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Article  |   November 2012
Further evidence that local cues to shape in RF patterns are integrated globally
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Journal of Vision November 2012, Vol.12, 16. doi:10.1167/12.12.16
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      J. Edwin Dickinson, Jessica McGinty, Kathryn E. Webster, David R. Badcock; Further evidence that local cues to shape in RF patterns are integrated globally. Journal of Vision 2012;12(12):16. doi: 10.1167/12.12.16.

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Abstract
Abstract
Abstract:

Abstract  Radial frequency (RF) patterns, paths deformed from circular by a sinusoidal modulation of radius, have proved valuable stimuli for investigation of visual shape processing. Their utility relies upon evidence that thresholds for detection of modulation decrease, as cycles of modulation are added, at a rate that cannot be accounted for by the improving probability of detection of any single cycle (probability summation). This has been interpreted as indicative of global processing. Recently Mullen, Beaudot, and Ivanov (2011), using low contrast RF patterns viewed in cosine phase through a Gaussian window, demonstrated the existence of a local cue to modulation that was more salient than the global shape cue present in sectors of RF patterns. The experiments reported here investigate why this cue has not previously obscured global integration of shape information in RF patterns. Using stimuli modulated in sine phase, Experiment 1 showed that the presence of a circular sector of path, used to complete a partially modulated RF pattern, does not raise thresholds, contrary to the suggestion of Mullen et al. (2011). Experiment 2 demonstrated integration for high and low contrast RF patterns viewed in sine phase through a Gaussian window and Experiment 3 showed the same for patterns in cosine phase if the use of a local phase specific curvature cue was precluded. Effective use of local curvature in the test comparison, then, requires knowledge of pattern orientation to define the sign of curvature. Experiment 4 demonstrated global processing of shape information for a range of radial frequencies and also showed that the local maximum gradient with respect to circular within an RF pattern covaries with threshold. This implies that it is this cue, or one that covaries linearly with it, that is integrated across cycles of modulation by the global processing mechanism.

Introduction
Processing of spatial form in the human visual system is hierarchical (Felleman & Van Essen, 1991; Lennie, 1998). In the primary visual cortex (V1) the incoming responses of neurons with circularly symmetric receptive fields are assembled into linear spatial filters capable of encoding the positions and orientations of segments of the boundaries between objects (Ferster & Miller, 2000; Hubel & Wiesel, 1959, 1968; Reid & Alonso, 1995). The responses of approximately collinear and adjacent V1 neurons are facilitated through excitatory lateral interactions within the cortical sheet (Field, Hayes, & Hess, 1993; Li & Gilbert, 2002), enhancing the salience of the boundaries. Current models of object identification encode curvature as a function of polar angle relative to the object center (Pasupathy & Connor, 1999, 2001, 2002; Poirier & Wilson, 2006) within a population of curvature selective neurons. 
Curvature population models of shape require the processing of information around a boundary. The most constrained versions suggest that contours may be represented as a collection of curves, varying in magnitude and sign, at locations specified by their polar angles relative to the centre of the shape (Pasupathy & Connor, 2001, 2002; Poirier & Wilson, 2006). However, a more general representation might be a radial frequency spectrum (Poirier & Wilson, 2006) derived from a continuous measure of curvature around the path. In this less constrained case, periodic repetition of curvature might be expected to result in integration of signal within the spectrum. Evidence for the integration of repeated shape information around a boundary was provided by the demonstration of rapidly decreasing amplitude thresholds for the detection of sinusoidal modulation of the radius of a circle, as the number of cycles of a particular period of modulation was increased (Bell & Badcock, 2008; Dickinson, Han, Bell, & Badcock, 2010; Loffler, Wilson, & Wilkinson, 2003). If signal is assumed to be a linear function of amplitude of modulation then, for modulation with long periods of modulation (2π/3 or π/2 radians for example), integration of signal around these so called radial frequency (RF) patterns has been shown to be almost linear (the efficiency of integration approaches 100%). These studies have all employed closed contour stimuli where the unmodulated sector forms a smooth circular arc. 
A recent paper (Mullen et al., 2011), however, argued that if a progressively larger sector of an RF pattern is removed from the stimulus, retaining only the modulated sector, then no difference in threshold is observed until less than a single cycle of modulation can be seen. Their thesis was that, for all RF patterns, detection of deformation is effected by relatively local processing mechanisms requiring information from a single cycle alone. A potential explanation offered by Mullen et al. (2011) for the rapid decrease in threshold with number of cycles of modulation seen in other studies is that the circular, unmodulated sector of a pattern that is present in patterns with incomplete modulation interferes with detection of modulation in the modulated section. The study of Mullen et al. (2011), however, was also different from earlier studies in the contrast of the stimuli employed. Earlier studies used stimuli with high luminance contrast, while those of Mullen et al. (2011) were predominantly at five times the threshold contrast for detection. In a subsequent study Ivanov and Mullen (2012) proposed, on the basis of a complex dependency of modulation detection thresholds on stimulus size, spatial frequency content and contrast, and a monotonic increase in threshold with decrease in contrast, that at low contrast shape discrimination is limited by nonglobal processing. This latter study, however, did not vary the number of cycles of modulation of patterns of particular radial frequencies and, therefore, did not test directly the efficiency of integration of modulation information around the path. Because of these alternative accounts it is not clear whether the explanation for the lack of global processing reported by Mullen et al. (2011) should be extrapolated to high contrast stimuli, nor whether the increase in threshold with contrast reported in Ivanov and Mullen (2012) is a consequence of a failure of global processing at low contrast. The experiments reported here sought to resolve these issues by measuring the efficiency of integration over a range of stimuli. Compared with experiments that have shown evidence for global processing of shape, the stimuli used by Mullen et al. (2011) differ in that incomplete modulation of an RF pattern is represented by an incomplete path, the patterns are presented at low contrast, and the modulation is presented in cosine rather than sine phase. These three manipulations of RF pattern stimuli are examined in Experiments 1, 2, and 3 of this study. 
In Experiment 1 of this study we used stimuli of high contrast (unlike Mullen et al., 2011, who used stimuli at five times the contrast detection threshold). The stimuli were RF patterns with a frequency of three cycles of modulation per 2π radians. The thresholds for detection of patterns with three cycles of modulation, one cycle of modulation of a complete path, and one cycle of modulation on a semi-circular path were compared to test the hypothesis that the unmodulated sector of path interferes with detection of modulation in the modulated sector. The stimuli containing only single cycles of modulation had the modulation presented in sine phase rather than the cosine phase used by Mullen et al. (2011) in order to test the hypothesis against stimuli that had previously been shown to provide evidence for global processing. We found that the interference by the presence of an unmodulated sector of path could not explain previously reported results indicating integration of signal around the path. The stimuli used in our study, however, had hard edges to the sectors of visible stimulus, rather than the Gaussian windowed sectors used by Mullen et al. (2011). Mullen and Beaudot (2002) demonstrated that the threshold for detection of modulation in RF patterns was strongly dependent on contrast. By modulation of contrast around the pattern they went on to show that, for achromatic stimuli, sides of RF patterns were more important for discrimination tasks than corners. This naturally led to the use of a smooth contrast envelope to constrain the proportion of RF pattern visible in each condition. Experiment 2 of this study, therefore, used Gaussian windowed stimuli with both high and low contrast to measure the rate of decrease in threshold as the number of visible cycles of modulation was added. The stimuli used were presented in sine phase to maintain consistency with previous experiments that have provided evidence for global processing of shape information. This phase for the modulation differs from that used by Mullen et al. (2011) who used cosine phase (this parameter change is subsequently shown to affect the results obtained and is explored in detail in Experiment 3). In contrast to the Mullen et al. (2011) study, evidence for integration of shape information across cycles was found for both patterns with high contrast and for patterns with a contrast of only five times the contrast for detection of the stimuli. One substantial difference between the two studies was that stimuli used in Experiment 2 of this study were viewed in sine phase through the Gaussian window and those used by Mullen et al. (2011) were in cosine phase. When viewed in cosine phase a small sector of an RF pattern can appear more or less tightly curved than a sector of a circle of the same radius. If the observer is aware of the phase of the pattern, as in the study of Mullen et al. (2011) due to blocking of conditions and the positioning of the point of maximum contrast at the maxima and minima of the modulation, then discrimination of the test sector from a circular arc can be made on the basis of local curvature. In Experiment 3, we show that this cue is more salient than the local cue to global deformation, as shown and claimed by Mullen et al. (2011). We also show, however, that if this cue is made unreliable by interleaving patterns viewed in positive and negative cosine phase then observers revert to use of the cue used by the global integration mechanism when performing the task. No evidence was found that a similarly salient local cue existed in a pattern viewed in sine phase. It appears, then, that Mullen et al. (2011) have indeed revealed a local cue that is more salient than the local cue integrated to provide the global cue to deformation, but one that can only be exploited if the phase of the pattern is known. Experiment 4 measured thresholds for differing numbers of cycles of RF2 (where RF2 denotes a frequency of two cycles of modulation in 2π radians), RF3, RF4, and RF6 modulation. Thresholds for patterns with the same number of cycles of modulation (rather than the same frequency of modulation) were shown to be inversely proportional to frequency of modulation. A quantity that conforms to this relationship for low amplitude RF patterns is the maximum orientation difference from circular. It might be expected that the local cue integrated across cycles of modulation is this cue or one that covaries linearly with it. 
General methods
Procedure
Stimuli for the high contrast studies were created in Matlab 5.3 (Mathworks, Natick, MA, USA) and presented to a Hitachi Accuvue 4821 monitor from the frame buffer of a Cambridge Research Systems VSG 2/4 graphics card housed in a PC (Pentium II, 400 MHz). For the low contrast conditions of Experiment 2 stimuli were created in Matlab 7.0 and presented to a Sony Trinitron G520 monitor from a Cambridge Research Systems Visage visual stimulus generator. Monitor frame rate in both cases was 100 Hz. Luminance calibration was performed using a Cambridge Research Systems OPTICAL OP 200-E photometer (Head model number 265) and associated software. Stimuli were embedded in a uniform field with a luminance of 45 cd/m2. The horizontal and vertical dimensions of the pixels were 1′ of visual angle for both sets of apparatus and the screen dimensions were 1024 × 768 pixels (17.07° × 12.80°). Test and reference stimuli were presented in two separate intervals with a 500-ms interstimulus interval interposed. Test stimuli were modulated in radius, and in all experimental conditions the reference stimuli were unmodulated (circles or circular arcs). High-contrast stimuli had a maximum luminance of 90 cd/m2 and were, therefore, presented at a Weber contrast of 1 (minimum luminance was approximately 8 cd/m2 and so stimuli had a Michelson contrast of 0.83) for 160 ms. The low-contrast stimuli had a Gaussian temporal profile with a standard deviation of 250 ms. Maximum contrast at the center of the temporal window was five times each observer's contrast detection threshold for each stimulus. The order of presentation of the test and reference stimuli was randomized and the task of the observer was to report which interval contained the test stimulus, a two-interval forced choice task. The method of constant stimuli was employed, with nine different amplitudes of modulation of the test pattern used for each stimulus condition. Three blocks of 180 trials were performed for each condition, a total of 60 trials for each of the nine amplitudes. The probability of correct response, p, was calculated for each of the amplitudes, A, and a Quick function (Equation 1) fitted to these data. The Quick function describes the probability of correct response as a function of amplitude thus, where Δ is the threshold for 75% correct response and Q is a quantity related to the slope of the function (Loffler et al., 2003; Quick, 1974; Wilson, 1980). 
Observers
Seven experienced psychophysical observers participated in the study. All had normal or corrected to normal visual acuity. Observers ED, JM, and KW are authors; RO, VB, TM, and KT were naïve to the purpose of the experiments. The study was approved by the University of Western Australia ethics committee and was, therefore, conducted in accordance with the Declaration of Helsinki. 
Experiment 1: The unmodulated connecting contour in an RF3 pattern does not impair modulation thresholds
Introduction
This experiment tested the hypothesis that when an RF pattern has less than the required complement of cycles of modulation to complete the pattern over 2π radians the presence of the circular sector completing the contour reduces the detectability of the modulation of the modulated arc. That is to say that the thresholds for the detection of modulation in patterns with the circular sector present and absent are compared. Should the thresholds be found to be higher when the circular sector is present then the hypothesis is supported. 
Methods
Three types of stimuli were used. All stimuli were based on a sinusoidal modulation of radius defined by where R0 is the unmodulated radius, ω is the frequency of the modulation (in this case ω = 3 cycles per 2π radians and the pattern will be referred to as an RF3), θ is the polar angle (the phase of the pattern was held constant so that observers were equally confident in the position of the modulated arc for both stimuli with a single cycle of modulation), and A is the amplitude of modulation. The first type of stimulus was a 180° sector of pattern (θ = −30° to θ = 150°) with a single cycle of modulation present (Figure 1A). Following Loffler et al. (2003) the single cycle of sinusoidal modulation was approximated by a D1 (first derivative of a Gaussian) function matched in amplitude and maximum gradient with the single cycle of sinusoidal modulation it replaces. A D1 was used to allow a smooth transition between the modulated and circular arcs. The single cycle of modulation was, consequently, in sine phase rather than the cosine phase used by Mullen et al. (2011). The modulation was always in zero sine phase and the patterns always contained the first cycle of modulation (between θ = 0 and θ = 120°). The second type of stimulus was identical to the first but with the sector of the pattern from θ = 150° to θ = 330° completed with a circular arc (Figure 1B). The third pattern was a complete RF3 pattern (Figure 1C). The patterns had a mean radius of 1° and a D4 luminance profile in section (a fourth derivative of a Gaussian) with a peak spatial frequency of 8 cycles per degree (c/deg). The spatial frequency content of the stimuli was matched to that used by Loffler et al. (2003) but integration of signal has been observed using stimuli with a Gaussian luminance profile (Bell & Badcock, 2008) so the D4 profile is not considered critical. Maximum Weber contrast was 1. Examples of the three types of stimuli are displayed in Figure 1
Figure 1
 
Examples of the three types of stimuli used in Experiment 1. (A) and (B) have one cycle of modulation and (C) has three. (A) has a semicircular sector of the pattern removed and is, therefore, an open contour.
Figure 1
 
Examples of the three types of stimuli used in Experiment 1. (A) and (B) have one cycle of modulation and (C) has three. (A) has a semicircular sector of the pattern removed and is, therefore, an open contour.
Results
Threshold amplitudes for detection of modulation in the three types of stimuli are displayed in Figure 2
Figure 2
 
Detection thresholds for the three types of stimuli used in Experiment 1. Error bars represent 95% confidence intervals. The threshold amplitude is expressed as a fraction of the unmodulated radius of the pattern and is, therefore, dimensionless. Threshold amplitudes are highest for the pattern with the circular sector removed and lowest for the RF3 pattern with three cycles of modulation.
Figure 2
 
Detection thresholds for the three types of stimuli used in Experiment 1. Error bars represent 95% confidence intervals. The threshold amplitude is expressed as a fraction of the unmodulated radius of the pattern and is, therefore, dimensionless. Threshold amplitudes are highest for the pattern with the circular sector removed and lowest for the RF3 pattern with three cycles of modulation.
The primary aim was to determine whether the thresholds for one cycle closed contours were higher than for one cycle open contours because this might reflect interference by the unmodulated sector. Inspection of the data for the one cycle open and closed conditions shows them to be highly variable, and because the thresholds were constrained to be positive, they might reasonably be expected to be log-normally distributed. In order to perform an analysis of variance (ANOVA) the data were, therefore, normalized by taking the logarithm of the thresholds. An ANOVA was then performed on the transformed data with condition as a within subjects factor and this revealed a main effect of condition, F(2, 6) = 13.42, p = 0.0061. A post hoc Tukey multiple comparison test showed that for the normalized data the thresholds for the RF3 pattern were significantly different to those for the one cycle open (q = 6.92, p < 0.05) and closed (q = 5.54, p < 0.05) conditions. The one cycle open and closed conditions, however, were not significantly different (q = 1.38). Thus, we find no evidence that the thresholds for the one cycle closed condition were impacted by the presence of the circular arc. As shown in Figure 2, three out of four observers had higher thresholds for the open rather than the closed contour. The lowest thresholds were observed for all observers for the complete RF3 pattern with three cycles of modulation (μ = 0.0030) and the highest were observed for the open contour stimulus (μ = 0.0132). 
The average ratio between the thresholds for the complete contour with a single cycle of modulation and that for the RF3 pattern with three cycles of modulation was 2.8 (3.0: ED, 3.7: KW, 2.1: RO, and 2.4: VB), which could arise from addition of the available signal (i.e., global integration). The average ratio between the thresholds for the open contour with a single cycle of modulation and those for the RF3 pattern with three cycles of modulation was higher, at 4.3 (individual values of this ratio were highly variable at 4.1 for ED, 8.7 for KW, 2.4 for RO and 1.8 for VB). The average ratio was greater than the number of cycles on the pattern, a situation impossible to achieve through the addition of signal. These ratios, however, varied substantially across observers. 
Conclusions
Of the two patterns with the single cycle of modulation, the pattern that included the circular sector had the lower threshold for three of the four observers. The results of statistical tests, however, showed that the thresholds for these two conditions were not significantly different. The thresholds were somewhat variable but the fact that thresholds for the one cycle open condition were higher than the one cycle closed condition, the direction opposite to that predicted by the hypothesis that the circular arc reduces the detectability of the modulation of the modulated arc, for three of the four observers makes it less likely that an effect was masked by the variability of the data. We conclude, therefore that the results do not support the hypothesis that the inclusion of the unmodulated sector of the contour interferes with the detection of the modulated sector. Moreover, the ratio of thresholds between the complete patterns with one cycle and three cycles, at 2.8, was consistent with efficient integration of signal around the whole pattern. This experiment, however, was performed using stimuli that were presented in the same orientation across trials. This contrasts with the stimuli that were used by Mullen et al. (2011), which were presented in random orientation. Experiment 2 therefore was designed to approach the stimuli of Mullen et al. (2011) more closely by using Gaussian windowed patterns in random orientation but examine a significant difference between their stimuli and the stimuli used in the paper that most comprehensively demonstrated integration across a number of frequencies of RF pattern, Loffler et al. (2003). The Gaussian windows employed by Mullen et al. (2011) revealed the modulation of the contour in cosine or negative cosine phase while the sectors of modulation used by Loffler et al. (2003) were in sine phase. Experiment 2 of this study, therefore, used Gaussian windows that, on subjective inspection, revealed one and two cycles of RF4 (four cycles of modulation in 2π radians) modulation in sine phase with respect to the center of the window, but with the pattern oriented at random. The power function that described the decrease in threshold from one to two and then four cycles of modulation in an RF4 pattern was derived. Given the results of Experiment 1 and several previous studies (Bell & Badcock, 2008; Dickinson et al., 2010; Hess, Wang, & Dakin, 1999; Loffler et al., 2003), the decrease in threshold with increasing numbers of cycles of modulation should be steeper than that predicted by probability summation. 
Experiment 2: A demonstration of global integration using Gaussian windowed stimuli
Introduction
The results of Experiment 1 demonstrated that removing the unmodulated sector of a contour does not account for the decrease in threshold for detection of a single cycle of modulation. The typical rapid decrease in threshold with increase in the number of cycles of RF modulation was reproduced. The window of Experiment 1 of this study was a sector with a hard edge, whereas the window used by Mullen et al. (2011) tapered towards zero contrast and so an attempt was made to demonstrate integration of signal as integer cycles of modulation were added to stimuli with tapered edges. Mullen et al. (2011) suggest that their results are independent of maximum contrast (caption to their Figure 1, although see Ivanov & Mullen, 2012) and so in the first instance high-contrast stimuli were used to maintain consistency with Experiment 1. Application of their Equations 5 and 6 (assuming that the contrast term C(r) in Equation 6 is actually the contrast as a function of θ given by Equation 5) results in a Gaussian window on the path. Mullen et al. (2011) used standard deviations for the window of 30°, 60°, and 90° to reveal one, two, and three cycles of modulation. However, for stimuli with substantial contrast the majority of the stimulus remains visible (see Figure 3A to 3C). Although this is a subjective observation it illustrates a problem with using a mask with a gently tapered edge. In Experiment 2 we initially used a high contrast stimulus viewed through Gaussian windows in θ with smaller standard deviations of 20° and 40° (see Figure 3D and 3E), through which the pattern modulation is viewed in sine phase, rather than the cosine phase of Mullen et al. (2011); the question of whether the phase of presentation of the patterns is important is addressed in Experiment 3. These narrower windows were chosen as they subjectively reveal one and two cycles of modulation. A window with a standard deviation of 60° (see Figure 3F) was not used as it appeared to reveal the whole pattern. For the condition where all four cycles were required to be visible the Gaussian window was not applied. 
Figure 3
 
An illustration of stimuli. The left hand column of stimuli shows the stimuli created by applying Gaussian windows in θ with standard deviations of 30°, 60°, and 90° (A, B, and C, respectively) to an RF4 pattern, as used by Mullen et al. (2011) at a lower contrast. The window is centered on a point where the radius is a minimum and so the modulation appears in negative cosine phase; Mullen et al. (2011) also used patterns viewed in positive cosine phase. The column of the right shows the type of stimuli used in Experiment 2 of this study. A Gaussian window is centered on a point of zero modulation of the radius and has a standard deviation of 20°, 40°, and 60° (D, E, and F, respectively), revealing one, two, and three cycles of modulation respectively in sine phase. Of course the windows on the stimuli in the left hand column would not have appeared as broad for the low contrast (five times detection threshold) stimuli of Mullen et al. (2011) The RF4 patterns illustrated have an amplitude of 0.0588, which is the largest amplitude for an RF4 that does not have concave regions of path (see Appendix A). Experiment 2 of this study used high contrast (Weber fraction 1) and low contrast (five times detection threshold) patterns.
Figure 3
 
An illustration of stimuli. The left hand column of stimuli shows the stimuli created by applying Gaussian windows in θ with standard deviations of 30°, 60°, and 90° (A, B, and C, respectively) to an RF4 pattern, as used by Mullen et al. (2011) at a lower contrast. The window is centered on a point where the radius is a minimum and so the modulation appears in negative cosine phase; Mullen et al. (2011) also used patterns viewed in positive cosine phase. The column of the right shows the type of stimuli used in Experiment 2 of this study. A Gaussian window is centered on a point of zero modulation of the radius and has a standard deviation of 20°, 40°, and 60° (D, E, and F, respectively), revealing one, two, and three cycles of modulation respectively in sine phase. Of course the windows on the stimuli in the left hand column would not have appeared as broad for the low contrast (five times detection threshold) stimuli of Mullen et al. (2011) The RF4 patterns illustrated have an amplitude of 0.0588, which is the largest amplitude for an RF4 that does not have concave regions of path (see Appendix A). Experiment 2 of this study used high contrast (Weber fraction 1) and low contrast (five times detection threshold) patterns.
Methods
RF4 patterns with a mean radius of 2.4° and a D4 luminance section in profile (with a peak in the spatial frequency spectrum of 3 c/deg) were used. These radius and frequency spectra characteristics matched those of a subset of the stimuli used by Mullen et al. (2011). The radius used was the default radius for the experiments of Mullen et al. (2011), and the spatial frequency was towards the middle of the range they employed. The exact location of the stimulus was allowed to vary at random within a vertical and horizontal range of 20% of the pattern radius but the subjects were asked to maintain fixation on a dark marker at the centre of the screen. Maximum stimulus Weber contrast was 1. Two Gaussian windows (σ = 20° and 40°) were used to constrain the visible part of two stimuli to one and two cycles of modulation respectively. The third stimulus was the complete RF4 pattern. This allowed for two doublings in the number of cycles present. The Gaussian windows were centered on a point of zero modulation in the pattern and so the positive and negative modulation was represented symmetrically; the patterns were, therefore, viewed in sine phase rather than the cosine phase used by Mullen et al. (2011). The two windowed stimuli used are shown in Figure 3D and 3E. Also shown is a stimulus with a Gaussian window defined by σ = 60° (Figure 3F). This stimulus was not used as all four cycles of modulation were clearly visible at high contrast. This illustration gives the appearance that the proportion of the stimulus visible increases nonlinearly with increase in σ. This impression may be a result of collinear facilitation of path elements (Li & Gilbert, 2002), another potential problem with the ill-defined edges to the stimuli. A stimulus duration of 160 ms was used, typical of the stimulus durations employed previously for these types of investigation (Bell & Badcock, 2008; Loffler et al., 2003) and the same as that used in Experiment 1. The window always revealed a sector of the RF4 pattern in sine phase, but because the patterns could appear in any orientation the visible sector could be centered at any polar angle. 
An additional control study was also performed matching the stimulus properties more closely to those of Mullen et al. (2011) in terms of contrast and the temporal window on the stimulus. All spatial properties of the stimuli were identical to those previously used in the first part of Experiment 2, but a Gaussian temporal contrast window was applied. The temporal window had a standard deviation of 250 ms and was applied to the total stimulus duration of 1.1 s (11 frames of 100 ms each). Maximum contrast within the temporal window was five times the contrast detection threshold for each stimulus (the contrast detection threshold was obtained for all three stimuli individually for each observer). These thresholds were determined using a two-interval forced choice detection task in which the observers were required to report the interval that contained a test stimulus. The reference interval contained no stimulus. The intervals were indicated by an increase in the size of a fixation point. The method of constant stimuli was employed using nine contrast levels. The probability of correct response was calculated for each contrast and the Quick function fitted to the data. The threshold obtained was the contrast for 75% correct identification of the interval that contained the test stimulus. 
Results
Threshold amplitudes for detection of modulation in the high contrast stimuli with one, two, and four visible cycles of modulation are displayed in Figure 4
Figure 4
 
Amplitude thresholds (expressed as a fraction of the unmodulated radius of the pattern) for the detection of modulation in high contrast RF4 patterns with varying numbers of cycles of modulation visible. Power functions are fitted to the data for comparison with the function predicted by probability summation (a power function with an index of the negative value for the average of 1/Q for the three data points). Error bars are 95% confidence intervals.
Figure 4
 
Amplitude thresholds (expressed as a fraction of the unmodulated radius of the pattern) for the detection of modulation in high contrast RF4 patterns with varying numbers of cycles of modulation visible. Power functions are fitted to the data for comparison with the function predicted by probability summation (a power function with an index of the negative value for the average of 1/Q for the three data points). Error bars are 95% confidence intervals.
Power functions were fitted to the data of the four observers. The indices of the power functions for the four observers were −0.97 for ED, −0.77 for KW, −0.85 for RO, and −0.63 for VB. These indices suggest a rapid decrease in threshold with increasing number of cycles of modulation, implying integration of information. A decrease in threshold was, however, expected with increasing number of cycles simply due to the improving probability of detection of a single cycle. This effect is known as probability summation and integration of information is typically only claimed if the magnitudes of the observed indices are greater than those predicted for probability summation (the negative reciprocal of the variable Q in the Quick function; Equation 1). The indices predicted by probability summation were −0.52 for ED, −0.46 for KW, −0.60 for RO, and −0.48 for VB. A paired t-test shows that the observed index was steeper than the prediction of probability summation for the group, t(3) = 4.601, p = 0.0097 (one tailed). A one-tailed test was used as the direction of the predicted effect was known. 
The results of the low contrast (five times contrast detection threshold) control experiment are presented in Figure 5. The rate of decrease of threshold with increasing number of cycles of modulation was, again, faster than that predicted by probability summation. The probability summation prediction of −0.44 for the index of the power function is not contained by the upper and lower bounds of the 95% confidence interval (−0.67 to −1.10) in the fit to the data of the two observers when treated as a group. 
Figure 5
 
Amplitude thresholds for the detection of modulation in low contrast (five times threshold contrast for detection) RF4 patterns with varying numbers of cycles of modulation visible. Power functions are again fitted to the data (solid line) for comparison with the functions predicted by probability summation (dashed red line). Functions are fitted to the data of the two individual observers and the combined data treated as a group. Error bars are 95% confidence intervals. In the bottom graph, where the data of the two observers are fitted by a single function, the upper and lower bounds of the 95% confidence interval in the index of the fitted power function are plotted as dashed black lines. The index of the probability summation prediction for the combined data is not contained by the 95% confidence interval.
Figure 5
 
Amplitude thresholds for the detection of modulation in low contrast (five times threshold contrast for detection) RF4 patterns with varying numbers of cycles of modulation visible. Power functions are again fitted to the data (solid line) for comparison with the functions predicted by probability summation (dashed red line). Functions are fitted to the data of the two individual observers and the combined data treated as a group. Error bars are 95% confidence intervals. In the bottom graph, where the data of the two observers are fitted by a single function, the upper and lower bounds of the 95% confidence interval in the index of the fitted power function are plotted as dashed black lines. The index of the probability summation prediction for the combined data is not contained by the 95% confidence interval.
Conclusions
Experiment 2 used RF4 patterns that had the same size and spatial frequency content as those used by Mullen et al. (2011), but the patterns were viewed in sine rather than cosine phase. The main body of the experiment used high-contrast stimuli, but a control experiment replicated the results for low-contrast stimuli. Stimuli that had fewer than the requisite number of visible cycles of modulation were incomplete paths created by windowing sectors of the complete pattern. The contrast of the sector was a Gaussian function of θ. The rate of change in threshold observed as the number of cycles of modulation of a specific period was increased was, however, consistent with the integration of information across cycles. These results conflict with the comparable results of Mullen et al. (2011), reported in Figure 3 of their paper, which showed no evidence of integration of information. 
The high contrast part of Experiment 2 of this study used subjectively chosen narrower windows than the Mullen et al. study. It might be argued that this resulted in less than a single cycle being visible for the one cycle condition. However, the contrast used was approximately 25 times the Weber contrast detection threshold for the single cycle for observer ED and approximately 50 times for KT. The full width of the window on a single cycle at the contrast detection threshold for these stimuli could be shown to be 101° for ED and 112° for KT (the width at five times detection threshold is 72° for this study and 108° for Mullen et al., 2011). The high contrast stimuli of this study used window widths that approximated those used for the low-contrast stimuli of Mullen et al. (2011) and, therefore, at least one cycle of modulation was visible for the nominal single cycle condition. 
The remaining fundamental difference between the stimuli of Experiment 2 of this study and those used by Mullen et al. (2011) pertains to the phase of the visible sector of the pattern with respect to the point of maximum contrast. The contrast envelope used in this study was centered on a zero crossing of the modulation (sine phase), while that used by Mullen et al. (2011) was centered on a point of maximum or minimum curvature (cosine phase). Experiment 3 examined this difference between the two studies. 
Experiment 3: A sector of RF pattern viewed in cosine phase contains a local cue more salient than the difference from a circular arc
Introduction
Experiment 2 revealed the integration of modulation information around a path that is typically observed in low frequency RF patterns, but on an incomplete path. This result differed from the comparable experiment of Mullen et al. (2011), and the remaining fundamental difference between the stimuli used in this study and those used by Mullen et al. (2011) was the phase in which the incomplete patterns were viewed. Experiment 3 measured discrimination thresholds for RF4 patterns viewed in negative cosine phase at five times contrast detection threshold (the same temporal window for contrast was used as for the low contrast conditions of Experiment 2). To avoid the assumption that particular widths of window correspond to a certain number of cycles of modulation for these stimuli, the parameter chosen as the independent variable for this experiment was the standard deviation of the Gaussian window. Window widths of 20°, 30°, 40°, 60°, and 90° were used, incorporating the window widths used in Experiment 2 of this study and those used by Mullen et al. (2011). The full widths (between the points at which the pattern falls below contrast detection threshold at either side of the contrast peak) of these windows at five times contrast detection threshold were 72°, 108°, 144°, 215°, and 323°, respectively. Because the incomplete paths viewed in cosine phase were symmetrical about the point of maximum contrast, modulation would produce an increase in curvature (in the plane) at that point for patterns viewed in positive cosine phase and a decrease in curvature for patterns in negative cosine phase. This potentially provided a curvature (or angle) cue that was signed. It is possible that observers were more sensitive to a directional curvature cue at this point than to the modulation from circular. Discrimination thresholds were, therefore, also measured in parallel for patterns viewed in sine, cosine, and negative cosine phase to render this cue ineffective. If the observer knows the phase of the test pattern at the point of maximum contrast (negative cosine phase for example) then the test pattern can be discriminated from the circular pattern on the basis of absolute curvature (the least curved sector locally). If the pattern might be in negative or positive cosine phase, or indeed sine phase, on a trial by trial basis then the discrimination must be made on the basis of shape with respect to circular. 
Methods
The methods used to measure the thresholds for the stimuli viewed in negative cosine phase were the same as those employed in Experiment 2 for the stimuli viewed in sine phase. For the thresholds for sine, cosine, and negative cosine phase stimuli measured in parallel the window widths were blocked, but each block comprised stimuli of all phase types presented in random order. Each block of trials comprised 10 examples of test stimuli at nine amplitudes of modulation for each of the three stimulus types. Six blocks of trials were performed for each window width resulting in 540 responses for each psychometric function. After the responses to each stimulus type were collated the treatment of the data was the same as for Experiment 2
Results
The results of this experiment are presented in Figure 6. Thresholds are plotted against the standard deviation of the Gaussian window width. 
Figure 6
 
Thresholds for RF patterns viewed in negative and positive cosine phase and in sine phase, derived independently and in parallel. The graphs in the left column present the thresholds for incomplete RF patterns viewed in negative cosine phase as a function of window width. The solid red line is the function fitted to the data presented in Figure 6. The graphs in the right column plot thresholds for incomplete RF patterns viewed in negative and positive cosine phase and in sine phase, when interleaved during measurement, against iwindow width. The solid red line in this column is the power function fitted to the thresholds for patterns viewed in sine phase. The dashed red line is the probability summation prediction. The threshold for a complete RF4 pattern is indicated by the horizontal dotted black line.
Figure 6
 
Thresholds for RF patterns viewed in negative and positive cosine phase and in sine phase, derived independently and in parallel. The graphs in the left column present the thresholds for incomplete RF patterns viewed in negative cosine phase as a function of window width. The solid red line is the function fitted to the data presented in Figure 6. The graphs in the right column plot thresholds for incomplete RF patterns viewed in negative and positive cosine phase and in sine phase, when interleaved during measurement, against iwindow width. The solid red line in this column is the power function fitted to the thresholds for patterns viewed in sine phase. The dashed red line is the probability summation prediction. The threshold for a complete RF4 pattern is indicated by the horizontal dotted black line.
When measured independently (left column of Figure 6), thresholds for incomplete RF patterns viewed in negative cosine phase were in general lower than the corresponding thresholds measured in parallel (right hand column of Figure 6) for patterns viewed in positive cosine phase, negative cosine phase, and sine phase. Moreover, the rate of decrease of threshold with increase in window width was substantially smaller than that measured for patterns viewed in sine phase using the same experimental paradigm. When the thresholds for patterns viewed in cosine and sine phase were measured in parallel, however, the rate of decrease in threshold with increase in window width was steeper. For both observers the functions describing threshold versus window width for the patterns viewed in sine phase (solid red lines) were steeper than the probability summation predictions (dashed red lines). The data for the patterns viewed in positive and negative cosine phase when conditions were interleaved were noisy and enhancement in performance in discrimination of one cosine phase pattern type resulted in a decrement in performance for the other for any particular window width condition. As can be seen from the graphs representing threshold as a function of window width measured independently for patterns in cosine phase, there is still some evidence for change in threshold with window width (thresholds for ED changed markedly while those for KT approximated the results for Mullen et al. [2011] other than in the complete RF condition). This still contrasts somewhat with the data of Mullen et al. (2011), which showed little change with window width. The rate of change of threshold with window width for ED was consistent with a probability summation prediction with the probability of detection of a single cycle of modulation increasing as the number of cycles of modulation was increased. For observer KT the relationship was quite flat and similar to the data of Mullen et al. other than the last point. This might have been due to a tendency for this observer (and potentially those in the study of Mullen et al.) to attend predominantly to the single cycle of modulation (in cosine phase) in the region of highest contrast. This cannot, however, explain the similar threshold for the complete RF pattern in the Mullen et al. study. An alternative explanation for the absence of a change in the threshold as window width was varied in the data of Mullen et al. (2011) is that the window widths chosen revealed a substantial proportion of the pattern in conditions assumed to be constraining visibility of the stimulus to particular numbers of cycles. That is, a substantial proportion of the stimulus might have been visible for a single cycle. 
Conclusions
The results of Experiment 3 indicate that Gaussian windowed RF stimuli viewed in cosine phase such that only a limited number of cycles of modulation are visible can be discriminated from similarly windowed circular arcs at a lower threshold than the same pattern viewed in sine phase. As suggested by Mullen et al. (2011) this indicates that a local cue exists on a modulated arc that is more salient than the local cue that is integrated across cycles of modulation by the global shape processing mechanism. Mullen et al. (2011), therefore, were justified in using cosine phase stimuli experimentally and in their modeling of this local cue, but it must be recognized that use of cosine phase stimuli with specified sign of modulation masks the global integration of signal around RF stimuli until a sufficient number of cycles of modulation is present to render the global cue the most salient. If, however, the task is constrained to be discrimination between a circular arc and a stimulus deformed from circular, as it is when the test stimulus can potentially be modulated in negative or positive cosine phase or sine phase in any one trial, then the global cue must be used and evidence for integration of information is restored. This can be understood by considering that a modulated arc viewed through a Gaussian window in negative cosine phase will always be straighter at the point of maximum contrast than a circular arc of the same radius. Thus, while Mullen et al. (2011) have identified a salient local cue, use of this easily located cue simply allows the observer to detect modulation of the path without recourse to the mechanism responsible for the global analysis of shape. Use of this local cue, however, is dependent on knowledge of the position of the peak or trough of modulation and the sign of the amplitude of modulation at that point. This knowledge is not available in the studies that have sought to demonstrate integration of shape information around a path. Typically such studies have no contrast cue to the phase of the pattern and the phase of the pattern is randomized. 
For patterns viewed in sine phase the decrease in threshold with increase in number of cycles of modulation is smooth and monotonic and conforms to a power function. Evidence for integration of information around the pattern, therefore, implies that the cue to deformation of the pattern in all conditions, one cycle to complete RF pattern, is object centric (the threshold for the detection of a single cycle is consistent with extrapolation of the function describing thresholds vs. number of cycles of modulation for multiple cycles). Threshold amplitudes for single cycles of RF modulation of different frequencies are different (Loffler et al., 2003), thus we can say that single cycles of different frequencies of RF modulation of the same amplitude are not equally salient. This effectively rules out amplitude of deformation from circular as the local cue to deformation. Another local, but object centric, cue that is present in a single cycle of modulation is the maximum angular deviation from the circular arc (for the small amplitudes of deformation at threshold for detection this is close to the zero crossing of the sinusoid). This is a parameter that varies systematically with the frequency of the RF pattern and so we might expect the thresholds for discrimination of RF patterns to vary linearly with frequency. This possibility is investigated in Experiment 4
Experiment 4: Discrimination thresholds for RF patterns are inversely proportional to frequency
Introduction
RF patterns are defined by a deformation from circular by a sinusoidal modulation of radius (see Equation 2). Experiments 1, 2, and 3 of this study confirm that the shape information inherent in the deformation of RF patterns integrates around that pattern. Single cycles of RF modulation of different frequencies of modulation are not equally salient at the same amplitude of modulation (Loffler et al., 2003). This experiment examined if a different cue to shape that is present in a single cycle of modulation, the maximum orientation difference from circular, was equated at the threshold for detection for different frequencies of sinusoidal modulation. The difference in local orientation between an RF pattern in sine phase and a circle is given by Equation 3 below. where θ is the polar angle, A is the amplitude of modulation, ω the frequency of modulation, R0 is the unmodulated radius, and r is the equation describing the radius of the pattern as a function of theta (Equation 2). 
For small amplitude RF patterns, such as those used in detection threshold experiments (see General discussion), the ratio r/R0 approximates 1 and the gradient of the pattern with respect to circular is at a maximum near the zero crossing of the sinusoidal modulation. The gradient (rate of change of r with θ) of the path relative to an unmodulated circle at a zero crossing of the sinusoid is given by the differential of the modulating sinusoid.  
The gradient at the point is, therefore, proportional to the frequency of modulation (ω) and is at a maximum when θ is zero. The gradient decreases with the reciprocal of the frequency of modulation. 
It is clear, therefore, that if threshold is determined by the maximum local gradient with respect to the circle (or the angle the tangent to the path makes with the circle for the small angle approximation appropriate for low amplitudes of modulation), the threshold amplitude for detection of modulation should decrease linearly with frequency. This experiment measured the thresholds for detection of RF2, RF3, RF4, and RF6 patterns with single and multiple cycles of modulation, including patterns with modulation around the whole 2π radians. 
Methods
The radii of the RF patterns as a function of theta, r(θ), were defined as follows, where R0 is the unmodulated radius, ω is the frequency of the modulation (cycles per 2π radians), θ is the polar angle, φ is the phase of the modulation (phase was randomized) and A is the amplitude of modulation (expressed as a proportion of the unmodulated radius). In the cases where a single cycle of modulation was present the sinusoid was replaced with a D1 (first derivative of a Gaussian) function matched in amplitude and maximum gradient with the single cycle of sinusoidal modulation it replaces (Loffler et al., 2003). Patterns with more than one cycle, but fewer than the full complement of cycles of modulation required to modulate 2π radians, had half cycles at the end of the chain of cycles of modulation replaced with the approximating D1 function (following Loffler et al., 2003). The patterns had a radius of 1° and a D4 luminance profile in section with a peak spatial frequency of 8 c/deg. Maximum Weber contrast was 1. Again, a stimulus duration of 160 ms was used. 
Results
The results are displayed in Figure 7
Figure 7
 
Thresholds for the detection of deformation of RF patterns as a function of number of cycles of modulation. The graphs in the left hand column plot the raw thresholds for RF2, RF3, RF4, and RF6 patterns for three observers. In the graphs on the right the data have been normalized for the maximum local gradient of the patterns, relative to circular, for any small amplitude of modulation (the thresholds have been scaled by the ratio RFn:RF3). In the right hand column the thresholds are coincident at each value for number of cycles of modulation. The solid black lines in this column are the functions that describe the normalized threshold as a function of number of cycles of modulation for each observer. The dashed lines are the predictions for probability summation. Error bars are 95% confidence intervals.
Figure 7
 
Thresholds for the detection of deformation of RF patterns as a function of number of cycles of modulation. The graphs in the left hand column plot the raw thresholds for RF2, RF3, RF4, and RF6 patterns for three observers. In the graphs on the right the data have been normalized for the maximum local gradient of the patterns, relative to circular, for any small amplitude of modulation (the thresholds have been scaled by the ratio RFn:RF3). In the right hand column the thresholds are coincident at each value for number of cycles of modulation. The solid black lines in this column are the functions that describe the normalized threshold as a function of number of cycles of modulation for each observer. The dashed lines are the predictions for probability summation. Error bars are 95% confidence intervals.
In the column of graphs on the left of Figure 7, the thresholds for detection of modulation are plotted for the four radial frequencies of modulation tested. The column of graphs on the right plots the same data transformed to account for the differences in maximum gradient (relative to a circle), for any small amplitude of modulation, normalized to the RF3 data (for example the RF4 thresholds are multiplied by 4/3). After normalization the functions describing gradient at a zero crossing versus number of cycles of modulation were coincident. The indices of the power functions describing the decrease of threshold with increasing cycles of modulation for the normalized data were −0.87 ± 0.12 (95% CI) for observer ED, −0.77 ± 0.21 for JM, and −0.72 ± 0.16 for TM. These indices compared with predictions of probability summation of −0.51 for ED, −0.55 for JM, and −0.53 for TM. The 95% confidence intervals do not contain the probability summation prediction for any of the three observers and a one-tailed paired t-test also demonstrates that the slopes of the fitted functions were steeper than those predicted by probability summation, t(2) = 4.899, p = 0.0196. 
Conclusions
The results demonstrate that the information present is integrated across the repeated cycles of modulation; i.e., global integration. In addition to this demonstration of integration the normalized data in the right hand column shows that the thresholds for different frequencies of RF pattern with the same number of cycles of modulation scale with frequency. One quantity present in a single cycle of modulation that scales linearly with frequency is the maximum orientation difference from circular. 
General discussion
Mullen et al. (2011) argued that thresholds for the detection of modulation in RF patterns can be predicted from local orientation change maxima introduced by the modulation of the radius of the patterns. In describing where these maxima occur they introduced the concept of a point of inflection on the path. Cartesian points of inflection can occur on the path when the amplitude is sufficient [if A > 1/(1 + ω2); Mullen et al. suggested that they occur when A > π/2(1+ω2) but see Appendix A]. For an RF4 pattern, Cartesian points of inflection (points where the rate of change of gradient of the path is zero and curvature changes sign) occur when A > 0.0588 (or 5.88% radius amplitude modulation (RAM). There were no thresholds in the paper that exceeded this amplitude. Rather than using a Cartesian point of inflection, an intrinsically local cue, Mullen et al. (2011) defined a point of inflection on an RF pattern as occurring at a point of maximum orientation deviation of the path from circular (although they actually calculated the orientation difference between the local radius and the normal to the path). Figure 8 illustrates how the maximum local orientation difference from circular (and approximations to this quantity) changed with the amplitude of an RF4 pattern 
Figure 8
 
Maximum orientation differences from circular (and different approximations) as a function of the amplitude (expressed as a proportion of the radius) of an RF4 pattern. The dashed blue line is derived numerically from Equation 3 and represents that maximum angle the tangent to the path makes with the unmodulated circle. The dashed red line is the angle complementary to that of equation A6.2 of Mullen et al. (2011), which represents the maximum angle the tangent makes with the local radius. The red and blue lines are coincident and represent the genuine maximum angular difference from circular. The black line represents the angle a tangent at a zero crossing of the sinusoid makes with a circle. The dashed green line is the approximation of the maximum angle the tangent makes with the unmodulated circle which is used in the modeling of Mullen et al. (2011).
Figure 8
 
Maximum orientation differences from circular (and different approximations) as a function of the amplitude (expressed as a proportion of the radius) of an RF4 pattern. The dashed blue line is derived numerically from Equation 3 and represents that maximum angle the tangent to the path makes with the unmodulated circle. The dashed red line is the angle complementary to that of equation A6.2 of Mullen et al. (2011), which represents the maximum angle the tangent makes with the local radius. The red and blue lines are coincident and represent the genuine maximum angular difference from circular. The black line represents the angle a tangent at a zero crossing of the sinusoid makes with a circle. The dashed green line is the approximation of the maximum angle the tangent makes with the unmodulated circle which is used in the modeling of Mullen et al. (2011).
The blue and red dashed lines represent the genuine maximum angular, or orientation, difference between the RF pattern and the unmodulated circle. The orientation difference from circular at any point of an RF pattern in sine phase is given by Equation 3. The function described by Equation 3 was sampled over the range 0 to 2π radians for each value of amplitude (A) sampled (in the range 0 to 1), to derive the maximum value of α for that amplitude. The numerically derived data are represented by the dashed blue line. 
The data represented by the dashed red line were calculated from which is the angle complementary to that given by equation A6.2 in Mullen et al. (2011). Their equation A6.2 is the analytically derived parametric equation that describes the angle between the local radius and the path. The dashed blue and red lines should therefore be coincident, and they are. 
The angular difference at the zero crossing (at θ = 0) can also be derived from Equation 3. The quantity r becomes R0 and cos(ωθ) is 1. Equation 3 then reduces to α = arctan () for any particular value for A. The black line describes the angle at a zero crossing of the modulation as a function of amplitude and is given by It is readily apparent that this describes the maximum orientation difference between a straight line and a sinusoid. 
The green dashed line is given by the function which is derived from equation A9 of Mullen et al. (2011). Mullen et al. used their equation to calculate model threshold amplitudes for RF patterns based on a minimum angular difference in a single cycle of an RF pattern. The equation has been rearranged to give the angular difference as a function of amplitude for an RF4 pattern. In the model used by Mullen et al. (2011) threshold is determined by four times the angle determined by Equation 8 because the angle occurs twice in the definition of a peak and twice in a trough. The angle is seen to approximate the other functions for the small amplitudes of modulation seen for the detection thresholds for RF patterns. 
Because all of the functions converge for low amplitudes, both approximations were valid for experiments at detection threshold. The approximation using the angular difference at the zero crossing did, though, have the important merits of being easy to visualize and having precedent in the literature. It also allowed modeling of sensitivity to line and RF stimuli to be easily compared. The modeling of Mullen et al. (2011) can be approximated more accurately as, for low amplitudes of modulation, thresholds for the detection of modulation in a line and an RF pattern as the frequency of modulation varies are given by constant angles. These angles are 2arctan() and 4arctan(), respectively. In both cases the amplitude at threshold is inversely proportional to the frequency of modulation. 
In Experiment 1 of this study observers did not have higher thresholds for single cycles on complete contours than on partial contours. This result is contrary to the hypothesis of Mullen et al. (2011) that the higher thresholds previously reported for patterns with incomplete RF modulation in comparison with patterns with complete modulation are due to interference by the circular sectors of the incomplete patterns. Furthermore the thresholds for detection of a complete, three-cycle RF3 pattern were lower than those for detection of a single cycle of modulation, suggesting global integration. 
Experiment 2 demonstrated that integration of information across cycles is observed for Gaussian windowed RF patterns if the pattern is viewed in sine phase. This result was consistent with the results of other studies that employed closed (Bell & Badcock, 2008; Loffler et al., 2003) or sampled (Dickinson et al., 2010) contours. 
Experiment 3 revealed that if patterns are viewed in negative cosine phase, discrimination of a modulated arc from a circular arc is achieved at lower thresholds than for test patterns viewed in sine phase. This suggests that the most probable explanation for the differences between the results of Mullen et al. (2011) and previous studies that have reported global summation of shape information is that a more salient local cue is present under these circumstances and the position of that cue is identified by having higher contrast than the rest of the contour in the partial contour figures. The demonstration of integration in all three types of stimulus, when the thresholds are determined in parallel and the observer does not know which phase of modulation to expect in each particular trial, indicates that the local cue is no longer sufficient to discriminate between the modulated pattern and a circular arc. Most probably the local cue allows the judgment of which of the two arcs is most or least curved at the point of greatest contrast. This information is not adequate to determine which of the two stimuli of a test is modulated when the possibility exists that the modulation could be in negative or positive cosine phase. 
Experiment 4 showed that integration of information across cycles can also be inferred from the reduction in threshold across different frequency RF patterns. When only a single cycle of modulation is present the threshold for detection of modulation is indeed defined by the maximum gradient (or angular difference from the unmodulated circle). However, the index of the power function that describes the decrease in threshold as the number of cycles of modulation increases after normalizing for maximum local orientation difference from circular in different RF patterns is again significantly greater in magnitude than the prediction of probability summation. This again indicates that replication of the same information in the path results in a rate of threshold reduction that is consistent with integration of that information in the process of detection. 
Overall the conclusion of this study coincides with most previous investigations with these patterns. That is, information is globally integrated to determine threshold when low radial frequency patterns are employed. Performance using the information content of a single cycle cannot explain the data and therefore, with appropriate stimulus choices these patterns are suitable for studying the properties of global shape-integration mechanisms. 
Acknowledgments
This research was supported by Australian Research Council Grants DP0666206, DP1097603, and DP110104553 to DRB. 
Commercial relationships: none. 
Corresponding author: J Edwin Dickinson. 
Email: edwin.dickinson@uwa.edu.au 
Address: School of Psychology, University of Western Australia, Crawley, Perth, WA, Australia. 
References
Bell J. Badcock D. R. (2008). Luminance and contrast cues are integrated in global shape detection with contours. Vision Research, 48(21), 2336–2344. [CrossRef] [PubMed]
Dickinson J. E. Han L. Bell J. Badcock D. R. (2010). Local motion effects on form in radial frequency patterns. Journal of Vision, 10(3):20, 1–15, http://www.journalofvision.org/content/10/3/20, doi:10.1167/10.3.20. [PubMed] [Article] [CrossRef] [PubMed]
Felleman D. J. Van Essen D. C. (1991). Distributed hierarchical processing in the primate cerebral cortex. Cerebral Cortex, 1(1), 1–47. [CrossRef] [PubMed]
Ferster D. Miller K. D. (2000). Neural mechanisms of orientation selectivity in the visual cortex. Annual Review of Neuroscience, 23(1), 441–471. [CrossRef] [PubMed]
Field D. J. Hayes A. Hess R. F. (1993). Contour integration by the human visual system: Evidence for a local “association field”. Vision Research, 33(2), 173–193. [CrossRef] [PubMed]
Hess R. F. Wang Y.-Z. Dakin S. C. (1999). Are judgements of circularity local or global? Vision Research, 39(26), 4354–4360. [CrossRef] [PubMed]
Hubel D. H. Wiesel T. N. (1959). Receptive fields of single neurones in the cat's striate cortex. Journal of Physiology, 148, 574–591. [CrossRef] [PubMed]
Hubel D. H. Wiesel T. N. (1968). Receptive fields and functional architecture of monkey striate cortex. Journal of Physiology, 195(1), 215–243. [CrossRef] [PubMed]
Ivanov I. V. Mullen K. T. (2012). The role of local features in shape discrimination of contour- and surface-defined radial frequency patterns at low contrast. Vision Research, 52(1), 1–10. [CrossRef] [PubMed]
Lennie P. (1998). Single units and visual cortical organization. Perception, 27(8), 889–935. [CrossRef] [PubMed]
Li W. Gilbert C. D. (2002). Global contour saliency and local colinear interactions. Journal of Neurophysiology, 88(5), 2846–2856. [CrossRef] [PubMed]
Loffler G. Wilson H. R. Wilkinson F. (2003). Local and global contributions to shape discrimination. Vision Research, 43(5), 519–530. [CrossRef] [PubMed]
Mullen K. T. Beaudot W. H. A. (2002). Comparison of color and luminance vision on a global shape discrimination task. Vision Research, 42(5), 565–575. [CrossRef] [PubMed]
Mullen K. T. Beaudot W. H. A. Ivanov I. V. (2011). Evidence that global processing does not limit thresholds for RF shape discrimination. Journal of Vision, 11(3):6, 1–21, http://www.journalofvision.org/content/11/3/6, doi:10.1167/11.3.6. [PubMed] [Article]. [CrossRef] [PubMed]
Pasupathy A. Connor C. E. (1999). Responses to contour features in macaque area V4. Journal of Neurophysiology, 82(5), 2490–2502. [PubMed]
Pasupathy A. Connor C. E. (2001). Shape representation in area V4: Position-specific tuning for boundary conformation. Journal of Neurophysiology, 86(5), 2505–2519. [PubMed]
Pasupathy A. Connor C. E. (2002). Population coding of shape in area V4. Nature Neuroscience, 5(12), 1332–1338. [CrossRef] [PubMed]
Poirier F. J. A. M. Wilson H. R. (2006). A biologically plausible model of human radial frequency perception. Vision Research, 46(15), 2443–2455. [CrossRef] [PubMed]
Quick R. F.Jr. (1974). A vector-magnitude model of contrast detection. Kybernetik, 16(2), 65–67. [CrossRef] [PubMed]
Reid R. C. Alonso J. M. (1995). Specificity of monosynaptic connections from thalamus to visual cortex. Nature, 378(6554), 281–284. [CrossRef] [PubMed]
Wilson H. R. (1980). A transducer function for threshold and suprathreshold human vision. Biological Cybernetics, 38(3), 171–178. [CrossRef] [PubMed]
Appendix A
Above particular amplitudes of modulation RF patterns develop Cartesian points of inflection (points at which the gradient of the path is not changing in the plane defined by the X and Y axes). The amplitude of modulation at which this happens depends on the frequency of modulation of the pattern. The parametric equation that defines this amplitude is derived below. Figure A1 shows an RF3 pattern with negative cosine modulation (this solution generalizes to all frequencies of RF pattern). 
For a pattern with a mean radius of 1 the radius r at any point on the RF pattern is given by where A is the amplitude of modulation, ω is the frequency of modulation (cycles in 2π radians), and θ is the angle the radius makes with the x axis. 
The point of inflection first appears at a minimum of the modulating function (including at θ = 0). A point of inflection at this location occurs when the second derivative of x with respect to θ is zero.  At θ = 0, sinθ = 0, sin(ωθ) = 0, cosθ = 1, cos(ωθ) = 1, therefore following substitution Therefore at the point of inflection  
Figure A1
 
An RF3 pattern with negative cosine modulation.
Figure A1
 
An RF3 pattern with negative cosine modulation.
Figure 1
 
Examples of the three types of stimuli used in Experiment 1. (A) and (B) have one cycle of modulation and (C) has three. (A) has a semicircular sector of the pattern removed and is, therefore, an open contour.
Figure 1
 
Examples of the three types of stimuli used in Experiment 1. (A) and (B) have one cycle of modulation and (C) has three. (A) has a semicircular sector of the pattern removed and is, therefore, an open contour.
Figure 2
 
Detection thresholds for the three types of stimuli used in Experiment 1. Error bars represent 95% confidence intervals. The threshold amplitude is expressed as a fraction of the unmodulated radius of the pattern and is, therefore, dimensionless. Threshold amplitudes are highest for the pattern with the circular sector removed and lowest for the RF3 pattern with three cycles of modulation.
Figure 2
 
Detection thresholds for the three types of stimuli used in Experiment 1. Error bars represent 95% confidence intervals. The threshold amplitude is expressed as a fraction of the unmodulated radius of the pattern and is, therefore, dimensionless. Threshold amplitudes are highest for the pattern with the circular sector removed and lowest for the RF3 pattern with three cycles of modulation.
Figure 3
 
An illustration of stimuli. The left hand column of stimuli shows the stimuli created by applying Gaussian windows in θ with standard deviations of 30°, 60°, and 90° (A, B, and C, respectively) to an RF4 pattern, as used by Mullen et al. (2011) at a lower contrast. The window is centered on a point where the radius is a minimum and so the modulation appears in negative cosine phase; Mullen et al. (2011) also used patterns viewed in positive cosine phase. The column of the right shows the type of stimuli used in Experiment 2 of this study. A Gaussian window is centered on a point of zero modulation of the radius and has a standard deviation of 20°, 40°, and 60° (D, E, and F, respectively), revealing one, two, and three cycles of modulation respectively in sine phase. Of course the windows on the stimuli in the left hand column would not have appeared as broad for the low contrast (five times detection threshold) stimuli of Mullen et al. (2011) The RF4 patterns illustrated have an amplitude of 0.0588, which is the largest amplitude for an RF4 that does not have concave regions of path (see Appendix A). Experiment 2 of this study used high contrast (Weber fraction 1) and low contrast (five times detection threshold) patterns.
Figure 3
 
An illustration of stimuli. The left hand column of stimuli shows the stimuli created by applying Gaussian windows in θ with standard deviations of 30°, 60°, and 90° (A, B, and C, respectively) to an RF4 pattern, as used by Mullen et al. (2011) at a lower contrast. The window is centered on a point where the radius is a minimum and so the modulation appears in negative cosine phase; Mullen et al. (2011) also used patterns viewed in positive cosine phase. The column of the right shows the type of stimuli used in Experiment 2 of this study. A Gaussian window is centered on a point of zero modulation of the radius and has a standard deviation of 20°, 40°, and 60° (D, E, and F, respectively), revealing one, two, and three cycles of modulation respectively in sine phase. Of course the windows on the stimuli in the left hand column would not have appeared as broad for the low contrast (five times detection threshold) stimuli of Mullen et al. (2011) The RF4 patterns illustrated have an amplitude of 0.0588, which is the largest amplitude for an RF4 that does not have concave regions of path (see Appendix A). Experiment 2 of this study used high contrast (Weber fraction 1) and low contrast (five times detection threshold) patterns.
Figure 4
 
Amplitude thresholds (expressed as a fraction of the unmodulated radius of the pattern) for the detection of modulation in high contrast RF4 patterns with varying numbers of cycles of modulation visible. Power functions are fitted to the data for comparison with the function predicted by probability summation (a power function with an index of the negative value for the average of 1/Q for the three data points). Error bars are 95% confidence intervals.
Figure 4
 
Amplitude thresholds (expressed as a fraction of the unmodulated radius of the pattern) for the detection of modulation in high contrast RF4 patterns with varying numbers of cycles of modulation visible. Power functions are fitted to the data for comparison with the function predicted by probability summation (a power function with an index of the negative value for the average of 1/Q for the three data points). Error bars are 95% confidence intervals.
Figure 5
 
Amplitude thresholds for the detection of modulation in low contrast (five times threshold contrast for detection) RF4 patterns with varying numbers of cycles of modulation visible. Power functions are again fitted to the data (solid line) for comparison with the functions predicted by probability summation (dashed red line). Functions are fitted to the data of the two individual observers and the combined data treated as a group. Error bars are 95% confidence intervals. In the bottom graph, where the data of the two observers are fitted by a single function, the upper and lower bounds of the 95% confidence interval in the index of the fitted power function are plotted as dashed black lines. The index of the probability summation prediction for the combined data is not contained by the 95% confidence interval.
Figure 5
 
Amplitude thresholds for the detection of modulation in low contrast (five times threshold contrast for detection) RF4 patterns with varying numbers of cycles of modulation visible. Power functions are again fitted to the data (solid line) for comparison with the functions predicted by probability summation (dashed red line). Functions are fitted to the data of the two individual observers and the combined data treated as a group. Error bars are 95% confidence intervals. In the bottom graph, where the data of the two observers are fitted by a single function, the upper and lower bounds of the 95% confidence interval in the index of the fitted power function are plotted as dashed black lines. The index of the probability summation prediction for the combined data is not contained by the 95% confidence interval.
Figure 6
 
Thresholds for RF patterns viewed in negative and positive cosine phase and in sine phase, derived independently and in parallel. The graphs in the left column present the thresholds for incomplete RF patterns viewed in negative cosine phase as a function of window width. The solid red line is the function fitted to the data presented in Figure 6. The graphs in the right column plot thresholds for incomplete RF patterns viewed in negative and positive cosine phase and in sine phase, when interleaved during measurement, against iwindow width. The solid red line in this column is the power function fitted to the thresholds for patterns viewed in sine phase. The dashed red line is the probability summation prediction. The threshold for a complete RF4 pattern is indicated by the horizontal dotted black line.
Figure 6
 
Thresholds for RF patterns viewed in negative and positive cosine phase and in sine phase, derived independently and in parallel. The graphs in the left column present the thresholds for incomplete RF patterns viewed in negative cosine phase as a function of window width. The solid red line is the function fitted to the data presented in Figure 6. The graphs in the right column plot thresholds for incomplete RF patterns viewed in negative and positive cosine phase and in sine phase, when interleaved during measurement, against iwindow width. The solid red line in this column is the power function fitted to the thresholds for patterns viewed in sine phase. The dashed red line is the probability summation prediction. The threshold for a complete RF4 pattern is indicated by the horizontal dotted black line.
Figure 7
 
Thresholds for the detection of deformation of RF patterns as a function of number of cycles of modulation. The graphs in the left hand column plot the raw thresholds for RF2, RF3, RF4, and RF6 patterns for three observers. In the graphs on the right the data have been normalized for the maximum local gradient of the patterns, relative to circular, for any small amplitude of modulation (the thresholds have been scaled by the ratio RFn:RF3). In the right hand column the thresholds are coincident at each value for number of cycles of modulation. The solid black lines in this column are the functions that describe the normalized threshold as a function of number of cycles of modulation for each observer. The dashed lines are the predictions for probability summation. Error bars are 95% confidence intervals.
Figure 7
 
Thresholds for the detection of deformation of RF patterns as a function of number of cycles of modulation. The graphs in the left hand column plot the raw thresholds for RF2, RF3, RF4, and RF6 patterns for three observers. In the graphs on the right the data have been normalized for the maximum local gradient of the patterns, relative to circular, for any small amplitude of modulation (the thresholds have been scaled by the ratio RFn:RF3). In the right hand column the thresholds are coincident at each value for number of cycles of modulation. The solid black lines in this column are the functions that describe the normalized threshold as a function of number of cycles of modulation for each observer. The dashed lines are the predictions for probability summation. Error bars are 95% confidence intervals.
Figure 8
 
Maximum orientation differences from circular (and different approximations) as a function of the amplitude (expressed as a proportion of the radius) of an RF4 pattern. The dashed blue line is derived numerically from Equation 3 and represents that maximum angle the tangent to the path makes with the unmodulated circle. The dashed red line is the angle complementary to that of equation A6.2 of Mullen et al. (2011), which represents the maximum angle the tangent makes with the local radius. The red and blue lines are coincident and represent the genuine maximum angular difference from circular. The black line represents the angle a tangent at a zero crossing of the sinusoid makes with a circle. The dashed green line is the approximation of the maximum angle the tangent makes with the unmodulated circle which is used in the modeling of Mullen et al. (2011).
Figure 8
 
Maximum orientation differences from circular (and different approximations) as a function of the amplitude (expressed as a proportion of the radius) of an RF4 pattern. The dashed blue line is derived numerically from Equation 3 and represents that maximum angle the tangent to the path makes with the unmodulated circle. The dashed red line is the angle complementary to that of equation A6.2 of Mullen et al. (2011), which represents the maximum angle the tangent makes with the local radius. The red and blue lines are coincident and represent the genuine maximum angular difference from circular. The black line represents the angle a tangent at a zero crossing of the sinusoid makes with a circle. The dashed green line is the approximation of the maximum angle the tangent makes with the unmodulated circle which is used in the modeling of Mullen et al. (2011).
Figure A1
 
An RF3 pattern with negative cosine modulation.
Figure A1
 
An RF3 pattern with negative cosine modulation.
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