**Shape is a critical cue to object identity. In psychophysical studies, radial frequency (RF) patterns, paths deformed from circular by a sinusoidal modulation of radius, have proved valuable stimuli for the demonstration of global integration of local shape information. Models of the mechanism of integration have focused on the periodicity in measures of curvature on the pattern, despite the fact that other properties covary. We show that patterns defined by rectified sinusoidal modulation also exhibit global integration and are indistinguishable from conventional RF patterns at their thresholds for detection, demonstrating some indifference to the modulating function. Further, irregular patterns incorporating four different frequencies of modulation are globally integrated, indicating that uniform periodicity is not critical. Irregular patterns can be handed in the sense that mirror images cannot be superimposed. We show that mirror images of the same irregular pattern could not be discriminated near their thresholds for detection. The same irregular pattern and a pattern with four cycles of a constant frequency of modulation completing 2 π radians were, however, perfectly discriminated, demonstrating the existence of discrete representations of these patterns by which they are discriminated. It has previously been shown that RF patterns of different frequencies are perfectly discriminated but that patterns with the same frequency but different numbers of cycles of modulation were not. We conclude that such patterns are identified, near threshold, by the set of angles subtended at the center of the pattern by adjacent points of maximum convex curvature.**

*π*radians are completed with a circular arc. As cycles of modulation of a particular frequency are added to RF patterns, the amplitude of modulation required for observers to be able to detect deformation of a pattern decreases at a rate faster than predicted by the increasing probability of detecting single cycles of modulation (probability summation). This has been interpreted as indicative of integration of shape information around the patterns (Loffler et al., 2003). RF patterns, then, provide stimuli that have demonstrably global shape representations and precisely specified shapes. By manipulating the amplitude and frequency of modulation and the number of cycles of that modulation, the relative importance of local properties such as the deviation of the local orientation of the contour from a circle and curvature of the pattern can be examined. Moreover, the widely used choice of the sinusoid as the modulating function is perhaps only a convenience, and the RF pattern, therefore, provides an ideal baseline for the exploration of other modulating functions in the investigation of shape processing in the visual system.

*π*radians were shown to have the same maximum orientation difference from circular. The orientation distributions for the three stimuli are different, but the maximum deviation from the orientation profile of a circle is the same for all three stimuli. That is to say that if the path was sampled for orientation deviation from circular on the curve, the orientation with respect to circular would differ locally around the pattern but the content of the orientation difference from circular would be identical. The maximum orientation difference from circular is the same for the patterns, but locally, the rate of change of orientation differs, whether measured as the rate of orientation with path length or polar angle. This implies that shape analysis at threshold relies on orientation difference from circular but is not critically dependent on the magnitude of curvature. By extension, this suggests that some analytical representation of shape is arrived at that is indifferent to the local properties from which it is derived. This conjecture is examined in Experiment 2 by testing whether the three pattern types could be discriminated at their thresholds for detection. Patterns with the same frequency of modulation of radius were used in a two-decision, two-interval, forced choice (2x2IFC) task (Watson & Robson, 1981). Evidence that the patterns were perfectly discriminated might suggest that shapes are labeled for the nature of the points of maximum convex curvature (i.e., smoothly curved or angles) or indeed were identified by any other local property, perhaps including the existence of points of infinite curvature (orientation discontinuities). However, the patterns are found to be not discriminable near their thresholds for detection. This suggests that the representations of shape arrived at, which are demonstrated to be global in Experiment 1, are indifferent to the local properties of the pattern. That is not to say that the local properties are unimportant to the analysis, as Experiment 1 also demonstrates the significance of local orientation difference from circular to the threshold for detection, but that the local information cannot be used in the discrimination task. This might indicate that, in the representation of shape, a perceptual decision has been made allowing for a more concise representation (Lennie, 1998).

^{2}in each case, and the screens were viewed in darkened rooms with an ambient luminance of <1 cd/m

^{2}. Luminance calibration was performed using an Optical OP200-E photometer (head model number 265) and associated software (Cambridge Research Systems). A CB3 button box was used to record responses for Experiments 1 and 2, and the computer keyboard was used for Experiments 3 and 4.

*σ*in equation 1 of Wilkinson et al. [1998] was 3.376′). The Weber contrast of all stimuli was 1.

*A*is the modulation amplitude as a proportion of the radius of the unmodulated circle,

*Δ*is the threshold at the 75% correct performance level, and

*Q*is a measure of the slope of the psychometric function.

*RF*in figures) had a sinusoidally modulated radius given, as a function of polar angle, by Equation 2. where

*θ*is the angle made with the

*x*-axis,

*R*

_{0}the mean radius,

*A*the amplitude of modulation,

*ω*the frequency of modulation (cycles in 2

*π*radians), and

*ϕ*the phase of the modulation (all patterns were presented in random phase). The radius of the positively rectified stimuli (denoted +ve) is given by and the negatively rectified stimuli (denoted –ve) by

*θ*is given by where for the angle

*α*, the value zero is defined to correspond to vertical. The orientations for the rectified patterns are also derived from this function with the appropriate modifications of frequency, amplitude, and sign. The value for

*R*

_{0}, nominally 60 min of visual angle, was allowed to vary by ±5%, at random to preclude the use of average pattern radius as a reliable cue to pattern identity, and the pattern center allowed to move ± one eighth of the pattern radius in the horizontal and vertical dimensions to prevent the buildup of afterimages that might be used as references and also to prevent the edges of the screen being used as alternative cues to shape change.

*ϕ*= 0. The curvature measure was calculated numerically as the rate of change of orientation with polar angle. For the small amplitudes of modulation used, the path length for one radian of polar angle approximates the radius of the pattern. Curvature in the Cartesian frame, the rate of change of orientation with path length, scales in inverse proportion to size. Thus, when considering patterns of a fixed size, comparison of local curvature when expressed as rate of change of orientation with path length, is, other than a change of unit, equivalent to the comparison of curvature expressed as the rate of change of orientation with theta.

*Q*).

*Q*that describes the steepness of the Quick functions fitted to the psychometric data (Loffler et al., 2003; Quick, 1974; Wilson, 1980). The probability summation prediction for the index is given by −1/

*Q*(see Equation 1). For each of the three types of stimulus, the observed indices were compared with the indices predicted for probability summation in paired, one-tailed

*t*tests for the population of observers. The rate of decrease in threshold with increase in the number of cycles of modulation was found to be significantly steeper than that predicted by probability summation for RF patterns,

*t*(4) = 6.460,

*p*= 0.0015; positively rectified patterns,

*t*(4) = 5.208,

*p*= 0.0032; and negatively rectified patterns,

*t*(4) = 2.910,

*p*= 0.0218. Individual values for the indices and the means are presented in Table 1. Functions predicted by probability summation are plotted as dashed lines in Figure 3.

*Q*, where

*Q*is the average of the slopes of the Quick functions fitted to the probabilities of correct response in the 2IFC detection of modulation task. If the single-cycle conditions are excluded, then the indices of the power functions are very similar across the three conditions. A one-way repeated-measures analysis of variance comparing the indices obtained for one to five cycles showed that there was an effect of condition,

*F*(2, 8) = 5.188,

*p*= 0.0359, and Tukey's multiple comparison test indicated that this was due to a significant difference between the indices for the conventional RF patterns and the negatively rectified condition,

*q*(8) = 4.546. The same analysis for the indices considering the data only for two to five cycles showed no effect of condition,

*F*(2, 8) = 1.473,

*p*= 0.2854, and Tukey's multiple comparison test confirmed there was no significant difference in any pairwise comparison. This result indicates that there were no statistically significant differences between stimulus types in the efficiency of integration of information across cycles if the single-cycle thresholds (which cannot include integration across cycles) were excluded.

*π*/2 radians from the maxima on the sinusoid. The curvature defined as the rate of change of orientation in the Cartesian reference frame with polar angle at these points is the same as a circular arc at the same radius (although the measure of curvature employed in the Poirier and Wilson [2006] model is zero at these points, being defined as the curvature difference from this arc). Thus, we arrive at the seemingly paradoxical position that the threshold for detection of modulation in RF patterns is limited by the maximum angular deviation from circularity (although this maximum deviation does not remain constant as cycles are added as might be claimed by Mullen, et al. [2011] but declines), but shape is encoded in the relative positions of maximum convex curvature. Moreover, the results of Experiment 1 imply that orientation difference from circular predominates over curvature in determining the salience of modulation for rectified patterns as well as the conventional patterns. This raises the possibility that the positions of the points of maximum curvature are determined by detectors exploiting the maximum deviation from circularity on the shoulders of the points of maximum curvature. The indifference to the positions of these points of maximum curvature to the efficiency of integration suggests that this information is not retained in the shape representation of these patterns near their thresholds for detection. If the angle at such points were not encoded at threshold for detection of the modulation, then the patterns would be indistinguishable at this threshold. Demonstration that the three pattern types are not discriminable at their thresholds for detection might therefore indicate that in the formation of an economic shape representation, the local information used in the analysis is discarded. Lennie (1998) refers to this as a

*perceptual decision*in the hierarchical analysis of shape.

*A*used in Equations 2–5. Each of the three pattern types has the same maximum orientation difference from circular and the same orientation histogram when they have the same peak-to-peak amplitude. Under these circumstances, as demonstrated in Experiment 1, the different patterns have the same salience. Quick functions were therefore fitted to the detection and identification data across the pattern types of each pair to compare thresholds for detection and discrimination of the patterns (see Figure 4). Because the responses were ordered (i.e., detection first response and discrimination second), potentially leading to an overestimate in sensitivity to one or other of the tasks, a control experiment was performed for which detection and discrimination tasks were completed in separate blocks. Three of the four observers completed one each of the stimulus pairings.

*A*for the RF stimuli and 2

*A*for the rectified stimuli. To confirm that the patterns had the same salience at the same peak-to-peak amplitudes, the data were combined across the block to derive detection thresholds for each pattern type. The mean thresholds for detection of peak-to-peak modulation are 0.00506 ± 0.00087 (95% confidence intervals), 0.00582 ± 0.00116, and 0.00526 ± 0.00121 for the conventional RF and positively rectified and negatively rectified RF stimuli, respectively. Paired, two-tailed

*t*tests of the thresholds across the pairs RF/+ve, RF/−ve, and +ve/−ve showed that the thresholds were not significantly different,

*t*(3) = 2.709,

*p*= 0.0733;

*t*(3) = 1.332,

*p*= 0.2749; and

*t*(3) = 1.446,

*p*= 0.2438, respectively, when thresholds were expressed as peak-to-peak amplitudes. Given this result, the two sets of data for detection for each pair of stimuli within a block were combined, as were the thresholds for identification, for the comparison of thresholds for detection and discrimination of the pattern types.

^{1}

*F*(2, 104) = 70.00,

*p*< 0.0001; SC,

*F*(2, 104) = 64.65,

*p*< 0.0001; TM,

*F*(2, 104) = 61.55, p < 0.0001.

*A*is the amplitude at an angle

_{θ}*θ*and

*ω*is the RF. The gradient of this function is given by the first derivative with respect to

*θ*: and, for the small angles of departure from circular at detection thresholds, the orientation with respect to circular varies approximately linearly with the first derivative. The curvature, expressed as the rate of change of orientation with respect to circular, is therefore proportional to the second derivative of the modulation in radius:

*π*radians. An illustration of how the stimulus was composed is shown in Figure 6. The amplitude quoted in the results for the irregular pattern is the amplitude of the pattern with a frequency of 22/4 cycles per 2

*π*radians, with the amplitudes of each of the other frequencies scaled by the inverse of the ratio between their frequency and 22/4 (for example, when the amplitude of the cycle with frequency of 22/4 is 0.04, the amplitude of the cycle with a frequency of 22/8 is 0.08). The cycles with differing frequencies of modulation were joined smoothly at the zero crossing of the sinusoid. For the low amplitudes of modulation used in this experiment, the maximum deviation of orientation from circular is coincident with this point (Dickinson, McGinty et al., 2012). The radius of the pattern changes with frequency due to the scaling process used to equate the first derivatives at the zero crossings (see Equation 7), and if the constant circular curvature is removed, the curvature scales with an additional factor of frequency (see Equation 8). The radius of higher-frequency cycles is lower and the curvature of higher-frequency patterns is higher. To reiterate, the maximum orientation difference remains constant, the maximum radius decreases with frequency, and the maximum curvature increases with frequency.

*t*tests, thresholds for the conditions with radial frequencies of 22/8, 22/7, and 22/3 were not statistically different from the condition 22/4, to which they were normalized, for the original,

*t*(3) = 1.461,

*p*= 0.2402;

*t*(3) = 2.194,

*p*= 0.1158;

*t*(3) = 0.8896,

*p*= 0.4392, and mirror image,

*t*(3) = 0.7966,

*p*= 0.4839;

*t*(3) = 0.1902,

*p*= 0.1534; t(3) = 2.067,

*p*= 0.1306, conditions.

*Q*across all of the psychometric functions pertaining to the derivation of the power function describing the performance for a particular pattern (i.e., the original or the mirror image). Paired

*t*tests showed that the observed indices were significantly different from the predictions for probability summation for both the original,

*t*(3) = 4.753,

*p*= 0.0177, and mirror image,

*t*(3) = 12.25,

*p*= 0.0012, patterns but that the indices for the original and mirror image patterns did not differ significantly,

*t*(3) = 0.5269,

*p*= 0.6348.

*π*radians). The extra sum of squares F tests showed that the functions fitted to the data describing the probability of correct performance in the detection and identification of the irregular and RF4 patterns was consistent with being equal for each observer: SC,

*F*(2, 104) = 0.01584,

*p*= 0.9843; ED,

*F*(2, 104) = 1.291,

*p*= 0.2794; RG,

*F*(2, 104) = 0.7849,

*p*= 0.4588.

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^{1}This test compares two models, the first being that of a global fit and the second that the two curves are distinct. It assumes a null hypothesis that the global fit is the best and uses statistical hypothesis testing, using a criterion of

*p*< 0.05, to examine if there is sufficient evidence to reject this hypothesis, The degrees of freedom in the numerator in the calculation of the F ratio is 2, the difference in the number of degrees of freedom in the null hypothesis model and the alternative model. The degrees of freedom in the denominator is the number of degrees of freedom in alternative hypothesis (Motulsky & Christopoulos, 2004). For the F tests, the probabilities of correct detection across blocks of trials were treated as independent, and therefore, each fitted function has 54 samples of probability of correct response (six blocks of nine points on the psychometric function). There were two free parameters in the fit and therefore 52 degrees of freedom or 104 for the two curves. The global fit has two fewer degrees of freedom.