant (P < 0.0001) despite inter-subject variability.

Fig. 2.

Detection thresholds for spiral Glass patterns. (A) Mean thresholds for all subjects (error bars are ±1 S.E.M.). Open circles represent thresholds for parallel patterns, filled circles for spiral patterns. (B) Threshold for subjects with multiple runs. (C) Thresholds for subjects with a single run.

In agreement with previous work (Wilson & Wilkinson, 1998), we found that thresholds for concentric patterns were, on average, slightly lower than thresholds for radial patterns. The mean threshold for concentric patterns was 21.5% (± 0.8% S.E.M.) versus 25.5% (± 1.3% S.E.M.) for radial patterns. The concentric threshold was lower than the radial in 30/31 (97%) experiments. The mean difference between concentric and radial thresholds was 4.1% and was statistically significant (P < 0.01, paired t-test, df = 30).

Because our method of generating Glass patterns with paired noise dots differs from most previous work, we also measured thresholds for parallel (translational) patterns for four orientations (−45°, 0°, 45° and 90°) Four subjects performed one run each (1 run = 4 orientations × 11 signal levels [0.0–1.0; 0.1 increment] × 25 repetitions). Three of these subjects (VF, NM, and JW) had been tested with spiral patterns. The mean threshold over all subjects for parallel patterns was 51.5% (± 2.5% S.E.M.). These thresholds are comparable to previous measurements (Wilson & Wilkinson, 1998) using unpaired noise dots.

4. Discussion

Wilson and Wilkinson (1998) reported that thresholds for concentric and radial Glass patterns were generally lower than thresholds for parallel or hyperbolic patterns. They concluded that the human visual system was specialized to detect concentric and radial patterns. Others have found little or no difference between concentric, radial and parallel Glass patterns when measuring discriminability (Maloney, Mitchison, & Barlow, 1987) or critical displacement (Stevens, 1978). Spiral Glass patterns represent an important test of Wilson and Wilkinson’s hypothesis as sets of spiral patterns, which include concentric and radial, can be constructed so as to have identical orientation content. In fact, we have gone one step further and constructed patterns with orientation content that is constant across variations in both spiral angle and signal level. If the human visual system contained a range of Glass pattern detectors tuned to different spiral angles, then we might have found that thresholds for spiral patterns were the same or lower than either concentric or radial patterns. Instead, we found that thresholds were significantly higher for intermediate spiral angles, which supports the idea that the human visual system is specifically adapted to detect concentric and radial patterns.

It is possible that detection thresholds for Glass patterns are related to the degree of symmetry in the patterns. Radial and concentric Glass patterns have infinite axes of symmetry while intermediate spiral angles have no axes of symmetry. Perhaps more lines of symmetry are associated with lower detection thresholds. One argument against the symmetry hypothesis is that, due to the random placement of dot pairs, none of the Glass patterns we have constructed is perfectly symmetrical. At threshold, there were always more noise dots than signal dots, thus degrading the symmetry still further. If thresholds were related to symmetry, one would expect parallel Glass patterns, which have a single axis of symmetry, to have lower thresholds than spiral patterns. In agreement with Wilson and Wilkinson (1998), we found that parallel patterns had thresholds roughly 2.5 times higher than concentric patterns, whereas we found that the highest threshold for spiral patterns was less than 2.0 times greater than the average concentric pattern threshold. Finally, the symmetry hypothesis cannot account for the threshold difference between concentric and radial patterns.

Recently, Morrone, Burr, DiPietro, and Stefanelli (1999) measured detection thresholds for spiral motion using random dot kinematograms. They found that thresholds for spiral motions were higher than thresholds for rotation or expansion/contraction, a result which is directly analogous to that of the current study. This suggests either that the same neural substrate is responsible for detection of static and moving patterns, or that there are separate pathways that happen to share a common insensitivity to spirals.

5. Models

There are several models in the literature for Glass pattern detection and/or discrimination based on (1) an ideal observer (Maloney et al., 1987), (2) local orientation statistics (Dakin, 1999; Stevens, 1978), or (3) multistage filtering (Wilson & Wilkinson, 1998). We, therefore, constructed three models inspired by each of these approaches and applied them to the patterns used in our experiments. All models were simulated in Matlab.

The first model we tried was an ideal observer based on autocorrelation. First, a spiral pattern was constructed with signal = 1.0, and then the 2-D autocorrelation of the pattern was computed. The autocorrelation was normalized to the total number of white pixels in the pattern to avoid spurious differences due to random variation in the number of overlapping dots. Next, the normalized autocorrelation of a noise pattern (signal = 0) was subtracted, point-by-point, from the autocorrelation of the full signal pattern. This resulted in a 2-D difference image (signal–noise). The procedure was repeated 100 times with different randomly generated patterns and the difference images were summed and then squared pixel-by-pixel. The mean and variance of the pixel values were then computed across the squared sum of difference images. The same procedure was repeated with parallel patterns as the input. As shown in Fig. 3, the results of this procedure are completely at odds with the experimental results. The ideal observer model predicts that radial patterns should have the lowest thresholds, and concentric the highest, with spirals and parallel patterns halfway in between. It should be noted that as the orientation content and total energy of the 2-D Fourier transforms are constant across patterns and signal levels, the predictions of the ideal observer model must be based on differences in spatial frequency content.

Fig. 3.

Signal-to-noise characteristics of an ideal observer model. Closed symbols with heavy solid line are results for spiral Glass patterns (0°=radial, 45°=spiral, 90°=concentric). Open symbols with heavy dashed line are results for parallel patterns. Thin lines are results obtained when the signal pattern was replaced by noise.

The second model we considered was based on local orientation statistics. Rather than computing ‘virtual lines’ (Stevens, 1978) between pairs of dots, we assessed local orientation by convolving the patterns with oriented 2-D Gabor filters of various sizes and frequencies. Specifically, we constructed a family of 24 vertically oriented Gabor filters with horizontal spatial frequency = (0.25, 0.5, 1.0, 2.0, 4.0, 8.0) cpd and horizontal space constant = (0.25, 0.5, 1.0, 2.0)°. The vertical space constant was always a factor of 2.5 greater than the horizontal. Because spiral Glass patterns have rotational symmetry, the results should be independent of Gabor orientation. We computed spiral pattens (angles = 0°, 45°, and 90°) with signal = 1.0 plus one noise pattern and convolved each pattern with each Gabor. Each 2-D convolution was normalized according to the total number of white pixels in the original pattern, then squared pixel-by-pixel and summed over all pixels. The process was repeated 25 times with randomly generated patterns. The sums-of-squares were subjected to t-tests for all possible pairwise combinations of pattern (radial, spiral, concentric, noise) and for each set of Gabor parameters. Out of 144 total comparisons (6 pairs of patterns × 24 Gabors), only three were significant at the P < 0.05 level and none at the P < 0.01 level. We conclude that, like the global orientation distribution, local orientation statistics are constant across variations in spiral angle.

The third model was based on the multi-stage nonlinear filtering scheme proposed by Wilson and Wilkinson (1998). We first tried a model with only concentric and radial detectors, which was, in fact, identical to the original Wilson and Wilkinson model. We found that the responses of the model detectors were tuned as a function of spiral angle (Fig. 4A), but that neither detector responded well to patterns with spiral angles near 45°. We then recreated the stimuli for a single experiment and ran ‘trials’ with the model in place of a human observer. A correct response was scored if either the concentric or radial detector had a response to the signal pattern greater than 1.5 times the average of the responses to the noise pattern. As in the real experiment, 25 repetitions of each signal level and spiral angle were run and psychometric functions were constructed.

Fig. 4.

Predictions of Wilson and Wilkinson model. (A) Responses of concentric, radial (solid lines) and spiral (dashed line) detectors as a function of spiral angle. Thick lines are responses to 50% signal patterns. Dotted lines are responses to 0% signal. (B) Thresholds predicted by a model with concentric, radial, and spiral detectors. Filled circles are real data (positive and negative angles combined). Solid line is prediction of unequally-weighted model. Dashed line is prediction of equally-weighted model.

We found that a model consisting of only concentric and radial detectors predicted infinitely high thresholds for spiral patterns. The reason is that the response of the detectors to spiral patterns with non-zero signal is smaller than the response to noise patterns (Fig. 4A, dotted lines). Hence, for spiral patterns, the performance of the model actually tends toward 0% correct as the signal strength increases. One way to correct this problem might be to increase the tuning width of the detectors. This can be done by increasing the space constant of the second stage filters (in Wilson and Wilkinson’s model, the second stage filters are modeled as a difference-of-Gaussians). However, we found that even if we doubled the space constants, the tuning width did not increase sufficiently to raise the performance above 50% correct.

To model our results, then, we added a ‘spiral detector’. The preferred spiral angle of the model detectors is determined by the relative orientation of the first and second stage filters. A relative orientation of 0° produces a radial detector, 90° produces a concentric detector; intermediate values produce spiral detectors. We constructed a spiral detector with a relative angle of 45°. Using the three types of detectors, we were able to obtain a good fit to the experimental results (Fig. 4B) but only if the responses were weighted unequally. We obtained the best fit, as determined by least squares optimization, when the concentric detector had a weight of 1.083 and the radial and spiral detectors had weights of 0.996 and 0.915, respectively (Fig. 4B, solid line, r = 0.97). When the responses of the three detectors were weighted equally (Fig. 4B, dashed line), we obtained a poor fit (r = 0.49).

In conclusion, our results suggest that the human visual system is specialized for detecting concentric and radial patterns as these had lower thresholds than other spiral patterns with the same distribution of orientations. A model comprising a concentric, radial and spiral detector provided a good fit to the experiment results provided that the response of the spiral detector was about 84% as strong as the concentric detector. This suggests that neurons responding to spiral patterns may be fewer in number or have weaker responses than those preferring concentric patterns.