Near their thresholds for detection, shapes are discriminated by the angular separation of their corners

Abstract

Observers make sense of scenes by parsing images on the retina into meaningful objects. This ability is retained for line drawings, demonstrating that critical information is concentrated at object boundaries. Information theoretic studies argue for further concentration at points of maximum curvature, or corners, on such boundaries [1]-[3] suggesting that the relative positions of such corners might be important in defining shape. In this study we use patterns subtly deformed from circular, by a sinusoidal modulation of radius, in order to measure threshold sensitivity to shape change. By examining the ability of observers to discriminate between patterns of different frequency and/or number of cycles of modulation in a 2x2 forced choice task we were able to show, psychophysically, that difference in a single cue, the periodicity of the corners (specifically the polar angle between two points of maximum curvature) was sufficient to allow discrimination of two patterns near their thresholds for detection. We conclude that patterns could be considered as labelled for this measure. These results suggest that a small number of such labels might be sufficient to identify an object.

Figure 1. Example stimulus pairs.

The pairs of test stimuli are those used in the 2x2FC tasks. One of the two stimuli was presented in each trial of a block. Each trial had two intervals, with the test stimulus presented in one interval and a circular reference stimulus presented in the other. The observer was required to identify the interval in which the test stimulus was presented (detection) and then identify which of the pair of test stimuli was presented in a second response (identification). The amplitudes, A, of the stimuli shown are 0.05 and 0.0135 for the patterns containing RF3 and RF6. A pattern with an amplitude of greater than 1/(1+ω²), where ω is the frequency of the modulating sine function, has points of maximum concave curvature in addition to the points of maximum convex curvature. None of the patterns used in the experiments had amplitudes this large. The examples shown have half of this amplitude. The pairs are: (a) complete RF3 and RF6 patterns, patterns deformed from circular by a sinusoidal modulation of radius, with a frequency of 3 and 6 cycles of modulation in 2π radians respectively; (b) an RF3 and an RF6 pattern with a single, (1), cycle of modulation each; (c) an RF3 and an RF6 pattern with two, (2), consecutive cycles of modulation; (d) an RF6 pattern with one (1) cycle of modulation and an RF6 pattern with four (4) consecutive cycles of modulation; (e) an RF6 pattern with three (3) cycles of modulation and a complete RF6 pattern; (f) a complete RF3 pattern and an RF6 pattern with 3 alternate single cycles of modulation (3x1).

Figure 2. Psychometric functions describing performance of the group of observers.

Probabilities of correct detection and identification performance of the group of observers are plotted against the amplitude of modulation, A, of the test stimulus. Error bars are 95% confidence intervals. The solid lines are Quick functions fitted to the detection data and dashed lines the identification data. The red lines denote performance pertaining to the first test stimulus of the pair and blue lines the second. If the solid and dashed lines are coincident for both stimuli the patterns are perfectly discriminated. The results of F tests to determine whether the fits described by the solid and dashed lines of each pair are significantly different statistically are presented in Table 1.

Figure 3. Curvature and curvature frequency spectra for Pair 1(a), 1(b) and 1(c).

The graph (a)i plots the curvature (rate of change of orientation with respect to circular) of the complete RF3 (red) and RF6 (blue) patterns (Pair 1(a)) as a function of the polar angle. The graph (a)iii plots the rectified curvature. The curvature shown is that derived for the amplitudes of modulation at the threshold for detection of the patterns. When the curvature is tighter than a circular arc at the base radius R₀ the curvature is positive and when it is shallower it is negative. The graphs (a)ii and (a)iv in the right hand column are the frequency spectra of the curvature measures shown in (a)i and (a)iii respectively. In (b) modulation of curvature is constrained to a single cycle (for the rectified measure of curvature it is constrained to a half cycle). Data pertaining to the single cycle RF3 pattern is in red and RF6 in blue. For the un-rectified measure of curvature, (b)i, distinct peaks distinguishing the RF3 and RF6 modulation can be discerned in the frequency spectra, (b)ii. For the rectified measure, however, this is not so. In (c), data pertaining to Pair 1(c), modulation of curvature is constrained to two cycles (RF3 modulation in red and RF6 in blue). Distinct peaks distinguishing the RF3 and RF6 patterns can be seen across the frequency spectra for both the un-rectified and rectified curvature measures.

Figure 4. Example stimuli for the conditions represented by Pair 1(a), 1(b) and 1(c).

In (a), the RF3 patterns are shown in the left hand and RF6 patterns in the right hand column. Amplitude of modulation increases from top to bottom. The amplitudes of modulation used in the stimuli are those used in the experiment for the method of constant stimuli. It is evident from these examples that when the modulation can be detected the patterns can be discriminated. Example stimuli for the condition represented by Pair (b) are shown in (b). The single cycle RF3 patterns are shown in the left hand and single cycle RF6 patterns in the right hand column. The patterns are in random phase. Again the amplitudes shown are those used in the experiment. It can be seen from these examples that modulation can be detected before it is of sufficient amplitude to allow discrimination. In (c) example stimuli for the condition represented by Pair (c) are shown. Two cycle RF3 patterns are shown in the left hand and two cycle RF6 patterns in the right hand column. The figure illustrates that when two cycles of modulation are present in the stimuli the RF3 and RF6 patterns can be identified at their thresholds for detection.