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Article  |   November 2019
Sensitivity to curvature deformations along closed contours
Author Affiliations
  • Michael Slugocki
    Department of Psychology, Neuroscience and Behaviour, McMaster University, Hamilton, Ontario, Canada
    slugocm@mcmaster.ca
  • Allison B. Sekuler
    Department of Psychology, Neuroscience and Behaviour, McMaster University, Hamilton, Ontario, Canada
    Rotman Research Institute, Baycrest Health Sciences, Toronto, Ontario, Canada
    Department of Psychology, University of Toronto, Toronto, Ontario, Canada
    sekuler@mcmaster.ca
  • Patrick J. Bennett
    Department of Psychology, Neuroscience and Behaviour, McMaster University, Hamilton, Ontario, Canada
    bennett@mcmaster.ca
Journal of Vision November 2019, Vol.19, 7. doi:https://doi.org/10.1167/19.13.7
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      Michael Slugocki, Allison B. Sekuler, Patrick J. Bennett; Sensitivity to curvature deformations along closed contours. Journal of Vision 2019;19(13):7. doi: https://doi.org/10.1167/19.13.7.

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Abstract

Human observers are exquisitely sensitive to curvature deformations along a circular closed contour (Wilkinson, Wilson, & Habak, 1998; Hess, Wang, & Dakin, 1999; Loffler, Wilson, & Wilkinson, 2003). Such remarkable sensitivity has been attributed to the curvature encoding scheme used by V4 neurons, which typically are assumed to be equally sensitive to curvature at all polar angles (Pasupathy & Connor, 2001, 2002; Carlson, Rasquinha, Zhang, & Connor, 2011). To test the assumption that detection thresholds for curvature deformations are invariant across polar angles, we used a novel stimulus class we call Difference of Gaussian (DoG) contours that allowed us to independently manipulate the amplitude, angular frequency, and polar angle of curvature of a closed-contour shape while measuring contour-curvature thresholds. Our results demonstrate that (a) detection thresholds were higher when observers were uncertain about the location of the curvature deformation, but on average, thresholds did not vary significantly across 24 polar angles; (b) the direction and magnitude of the oblique effect varies across individuals; (c) there is a strong association between detecting a contour deformation and identifying its location; (d) curvature detectors may serve as labeled lines.

Introduction
The retinal image produced by a visual scene is processed by a dynamic network that organizes information into increasingly complex elements of perception, such as shapes and objects (Felleman & Van, 1991; Riesenhuber & Poggio, 1999). Feedforward connections transmit information from lower to higher visual areas, and recurrent, skip, parallel, and within-area horizontal connections all contribute to the image processing capabilities of the visual system (Lamme & Roelfsema, 2000). Neurons in primary visual cortex (V1) encode local contour orientation (Hubel & Wiesel, 1959, 1968; Hubel, Wiesel, & Stryker, 1978), whereas neurons in V2 combine local orientation information to represent angles and arcs (Hegdé & Essen, 2000; Ito and Komatsu, 2004; Anzai, Peng, & Van Essen, 2007). Neurons in areas V3 and V4 respond to increasingly complex stimulus relations, such as the curvature along closed contours (Gallant, Connor, Rakshit, Lewis, & Essen, 1996; Pasupathy & Connor, 1999, 2001, 2002), and IT neurons are selective for stimuli in particular object-level categories, such as faces (Tanaka, 2003). 
For objects to be well represented, visual mechanisms must adequately encode aspects of stimuli, such as curvature extrema (Attneave, 1954), that are diagnostic for shape and or identity. However, given the uncertainty of where maximally informative regions might appear along the contour of an object, representations of curvature should be well preserved across all polar angles. Previous neurophysiological studies have demonstrated that the tuning properties of V4 neurons in macaques often are well described by models that assume curvature is sampled uniformly across polar angles (Pasupathy & Connor, 2001, 2002; Carlson et al., 2011). A prediction that follows from such an assumption is that observers are uniformly sensitive to curvature deformations at all polar angles. This prediction is surprising when one considers that many aspects of vision are relatively poorer for oblique contours compared to horizontal and vertical contours: oblique effects have been found for grating acuity (Campbell, Kulikowski, & Levinson, 1966; Teller, Morse, Borton, & Regal, 1974; Berkley, Kitterle, & Watkins, 1975), orientation discrimination (Appelle, 1972; Mansfield, 1974; Heeley, Buchanan-Smith, Cromwell, & Wright, 1997; Westheimer, 2003), motion perception (Gros, Blake, & Hiris, 1998; Westheimer, 2003; Dakin, Mareschal, & Bex, 2005), and many other visual tasks (Appelle, 1972). The oblique effect is thought to reflect differences in the both the number and response properties of visual neurons that encode contours at oblique and cardinal orientations (Bonds, 1982; Furmanski & Engel, 2000; Li et al., 2003; Wang, Ding, & Yunokuchi, 2003; Xu, Collins, Khaytin, Kaas, & Casagrande, 2006). These findings suggest that it is at least plausible that observers are not uniformly sensitive to curvature at all polar angles, but rather are more sensitive to curvature at horizontal and vertical orientations compared to oblique orientations. 
Watt and Andrews (1982) found that efficiency of curvature discrimination for oblique stimuli was at least as high as efficiency for stimuli oriented along cardinal axes. Consistent with this observation, using a curvature discrimination task Schmidtmann, Logan, Kennedy, Gordon, and Loffler (2015) also observed isotropic sensitivity in processing curvature information across four cardinal orientations (i.e., 0°, 90°, 180°, 270°) relative to fixation, although no oblique orientations were tested. However, in contrast to these findings, Wilson (1985) found an oblique effect on curvature discrimination thresholds using similar stimuli and experimental procedures. The different results in the two studies might reflect differences among observers. Alternatively, the different findings could be a result of the two studies comparing thresholds across a very small set of orientations (i.e., a single cardinal orientation vs. one oblique orientation). A more comprehensive examination of curvature discrimination at multiple orientations is thus needed. 
Previous psychophysical studies investigating visual sensitivity for contour curvature have either randomized the polar angle at which deformities appear or kept the spatial location of the contour deformation fixed (Wilkinson et al., 1998; Hess et al., 1999; Jeffrey, Wang, & Birch, 2002; Loffler et al., 2003; Bell, Wilkinson, Wilson, Loffler, & Badcock, 2009; Bell & Badcock, 2009; Schmidtmann, Kennedy, Orbach, & Loffler, 2012; Baldwin, Schmidtmann, Kingdom, & Hess, 2016). Consequently, it is difficult to determine if observed changes in visual sensitivity to curvature are due to uncertainty about polar angle rather than polar angle per se. In a series of studies that examined the effect of spatial uncertainty on shape discrimination thresholds, Green and colleagues (2017, 2018a, 2018b, 2018c) found that sensitivity to curvature differences between two shapes was greatest when the phase (i.e., angular position) of the shapes varied from trial to trial. This finding suggests that spatial uncertainty affects sensitivity to contour curvature (Green et al., 2017, 2018a, 2018b, 2018c), but does not reveal how spatial uncertainty affects sensitivity to curvature. Baldwin et al. (2016) examined the effect of spatial uncertainty on curvature detection thresholds as a function of the number of modulated cycles appearing along the contour of a shape. With stimuli that contained only one cycle of modulation, the slopes of the psychometric functions were steeper when observers were uncertain about the location of curvature deformation. Baldwin et al. attributed the effect of slopes becoming slightly shallower as the number of modulated cycles increased due to a reduction in extrinsic uncertainty (see Pelli, 1985). However, because Baldwin et al. averaged thresholds across different spatial locations to estimate the effect of spatial uncertainty on curvature discrimination, it remains unclear whether thresholds vary at different angular orientations as a function of spatial uncertainty. 
Despite the prominent and consistent effect of stimulus orientation on visual performance, evidence for an oblique effect on curvature discrimination is mixed. The purpose of the current study, therefore, was to test whether human observers are equally sensitive to positive deformations of curvature across polar angles. We make use of a novel stimulus class, which we refer to as Difference of Gaussian contours that ease the process of generating closed contour stimuli with a single (signed) curvature deformity occurring along its contour. We believe that our results are the first to demonstrate that observers are indeed approximately uniformly sensitive to positive deformations of curvature at all polar angles tested, both in the presence and absence of spatial uncertainty, and across different angular frequencies. 
General methods
Participants
Three experienced psychophysical observers participated in the experiment. Two observers were naïve to the purpose of the study (LUC, KAT), and one was an author (MS). The mean age of the observers was 24.7 years (SD = 2.52, range: 22–27) and all observers had normal or corrected-to-normal visual acuity. Experimental protocols were approved by the McMaster University Research Ethics Board, and informed consent was obtained from each observer prior to the start of the experiment. 
Apparatus
An Apple G4 2.66 GHz Quad-Core Intel Xeon computer (Apple Inc, Cupertino, CA) generated and displayed stimuli using MATLAB 10.7.0 (MathWorks, Natick, MA) and Psychophysics Toolbox (Brainard, 1997; Pelli, 1997). Stimuli were displayed on a Sony Model GDM-F520 monitor (Sony Corporation, Tokyo, Japan) with a pixel resolution 1,024 × 768 (62.5 pixels/°) and a refresh rate of 100 Hz. The display was the only light source in the room during testing, and had a mean luminance of 69.2 cd/m2. The stimuli were viewed binocularly through natural pupils. A chin rest was used to ensure a viewing distance of 131 cm was maintained throughout the duration of the experiment. 
Stimuli
Many studies of curvature detection and discrimination have used radial frequency (RF) stimuli consisting of an integer number of cycles that cover the entire closed contour (Wilkinson et al., 1998). For our experiments, we wanted to use stimuli in which (a) curvature was restricted to a small portion of the contour; and (b) the radial frequency potentially could be manipulated as a continuous variable. Loffler et al. (2003) provided formulas for generating whole integer cycle or half-cycle stimuli sampled from RF contours but not for other radial frequencies. Schmidtmann et al. (2012) provided a more comprehensive set of formulae for defining shapes of different cycle numbers, but the formulas are difficult to implement to create stimuli consisting of spatially-restricted deformations. Lastly, Dickinson, Cribb, Riddell, and Badcock (2015) provided formulae and textual descriptions for rectified contours sampled from RF patterns. Because these patterns had constraints that made them less than ideal for our experiments, we developed a new type of closed-contour stimulus, Difference of Gaussian (DoG) contours, that consist of a single, signed curvature deformity defined as:  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}\tag{1}r(\theta ) = \bar r(1 + {DoG}(\theta ))\end{equation}
where Display Formula\(\bar r\) is the mean radius of the base circle, and DoG corresponds to the deformed segment along the circumference of the base circle. DoG was defined as:  
\begin{equation}\tag{2}{ {DoG}(\theta ) = {G_{{\sigma _1}}}(\theta ;{\theta _c};{\sigma _1}) - {G_{{\sigma _2}}}(\theta ;{\theta _c};{\sigma _2}) = {1 \over {\sqrt {2\pi } }}\left( {{1 \over {{\sigma _1}}}{e^{ - (\theta - {\theta _c})/2\sigma _1^2}} - {1 \over {{\sigma _2}}}{e^{ - (\theta - {\theta _c})/2\sigma _2^2}}} \right)} \end{equation}
where Display Formula\({G_{{\sigma _1}}}\) and Display Formula\({G_{{\sigma _2}}}\) are Gaussian probability density functions. Parameter θc and variance (σ2) of each Gaussian distribution determine the angular frequency, amplitude, and location of the deformed segment along the circumference of a base circle. The DoG portion of the contour smoothly transitions into a base circle with a radius equal to Display Formula\(\bar r\). To ease the process of choosing parameters for the Gaussian functions that compose the DoG function, we fit Equation 1 to one positive half-cycle of a radial frequency contour:  
\begin{equation}\tag{3}r(\theta ) = \bar r(1 + Asin(\omega \theta + \phi ))\end{equation}
where Display Formula\(\bar r\) is the base radius of the circular contour, A is the amplitude, ω is the radial (angular) frequency, and Display Formula\(\phi \) is the phase (also see equation 1 in Wilkinson et al., 1998) Parameter estimates were computed using iterative least squares estimation (i.e., nlinfit in MATLAB) at 500 linearly spaced points along an interval of Display Formula\( \pm 2\pi /4*\omega \) from the peak of maximum curvature for one positive-half cycle of a RF contour.  
One benefit of using DoG contours, as opposed to radial frequency (RF) patterns as originally described by Wilkinson et al. (1998), is that a single region of the close contour can be modulated by frequencies that are non-integer values. This feature makes it easier to investigate the perception of deformed regions that vary along a continuum of angular frequency. Nevertheless, in the present study we used only integer values to define the angular frequency of DoG contours to make it easier to compare our results with those obtained in previous studies using RF contours. 
Examples of DoG contours used in the present study, fit to angular frequencies of three and six cycles of modulation per circumference, are shown in Figure 1. In the current experiments, we measured detection thresholds with stimuli in which the location of the DoG region varied across 24 polar angles (in 15° increments). 
Figure 1
 
Example of DoG contours used in this study. The amplitude of modulated DoG contours shown are set to 10% of the radius of the base circle for illustrative purposes. (a) DoG contour fit to an angular frequency of 3 cycle/2π. (b) DoG contour fit to a angular frequency of 6 cycle/2π. In the experiments, the luminance profile of the contour was a D4 pattern, with a peak spatial frequency of 8 cpd and a luminance contrast of 99%.
Figure 1
 
Example of DoG contours used in this study. The amplitude of modulated DoG contours shown are set to 10% of the radius of the base circle for illustrative purposes. (a) DoG contour fit to an angular frequency of 3 cycle/2π. (b) DoG contour fit to a angular frequency of 6 cycle/2π. In the experiments, the luminance profile of the contour was a D4 pattern, with a peak spatial frequency of 8 cpd and a luminance contrast of 99%.
The radial luminance profile of each contour was defined by a fourth derivative Gaussian (D4; see Wilkinson et al., 1998) with a peak spatial frequency of 8 cpd and a luminance contrast of 99%. All DoG contours had a base radius (Display Formula\(\bar r\)) of 1.14°. 
Psychophysical procedure
Thresholds were obtained using a two-interval forced choice (2IFC) task. On each trial, observers were presented with a modulated DoG contour in one stimulus interval and a circular (i.e., nonmodulated) contour in the other interval. Observers were asked to judge whether the first or second interval contained the modulated stimulus. A 60 s light adaptation period preceded the start of every testing session to ensure observers' eyes were adapted to the average luminance of the display. Each trial began with a small, high-contrast fixation dot drawn in the middle of the display. Observers initiated trials by pressing the spacebar, after which the fixation dot flickered at 10 Hz for 200 ms. The fixation point was extinguished and, after a 300 ms delay, was followed by the two 150 ms stimulus intervals separated by a 500 ms interstimulus interval (ISI). Across stimulus intervals and trials, the central position of each contour was jittered randomly 0.17° within a circular region originating from the center of the screen. Observers received 50 practice trials prior to the start of their first experimental session and 10 practice trials at the start of subsequent testing sessions. Auditory feedback informed observers as to the correctness of their response at the end of every trial. 
In the fixed angle condition, the polar angle of the DoG function was held constant within a single testing session but varied across sessions. In the random angle condition, the polar angle of the DoG function varied randomly across 24 angles within each testing session. Across trials, the amplitude of the DoG function for each polar angle was adjusted with a two-down, one-up staircase that converged on 71% correct performance. Each staircase terminated after 16 reversals, and thresholds were estimated by averaging the modulation amplitude of the last eight reversals. Observers completed two sessions per each combination of DoG, polar angle, and angle uncertainty condition, for a total of eight testing sessions. 
Data analysis
Unless stated otherwise, statistical analyses were performed in R (R Core Team, 2017). Mixed linear models were used to analyze thresholds using the lme4 package (Bates, Mächler, Bolker, & Walker, 2015). The Kenward-Roger method (Kenward & Roger, 1997) was used to approximate the degrees of freedom for each mixed model, as this method better approximates F-distributions for linear mixed models (Judd, Westfall, & Kenny, 2012). For brevity, we report only the F tests from the linear mixed-effects regression analyses (i.e., the analysis of variance or ANOVA of Type III sums of squares with Kenward-Roger approximation for degrees of freedom). To increase the power of our statistical analyses, we performed a more focused comparison of detection thresholds across four visual quadrants (see Figure 3). Each quadrant was composed of detection thresholds collected across 6 polar angles that split the visual field both horizontally (top and bottom), and vertically (left and right). 
Results
Results for each observer are shown in Figure 2. Thresholds were lower for curvature deformations of high angular frequency (DoG 6) than those for contours of low angular frequency (DoG 3). Furthermore, detection thresholds were lower when the polar angle of peak curvature was fixed within a block of trials, compared to when the angle varied randomly across trials. Finally, in all three observers sensitivity to curvature was surprisingly similar across all polar angles, with slight departures from uniformity occurring at select polar angles for each observer. Aggregate results with polar angles grouped according to visual quadrant are shown in Figure 3. The data were analyzed with a mixed linear model that included Visual Quadrant, DoG angular frequency, and Angle Uncertainty as fixed effects, and Observer and Session as random effects. The analysis of variance revealed a significant effect of DoG angular frequency, F(1, 555) = 300.25, p < 0.0001, and Angle Uncertainty, F(1, 555) = 19.34, p < 0.0001, with no other terms reaching significance (p > 0.35 in each case). 
Figure 2
 
(a) Detection thresholds for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is represented by symbol color and shape. Light and dark colors represent thresholds collected with different levels of polar angle uncertainty. Error bars represent ±1 SEM. (b) Polar plots of mean detection thresholds for each observer for all conditions.
Figure 2
 
(a) Detection thresholds for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is represented by symbol color and shape. Light and dark colors represent thresholds collected with different levels of polar angle uncertainty. Error bars represent ±1 SEM. (b) Polar plots of mean detection thresholds for each observer for all conditions.
Figure 3
 
Average detection thresholds grouped according to visual quadrant. Each visual quadrant contains thresholds from six separate polar angle conditions. Light and dark bars represent conditions where polar angle was fixed and varied across trials, respectively. Angular frequency is represented by the color of each bar (DoG 3: blue; DoG 6: red), and errors bars represent ±1 SEM.
Figure 3
 
Average detection thresholds grouped according to visual quadrant. Each visual quadrant contains thresholds from six separate polar angle conditions. Light and dark bars represent conditions where polar angle was fixed and varied across trials, respectively. Angular frequency is represented by the color of each bar (DoG 3: blue; DoG 6: red), and errors bars represent ±1 SEM.
Oblique effect
By grouping contour-curvature thresholds according to visual quadrant as shown in Figure 3, we removed any possibility of observing an oblique effect. Therefore, to estimate an oblique effect, we used a linear contrast to compare thresholds measured with oblique (45°, 135°, 225°, 315°) and cardinal (0°, 90°, 180°, 270°) orientations. The linear contrast essentially computes the difference between the average thresholds for the oblique and cardinal orientations. The average thresholds are shown in Figure 4A: for observers KAT and LUC, thresholds were 22% and 28% higher for oblique orientations than cardinal orientations, whereas observer MS exhibited nearly identical thresholds for the two sets of orientations. On average, thresholds were 14.5% higher for oblique orientations, a difference that was not statistically significant, t(2) = 1.61, p = 0.25. 
Figure 4
 
Thresholds measured at oblique and cardinal orientations in Experiments 1 and 2. Average thresholds are shown individual observers as well as averaged across observers. Error bars represent ±1 SEM.
Figure 4
 
Thresholds measured at oblique and cardinal orientations in Experiments 1 and 2. Average thresholds are shown individual observers as well as averaged across observers. Error bars represent ±1 SEM.
Given interobserver variability in the magnitude of the oblique effect, for each observer we computed Jeffreys-Zellner-Siow (JZS) Bayes factors (BF; Morey & Rouder, 2011) to obtain a level of confidence that thresholds differed between cardinal and oblique angles. The BF represents the ratio between the marginal likelihood for the alternative hypothesis (i.e., H1) that “thresholds differ across different angular conditions” divided by the marginal likelihood for the null hypothesis (i.e., H0) that there is “no or negligible difference in thresholds between oblique versus cardinal angle conditions.” Therefore, BF values larger than one reflect a greater probability (i.e., belief) that thresholds differ between cardinal and oblique angular orientations. According to the classification scheme described by Raftery (1995), BFs between 1 and 3 constitute weak evidence in favor of an oblique effect, BFs between 3 and 20 constitute positive evidence, and BFs greater than 20 constitute strong or very strong evidence. 
All BF computations were performed using the BayesFactor package by Morey and Rouder (2015), with the scaling factor in the Cauchy prior set to equal one. The BFs for each observer are shown in Table 1. For observer LUC, the computed BF value was approximately 2.7, which constitutes weak evidence (Raftery, 1995) in favor of the alternative hypothesis for the existence of an oblique effect. Conversely, the BF value for observer MS was below 1, suggesting that for this observer there is positive evidence in favor of the null hypothesis that sensitivity does not differ between cardinal and oblique orientations. Lastly, a BF value near one was computed for observer KAT, suggesting evidence for both the null and alternative hypotheses. To summarize, our analysis of Bayesian factors found weak evidence of an oblique effect in only one of three observers in Experiment 1. 
Table 1
 
Jeffreys-Zellner-Siow (JZS) Bayes factors (BF) quantifying the strength of evidence that thresholds differed between cardinal and oblique orientations for each observer in Experiments 1 and 2.
Table 1
 
Jeffreys-Zellner-Siow (JZS) Bayes factors (BF) quantifying the strength of evidence that thresholds differed between cardinal and oblique orientations for each observer in Experiments 1 and 2.
Multiple- versus single-channel models
Many models of visual processing assume that performance is based on the responses of multiple, stimulus-selective channels (Graham, 1989). Within this multiple channels framework, blocking or interleaving the location of curvature deformations across trials could have a significant effect on the number of mechanisms an observer must monitor. In the framework of signal detection theory (SDT) and probability summation (PS) models, which have been used to account for the detection of localized RF contours (i.e., single cycle or less) (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen, Beaudot, & Ivanov, 2011; Schmidtmann et al., 2012; Green et al., 2017; Schmidtmann & Kingdom, 2017; Green et al., 2018a), blocking trials potentially allows observers to focus attention on relevant channels and ignore channels that do not convey task-relevant information (Kingdom, Baldwin, & Schmidtmann, 2015; Kingdom & Prins, 2016). Alternatively, when locations of curvature deformation are varied from trial to trial (i.e., interleaved), observers may monitor many or perhaps all channels that are potentially relevant to the task, even though on any single trial only a few of those channels will convey task-relevant information and the rest will contain only noise (Kingdom & Prins, 2016). The addition of irrelevant sensory noise from monitoring irrelevant channels will degrade performance in the interleaved condition, particularly on trials when the signal level is low, and in SDT-PS models this will be reflected as an increase in the slope of the psychometric function with increasing number of channels monitored (Kingdom & Prins, 2016). 
To determine whether data from Experiment 1 are consistent with observers monitoring different numbers of sensory channels in the blocked and interleaved conditions, we fit our data with two SDT-PS models using the Palamedes Toolbox (Prins & Kingdom, 2018) in MATLAB 10.7.0. Specifically, we collapsed data across observers and irrelevant experimental variables (i.e., DoG angular frequency and test session) to obtain probabilities of correct detection as a function of deformation amplitude for both blocked and interleaved conditions. We then computed the best-fitting values for gain (g) and transducer (τ) parameters separately for the blocked and interleaved conditions, and used these values to fit two SDT-PS models to each condition that assumed that the total number of monitored channels was one (Q = 1) or many (Q = 24). The number of channels activated by our stimuli (n) on each trial was set to one and the number of alternatives (M) presented on each trial was set to two. 
Differences between models fit to each condition are shown in Figure 5, and a summary table of relevant parameters for fitted models is shown in Table 2. As demonstrated by both the closer approximation to data and lower mean-squared error (MSE), models where the total number of monitored channels was set to one better predicted observed data for both blocked and interleaved conditions. This result supports the hypothesis that observers monitored a single sensory channel in making detection judgments of curvature across stimulus intervals, or that they were matching each stimulus to a single internal template. 
Figure 5
 
Detection accuracy plotted against log-transformed modulation amplitude for conditions in which the location of curvature deformations was either (a) blocked or (b) interleaved across trials. In both figures, data were collapsed across observers and irrelevant experimental variables, and the number of trials for each point is represented by symbol size. The black lines are the best-fitting SDT–PS model for each condition with Q = 1, while the dashed red lines are the best fitting SDT–PS model for each condition with Q = 24. The horizontal dotted lines represent chance performance.
Figure 5
 
Detection accuracy plotted against log-transformed modulation amplitude for conditions in which the location of curvature deformations was either (a) blocked or (b) interleaved across trials. In both figures, data were collapsed across observers and irrelevant experimental variables, and the number of trials for each point is represented by symbol size. The black lines are the best-fitting SDT–PS model for each condition with Q = 1, while the dashed red lines are the best fitting SDT–PS model for each condition with Q = 24. The horizontal dotted lines represent chance performance.
Table 2
 
Parameter values of probability summation models fit to data using different number of monitored channels (Q) for the blocked and interleaved conditions in Experiment 1. Values for gain(g) and tau(τ) were estimated separately for each condition prior to generating function fits for models manipulating parameter Q.
Table 2
 
Parameter values of probability summation models fit to data using different number of monitored channels (Q) for the blocked and interleaved conditions in Experiment 1. Values for gain(g) and tau(τ) were estimated separately for each condition prior to generating function fits for models manipulating parameter Q.
Discussion
Experiment 1 demonstrated that, on average, sensitivity to curvature deformation was similar across polar angles, regardless of angular frequency or whether spatial uncertainty was introduced in the location of deformation. On average, detection thresholds were approximately equal for curvature on cardinal and oblique orientations, and our analysis of individual observers found weak evidence for an oblique effect in only one of three participants. We also found that detection thresholds were significantly lower for the higher angular frequency, and when polar angle was constant within a block of trials. This last result suggests that detection thresholds were affected by spatial uncertainty about the polar angle of the deformation. 
One possible explanation for the uniform sensitivity to curvature across polar angles is that observers performed the task by looking for a circle, not the deformed target. By identifying which interval contained the circle, observers could attenuate the effects of polar angle and spatial uncertainty on detection thresholds. For example, the human visual system could perform an operation similar to computing the cross-correlation between an internal template for circularity in both intervals, identifying which interval best matched the internal template. At first glance it would seem that the difference between thresholds in the blocked and intermixed conditions would be inconsistent with this hypothesis, because observers would be looking for one target (i.e., a circle) in both conditions, and thus contrary to our findings, thresholds should be similar across conditions. Nevertheless, results from our SDT analyses for Experiment 1 support this idea, as SDT-PS models whereby observers monitored fewer task irrelevant channels better accounted for observed data in both the blocked and interleaved conditions. However, if observers were adopting such a perceptual strategy, then they would be unable to identify the polar angle of the curvature deformation. If instead, observers are able to accurately identify angular position of curvature deformations as soon as these regions are detected, these results would be supportive of curvature detectors being labeled lines, as described by Watson and Robson (1981). 
Another possibility is that observers are using the aspect ratio of shapes to determine which interval contains the noncircle. Surprisingly, there is very little discussion surrounding the use of aspect ratio as a cue for curvature discrimination along closed-contour shapes, specifically with regards to studies using shapes sampled from RF patterns of a single-cycle or less (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen et al., 2011; Schmidtmann et al., 2012; Green et al., 2017; Schmidtmann & Kingdom, 2017; Green et al., 2018a) demonstrated that thresholds for discriminating shape based on aspect ratio can fall within the hyperacuity range of human vision. If aspect ratio is a salient cue used to detect isolated curvature deformations, observers should be able to identify the axis that contains the curvature deformation, but not its precise angle. In other words, observers should be able to identify the axis, but not which end of the axis is deformed. 
Experiment 2 examined these possibilities by measuring detection thresholds under conditions of spatial uncertainty using a dual-judgment task. Observers were asked to identify both the interval containing the curvature deformation as well as its orientation, thus allowing detection and identification thresholds for curvature to be measured. 
Experiment 2
Participants
Two new naïve observers (HW and JJ) participated in Experiment 2, along with the author (MS) who had participated in Experiment 1. The mean age of the observers was 22.0 (SD = 4.35, range: 19–27) and all observers had normal or corrected-to-normal visual acuity. Experimental protocols were approved by the McMaster University Research Ethics Board, and informed consent was obtained from each participant prior to the start of the experiment. 
Apparatus, Stimuli, and Procedure
The apparatus, stimuli, and psychophysical procedure were identical to those described for Experiment 1 except for the following minor alterations. First, the location of deformation was limited to eight polar angles (i.e., 0 to 315° in 45° increments), one of which was randomly selected on each trial. Hence, the different angles were interleaved; there was no blocked condition. Also, on each trial observers were asked to identify both the interval and location of the deformity using the number-pad on a computer keyboard. Observers completed two sessions per DoG condition, for a total of four testing sessions. 
Data analysis
Data were analyzed using mixed-linear models as in Experiment 1, except that polar angles were not grouped into visual quadrants. Analyses of localization judgments were restricted to the responses on trials occurring between reversals 8 and 16 on each staircase to ensure that judgements were made for stimuli that included curvature deformations that were near detection threshold. 
Results
Detection thresholds for Experiment 2 are shown in Figure 6. As was found in Experiment 1, at the group level, thresholds were approximately constant across polar angle, and were lower for the higher angular frequency (i.e., 6 cycles/2π vs. 3 cycles/2π). These observations were tested using a mixed linear model with polar angle and DoG angular frequency as fixed effects, and Observer and Session as random effects. The ANOVA revealed a significant effect of DoG angular frequency, F(1, 79.45) = 41.60, p < 0.0001, with no other terms approaching significance (p > 0.47 in each case). 
Figure 6
 
Detection thresholds measured in Experiment 2 for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is coded for by line color and symbol shape. Error bars represent ±1 SEM.
Figure 6
 
Detection thresholds measured in Experiment 2 for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is coded for by line color and symbol shape. Error bars represent ±1 SEM.
As in Experiment 1, we performed a linear comparison of detection thresholds measured with oblique versus cardinal orientations. The average thresholds at the two sets of orientations are shown in Figure 4B: for observers HW and MS, thresholds were 25% and 5% lower for oblique orientations than cardinal orientations, whereas observer JJ exhibited nearly identical thresholds for the two sets of orientations. On average, thresholds were 9% lower for oblique orientations, a difference that was not statistically significant, t(2) = 0.85, p = 0.48. 
As in Experiment 1, we computed JZS Bayes factors for each observer to quantify the strength of evidence in support of an oblique effect (Table 1). For observers JJ and MS, computed BF values were less than 1, indicating that the evidence favors the hypothesis that detection thresholds did not differ between cardinal and oblique orientations. The BF for observer HW was significantly greater than 1, and thus can be classified as positive evidence (Raftery, 1995) in support of the hypothesis that thresholds differed between cardinal and oblique orientations. However, recall that thresholds for observer HW were lower for oblique orientations, and therefore this analysis suggests there was an inverse oblique effect for this observer. Overall, we found positive evidence in support of an inverse oblique effect in one of three observers, with evidence for the remaining two observers favoring the hypothesis of no oblique effect. Interestingly, the values of the BFs for observer MS were very similar across experiments. The average of the mean BF for observer MS and the BF values for the other four observers across experiments was 1.54, which was not significantly greater than 1, t(4) = 0.874, p = 0.216, d = 0.391. Hence, across experiments we obtained very weak evidence for the existence of an oblique effect. 
The proportion of correct angle identification judgments is shown in Figure 7a. For both DoG angular frequencies and across all polar angles, observers were able to identify the orientation of the deformity at accuracies that were well above chance. The proportions of correct angle identification responses were analyzed with a mixed-linear model that included two fixed effects (DoG angular frequency and polar angle), and two random effects (Observer and Session). Results from the analysis were consistent with the observations described already, in that no effects reached significance (p > 0.20 in each case). 
Figure 7
 
Polar angle identification results from Experiment 2 on trials in which the curvature was near detection threshold. (a) Mean accuracy of angle identification judgements for each angular frequency and each polar angle. The dashed lines indicate chance performance. (b) Mean absolute error in angle identification responses (in degrees of polar angle) plotted as a function of DoG angular frequency and polar angle. Dashed lines represent the predicted error based on guessing. Error bars represent ±1 SEM.
Figure 7
 
Polar angle identification results from Experiment 2 on trials in which the curvature was near detection threshold. (a) Mean accuracy of angle identification judgements for each angular frequency and each polar angle. The dashed lines indicate chance performance. (b) Mean absolute error in angle identification responses (in degrees of polar angle) plotted as a function of DoG angular frequency and polar angle. Dashed lines represent the predicted error based on guessing. Error bars represent ±1 SEM.
The magnitude of angle identification error is plotted as a function of angular frequency and polar angle in Figure 7b. Error magnitude did not differ markedly across angular frequencies or polar angles, and an analysis with a mixed effect model that included the same factors as the one used to analyze identification accuracy revealed no significant effects (p > 0.11 in each case). 
We next considered whether the identification errors were consistent with the hypothesis that observers were guessing. If all identification errors were caused by guessing, and if all directions were equally likely to be guessed, then the average error ought to be 102°. Inspection of Figure 7b suggests that the errors were very close to the predicted value for guessing: the mean errors for the three observers were 101.35°, 104.44°, and 97.88°, and the grand mean was 101.2°. Hence, the observed errors were consistent with the hypothesis that most, if not all, errors were guesses. 
Relating detection and identification
Finally, we examined the association between detection and angle identification judgments. Detection and identification judgments on near-threshold trials (i.e., trials between staircase reversals 8 and 16) are shown in Table 3. For each subject, the probability of correctly identifying the polar angle was much greater when the detection judgment was correct, and the probability of correctly detecting the curvature deformation was much greater on trials on which the identification judgment was correct. These observations were supported by chi-square tests that rejected the hypothesis of no association between detection and identification (Display Formula\(\chi _{(1)}^2 \ge 78.1\), p < 0.0001 for each observer). 
Table 3
 
Contingency tables for detection and identification judgements for near-threshold trials (between staircase reversals 8 and 16) in Experiment 2.
Table 3
 
Contingency tables for detection and identification judgements for near-threshold trials (between staircase reversals 8 and 16) in Experiment 2.
The association between detection and identification is illustrated differently, and across a wider range of curvature amplitudes, in Figure 8. Figure 8a plots the proportion of correct detection responses (after collapsing across conditions, sessions, and observers) as a function of log-transformed curvature amplitude, separately for trials on which the identification response was correct and incorrect. For trials on which the identification judgments were correct, detection accuracy increased with increasing curvature amplitude, and the data were well-fit by a Weibull function. On the other hand, detection accuracy on trials where the identification judgment was incorrect hovered slightly above chance levels and did not change noticeably with increasing curvature amplitude. As a result, the psychometric function fit to the data from identification-correct trials is shifted to lower amplitudes and is noticeably shallower than the function fit to all of the data. Figure 8b shows identification accuracy plotted against modulation amplitude, separately for trials on which detection responses were correct and incorrect. As was found with detection accuracy, identification accuracy was much lower on trials when detection responses were incorrect; however, unlike what was found for detection, there is some suggestion that identification accuracy on detection-incorrect trials increased slightly with increasing deformation amplitude. Consequently, and unlike what was found for detection accuracy, the psychometric function fit to identification accuracy for detection-incorrect trials is similar to the function fit to all of the data. 
Figure 8
 
(a) Detection accuracy plotted against log-transformed modulation amplitude, separately for trials on which identification was correct or incorrect. (b) Identification accuracy plotted separately for trials on which detection was correct or incorrect. In both figures, the data were collapsed across conditions and observers, and the number of trials for each point is represented by symbol size. The green lines are the best-fitting (weighted least-squares) Weibull function fit to data from correct (a) identification or (b) detection responses; the black lines are the Weibull functions fit to all of the data. The horizontal dashed lines represent chance performance, and the dotted lines indicate points on the psychometric functions where d′ equals 2, which corresponds to 92% and 71% accuracy in the detection and identification tasks, respectively.
Figure 8
 
(a) Detection accuracy plotted against log-transformed modulation amplitude, separately for trials on which identification was correct or incorrect. (b) Identification accuracy plotted separately for trials on which detection was correct or incorrect. In both figures, the data were collapsed across conditions and observers, and the number of trials for each point is represented by symbol size. The green lines are the best-fitting (weighted least-squares) Weibull function fit to data from correct (a) identification or (b) detection responses; the black lines are the Weibull functions fit to all of the data. The horizontal dashed lines represent chance performance, and the dotted lines indicate points on the psychometric functions where d′ equals 2, which corresponds to 92% and 71% accuracy in the detection and identification tasks, respectively.
In relating detection to identification, Watson and Robson (1981) discussed the possibility that detectors could act as labeled lines that would allow observers to correctly identify a stimulus from a set of alternatives whenever it is detected. That is, two stimuli are perfectly discriminated if they are accurately identified whenever detected, as responses from different sets of detectors may be diagnostic in signaling for the presence of a particular stimulus feature (Watson & Robson, 1981). In the context of the current experiments, if response accuracy (pc) is related to curvature amplitude (c) by the psychometric function  
\begin{equation}\tag{1}pc = 1 - (1 - g - \gamma )\exp ( - {(c/\alpha )^{ - \beta }})\end{equation}
then the labeled lines hypothesis predicts that the functions fit to performance in the two tasks should differ only in the guessing rate, g, which would be 0.5 and 0.125 in the detection and identification tasks, respectively. To test this idea, we fit Equation 4 to all of the detection and identification data in Figures 8a and b by fixing the value of γ at 0.01, setting the guessing rate to the appropriate value for each task, and then varying the parameters α and β. The values best-fitting parameters in the detection task (α = −4.32, β = 5.74) were nearly identical to values in the identification task (α = −4.29, β = 5.70). Table 4 shows the values fit to the data from individual subjects: on average, the values of α obtained in the detection and identification tasks differed by less than 1%, and the values of β, which in our experience tends to be more variable, differed by only 6.4%. Hence, the data were consistent with a prediction of the labeled lines hypothesis, namely that the psychometric functions in the detection and identification tasks were essentially identical except for guessing rate.  
Table 4
 
Best-fitting values of α and β in Eq. 4 for each subject and task in Experiment 2. The fits were performed on data that were combined across stimulus conditions. The value of the lapse rate (γ) was fixed at 0.01, and the guessing rate (g) was 0.5 and 0.125 for the detection and identification tasks, respectively.
Table 4
 
Best-fitting values of α and β in Eq. 4 for each subject and task in Experiment 2. The fits were performed on data that were combined across stimulus conditions. The value of the lapse rate (γ) was fixed at 0.01, and the guessing rate (g) was 0.5 and 0.125 for the detection and identification tasks, respectively.
In summary, our results are consistent with the idea that detection and identification judgments were not independent: correct detection was associated with correct identification, and vice versa. This association was not an artifact caused by detection and identification accuracy both being positively correlated with deformation amplitude, because it was found even when we considered only trials near detection threshold (Table 3). When we examined the association between the two judgments across a wider range of stimulus levels, we found that detection and identification accuracy were both near chance and only weakly associated with deformation amplitude when responses on the other task were incorrect (Figure 6). This second results implies that observers had very poor information about polar angle when the curvature deformation was not detected, and very poor information about the stimulus interval containing the target when its polar angle was misidentified. We also found that the psychometric functions measured in the detection and identification tasks in Experiment 2 were nearly identical once the difference in guessing rate was taken into account, which is consistent with the hypothesis that detectors for curvature serve as labelled lines. Therefore, even though general uniformity in sensitivity to curvature could be accounted for by a circular template matching model in Experiment 1, such an explanation does not account for the close relation between detection and identification thresholds for curvature measured in Experiment 2
Aspect ratio and curvature discrimination
Regan and Hamstra (1992) found that observers were remarkably good at discriminating between shapes that differed in aspect ratio, even in the absence of size cues. The DoG contours and circular contours used in the current experiments necessarily differed in aspect ratio, and therefore it is possible that observers used aspect ratio as an additional cue to curvature in making shape discrimination judgments. This idea is consistent with results reported by Zanker and Quenzer (1999), who found that sensitivity to changes in aspect ratio was higher for circles than squares. Zanker and Quenzer argued that the higher sensitivity with circles was due to observers using local curvature in combination with aspect ratio to discriminate shapes. 
Although sensitivity to changes in aspect ratio might contribute to the high sensitivity in our experiments, it is unlikely that aspect ratio was the main cue used by observers to perform our task. At best, mechanisms that are sensitive to changes in aspect ratio could determine the axis of elongation of the closed contour shape, but it is not clear how such mechanisms could determine which end of the axis contained the curvature deformation. Yet Experiment 2 found that the psychometric functions for detection and identification were essentially identical after correcting for differences in guessing, and were consistent with the labeled lines hypothesis. The close coupling of detection with identification judgments argues against aspect ratio being the only cue used by observers, which would cause thresholds for identification to increase relative to those for detection. Therefore, results from Experiment 2 suggest that curvature is likely the principle cue used in discriminating between shapes in our present experiments and, consistent with previous views (Foster, Simmons, & Cook, 1993), suggests that mechanisms for curvature also might underlie the computation for aspect ratio for circular shapes. 
General discussion
The current study used a novel stimulus class we call DoG contours to examine the sensitivity of human observers to local curvature deformations along a closed circular contour as a function of polar angle. We found that thresholds were higher when observers were uncertain about polar angle. Furthermore, although the magnitude (and direction) of the oblique effect varied across individuals, average sensitivity to curvature deformations was remarkably similar across all polar angles tested (0°–345°). Overall, we found no group level effect of oblique orientations on detection thresholds for closed contour shapes, and only weak evidence in two of six observers for an oblique effect. Finally, observers in Experiment 2 accurately identified the polar angle in location of deformations along DoG contours when the magnitude of the deformation was near detection threshold and the location of the deformity was uncertain. Together, these results suggest that, on average, observers are approximately uniformly sensitive to curvature deformations across all polar angles, and that curvature detection mechanisms may serve as labeled lines. 
Angular frequency
Consistent with previous findings, we found that observers were more sensitive to curvature segments of higher angular frequency (Wilkinson et al., 1998; Hess et al., 1999; Loffler et al., 2003; Schmidtmann et al., 2012; Schmidtmann & Kingdom, 2017). Watt and Andrews (1982) were one of the first to demonstrate that efficiency for curvature discrimination performance was higher when curved line segments traversed smaller orientation ranges, with orientation range being defined as the product of curvature and contour length. Watt and Andrews (1982) argued that the decline in efficiency with increasing stimulus orientation range demonstrates a limitation on the quantity and range of orientations which may be used for curvature estimation and thus used for shape discrimination based on curvature deformations. 
More recent data from studies that measured detection thresholds for single cycles of sinusoidal deformation along closed contours also suggest that performance is constrained by mechanisms that encode local, rather than global, contour shape (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen et al., 2011; Schmidtmann et al., 2012; Green et al., 2017; Schmidtmann & Kingdom, 2017; Green et al., 2018a). In other words, although these studies disagree as to which model of summation best describes how shape discrimination performance varies as a function of the number of periodic cycles of modulation, they all support the idea that V1 (possibly V2) sampling limits performance for single-cycle contours (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen et al., 2011; Schmidtmann et al., 2012; Green et al., 2017, Schmidtmann & Kingdom, 2017; Green et al., 2018a). For example, Schmidtmann et al. (2012) found that detection thresholds decreased as the number of sinusoidal deformation cycles increased, and that thresholds obtained with subsampled patterns were well fit by probability summation models that combined responses across multiple, independent local mechanisms. Schmidtmann and Kingdom (2017) extended these results in demonstrating that detection thresholds also decrease for sinusoidally modulated lines of increasing frequency similar to that of angular frequency for closed contour stimuli. Therefore, increased sensitivity to curvatures of high frequency may reflect a general mechanism that can encode higher frequency curvatures better, whether the modulation is applied to a circle or a line stimulus (Schmidtmann & Kingdom, 2017). 
Regardless of how curvatures are summed along contours, local curvature detectors are thought to be based on analyses of local orientation that are performed in V1 or V2 (Hubel & Wiesel, 1965; Blakemore & Over, 1974) and form the input to shape-encoding mechanisms that operate over larger spatial scales (Felleman & Van, 1991). Computational models that incorporate such a hierarchical arrangement of visual analyses have been used to predict the population responses of V4 cells (Pasupathy & Connor, 2001, 2002; Carlson et al., 2011; Rodríguez-Sánchez & Tsotsos, 2012), which represent signed degrees of curvature as a function of polar angle. Our data are consistent with these types of models, as observers were more sensitive at both detecting and localizing curvatures that were restricted to smaller fractions along a closed contour. These findings are also consistent with computational models of V4 units that predict patterns of activation produced by contours of higher angular frequency should be less spread across polar angles, and thus are well localized along a population surface (Pasupathy & Connor, 2002; Carlson et al., 2011; Slugocki, Sekuler, & Bennett, 2019). Together, this collection of results supports the growing body of evidence suggesting that angular frequency is a critical geometric property limiting sensitivity to curvatures along closed contour shapes (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen et al., 2011; Schmidtmann et al., 2012; Green et al., 2017; Schmidtmann & Kingdom, 2017; Green et al., 2018a). 
Uniform sensitivity to curvature across orientations and localizability
The current experiments demonstrate that observers are, on average, approximately equally sensitive to local curvature across all polar angles. Neurophysiological evidence from macaques suggest this result may arise from the uniform sampling of polar angle by V4 neurons that are sensitive to positive curvature deformations (Pasupathy & Connor, 2001, 2002). However, it is unclear whether responses from V4 neurons need to be considered in characterizing performance on this task. Another possible explanation is that observers based their judgments on the responses of V1 and/or V2 neurons, which is consistent with psychophysical evidence from single cycle studies of RF patterns (Hess et al., 1999; Jeffrey et al., 2002; Loffler et al., 2003; Mullen et al., 2011; Schmidtmann et al., 2012; Green et al., 2017; Schmidtmann & Kingdom, 2017; Green et al., 2018a). For example, hypercomplex cells in primary visual cortex have been shown to be responsive to contour curvature (Hubel & Wiesel, 1965; Dobbins, Zucker, & Cynader, 1987, 1989), and therefore could serve as a mechanism for detecting local curvature, without the need to monitor responses from cells at higher visual areas. 
Data from neuroimaging studies have shown that neurons in human primary visual cortex oriented along cardinal meridians are over represented and exhibit stronger responses to stimuli relative to neurons tuned to oblique angles (Furmanski & Engel, 2000; Li et al., 2003). Interestingly, these differences between cardinal and oblique orientations appear to be less conspicuous in visual cortical areas beyond V1 (i.e., V2, V3) (Furmanski & Engel, 2000; Levitt, Kiper, & Movshon, 1994). Hence, differences between V1 neurons tuned to cardinal and oblique orientations are thought to underlie oblique effects, which are especially apparent for stimuli located near the fovea (Berkley et al., 1975; Vandenbussche, Vogels, & Orban, 1986), like the stimuli used in our experiments. Therefore, if neurons in primary visual cortex were being used to encode curvature deformations, sensitivity to curvature deformations should be better along cardinal than oblique orientations (Watt & Andrews, 1982). Contrary to this prediction, at the group level of analysis we found negligible evidence in support of an oblique effect for contour-curvature thresholds. 
At the individual level of analysis, we found only weak evidence in favor of models incorporating an oblique effect. Indeed, data for only one of the six observers tested across both experiments demonstrated a classic oblique effect, with one observer (JJ) in Experiment 2 actually demonstrating an inverse oblique effect. In comparing our findings with those from previous curvature discrimination studies, there appears to be little agreement across studies as to whether curvature discrimination is worse for oblique stimuli. In agreement with our findings, Watt and Andrews (1982) demonstrated that relative efficiency was unaffected, if not higher, for oblique stimuli compared to those in which the tangent at the point of maximal curvature was horizontal. In contrast, Wilson (1985) showed that discrimination thresholds for oblique curved lines were on average 1.57 times higher compared to those oriented along the horizontal. Given all of these studies were of small-N design, there appears to be appreciable heterogeneity in the underlying population, as we might expect greater consistency to otherwise eventuate across studies. What might account for such variability across observers in the oblique effect for curvature discrimination? 
One potential explanation is that meridional preferences arising from the responses of orientation-selective cortical units, those that may contribute to curvature discrimination performance, diminishes with increasing eccentricity at a rate that varies from observer to observer. In measuring detection of gratings at different orientations and eccentricities, Berkley et al. (1975) found remarkable agreement between observers in the magnitude of the effect of eccentric viewing on oblique target orientations until approximately 6°. However, beyond 6°, there is little agreement between observers in the effect of off-axis viewing angles with increasing eccentricity (Berkley et al., 1975). Given the mean radius of contours used in our experiments was 1.14°, well below 6° of viewing angle from fixation, we feel interobserver differences attributed to eccentric viewing of stimuli can be safely ruled out. 
Another possibility is that observers differ in the degree to which discrimination judgments are mediated by responses from orientation-selective cells in early visual cortex. If the magnitude of the oblique effect varies as a function of receptive field size, as has been put forth in explaining why the oblique effect varies in strength with viewing angle (Berkley et al., 1975), perhaps curvature discrimination might also use mechanisms with larger receptive field sizes. Because receptive field size of cortical units increases as one moves rostrally along the ventral visual pathway (Kravitz, Saleem, Baker, Ungerleider, & Mishkin, 2013), observers preferentially weighting the response from higher visual processing areas in making discrimination judgments might demonstrate diminished oblique effects. The results of the current study are consistent with this hypothesis, and help to explain the variability across observers and studies in observing an effect of oblique orientations on curvature discrimination thresholds. 
It is also important to consider that localization responses across observers from Experiment 2 suggest that once changes in curvature are reliably detected (i.e., reach threshold), curvatures can be localized well above chance, even when the location of the deformity varies from trial to trial. Although this finding alone cannot be used to differentiate between models described already in detecting curvature (i.e., early vs. late mechanisms), it does suggest that the mechanism(s) used to detect curvature are also sensitive to the location at which deformity occurred. Therefore, location of changes in curvature is an additional requirement that should be captured by a model aimed at describing how the visual system represents curvatures along closed contours. For example, the population response of V4 neurons as modelled by Pasupathy and Connor (2002) represents closed contour shapes along two dimensions, curvature and polar angle, that would allow an observer to both locate and detect curvature along a shape using the same mechanism. 
One attempt to dissociate the contribution of early and late visual filters in discriminating shapes based on curvature is in comparing the angular and radial resolution at which observers are able to locate deformations along targets. A difference in the resolution at which angular and radial positions of curvatures are coded for would help reveal what receptive field models, especially with regards to size and shape, might be most consistent with psychophysical measures of curvature discrimination performance. If early visual filters underlie curvature discrimination performance, we might predict that localizability of curvature deformations along both radial and angular dimensions will be high, as receptive fields of cells in primary visual cortex are well localized. If, however, late visual filters that integrate outputs from early visual processes mediate curvature discrimination, then information regarding the radial position of curvatures might be lost in generating more aggregate representations. Future work should further investigate whether, or when, radial and angular location specific information is lost in representing curvatures along shapes. 
Spatial uncertainty
The results of our study also demonstrate that spatial uncertainty, varied by fixing or randomizing the location of peak deformation along a contour, increased detection thresholds. These results are consistent with previous psychophysical studies whereby attenuating attentional effects through the removal of spatial uncertainty led to improved performance at detecting modulated contour fragments along circular shapes Pasupathy and Connor (2002). Attention has been shown to increase the discriminability of select types of spatial information, especially when focused toward a well localized region of space (Lee, Koch, & Braun, 1997). 
One possibility is that constricting visual attention to a well localized portion of the visual field is associated with an increase in processing power of information that falls within that region (Eriksen & Yeh, 1985). Studies of spatial selective attention in macaques have demonstrated an increase in the spontaneous firing rates of neurons when attention is directed inside the receptive field of a neuron, even in the absence of visual information (Luck, Chelazzi, Hillyard, & Desimone, 1997). Neurons within area V4, a candidate area for curvature processing, have been shown to increase their response to attended targets by as much as 51% (Mitchell, Sundberg, & Reynolds, 2007; Reynolds, Pasternak, & Desimone, 2000), and also increase their neural synchrony of firing rates to attended locations (Taylor, Mandon, Freiwald, & Kreiter, 2005). Therefore, attention may be acting as a mechanism that controls the gain of signals representing curvature information. Supportive of this view, our SDT analyses for Experiment 1 revealed the best fitting parameter for gain (g) in our SDT-PS models was higher in blocked conditions (95.9) than interleaved conditions (88.4). This result also helps to explain why observers are able to detect curvature deformities once well localized, such as the findings in Experiment 2, as attended locations receive a boost in signal strength. However, the functional mapping between the gain of curvature mechanisms and its impact on performance of labelled curvature detectors remains unclear, and thus additional research is needed to understand this relation. 
In the SDT framework, spatial uncertainty can be modeled as the number of sensory channels that observers monitor, such that task-irrelevant channels do not contribute additional noise in assessing sensory evidence (Kingdom et al., 2015; Kingdom & Prins, 2016). As the number of locations in which a stimulus can appear increases, so does the number of irrelevant channels that need to be monitored by an observer, as observers will be uncertain as to which channel will be activated by the stimulus. The penalty for monitoring nontarget channels is the addition of noise contributed by channels non-relevant to the task that perturb decision making processes in interpreting responses from activated mechanisms (Kingdom et al., 2015; Kingdom & Prins, 2016). Perhaps then, our manipulation of spatial uncertainty in Experiment 1 also led observers to monitor more sensory channels in the interleaved condition, which therefore contributed to overall worse detection performance relative to the condition where spatial location of curvature deformities was blocked across trials. Contrary to this prediction, SDT-PS models with fewer number of total monitored channels (i.e., Q = 1) accurately predict detection thresholds for both conditions where the location of curvature deformations was blocked or interleaved across trials. 
Instead, observers might be comparing each stimulus interval to an internal representation for circularity, and deduce which trial was non-circular following such an analysis. Habak, Wilkinson, Zakher, and Wilson (2004) have postulated that circularity may function as a “neural default” for shape, as closed contours that are perfectly circular fail to elevate detection thresholds for curvature beyond baseline conditions where no lateral mask is present. However, if observers adopted such a perceptual strategy, then thresholds between interleaved and blocked condition should be similar, as varying the location of curvature deformation from trial to trial should have little effect on the output of such an operation. Counter to this view, thresholds in Experiment 1 were higher for the interleaved condition relative to the blocked condition. One possibility is that observers did base their judgments on the output of a template matching operation, but because DoG contours in the interleaved condition activated different sets of shape-encoding neurons on each trial, it was more difficult for observers to use the same decision criterion in that condition. In other words, observers may have used a more variable criterion in the interleaved condition, which would result in poorer performance, particularly for weak signals. Our findings are consistent with this idea, as thresholds were higher for the interleaved condition, suggesting that observers made overall more errors in discriminating shapes based on curvature. Nevertheless, more research is needed to better understand what affect adopting a non-stationary criterion would have on curvature discrimination performance. 
Conclusion
The aim of our study was to test the hypothesis that observers are uniformly sensitive to positive curvature deformations across all polar angles. Using a novel stimulus class we call Difference of Gaussian (DoG) contours, we have shown observers are remarkably sensitive to local curvature deformations across all polar angles, regardless of spatial uncertainty or angular frequency of deformed parts. In addition, thresholds for curvature deformations at oblique and cardinal orientations were (on average) similar, although the direction and magnitude of the oblique effect varied across individuals. Finally, psychometric functions for the detection and identification of curvature deformations were nearly identical except for differences in guessing rate, a result that is consistent with the hypothesis that mechanisms that encode curvature deformations at different polar angles act as labeled lines. 
Acknowledgments
Supported by NSERC Discovery grants (ABS and PJB) and the Canada Research Chair Program (PJB). 
Commercial relationships: none. 
Corresponding author: Michael Slugocki. 
Address: Department of Psychology, Neuroscience and Behaviour, McMaster University, Hamilton, Ontario, Canada. 
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Figure 1
 
Example of DoG contours used in this study. The amplitude of modulated DoG contours shown are set to 10% of the radius of the base circle for illustrative purposes. (a) DoG contour fit to an angular frequency of 3 cycle/2π. (b) DoG contour fit to a angular frequency of 6 cycle/2π. In the experiments, the luminance profile of the contour was a D4 pattern, with a peak spatial frequency of 8 cpd and a luminance contrast of 99%.
Figure 1
 
Example of DoG contours used in this study. The amplitude of modulated DoG contours shown are set to 10% of the radius of the base circle for illustrative purposes. (a) DoG contour fit to an angular frequency of 3 cycle/2π. (b) DoG contour fit to a angular frequency of 6 cycle/2π. In the experiments, the luminance profile of the contour was a D4 pattern, with a peak spatial frequency of 8 cpd and a luminance contrast of 99%.
Figure 2
 
(a) Detection thresholds for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is represented by symbol color and shape. Light and dark colors represent thresholds collected with different levels of polar angle uncertainty. Error bars represent ±1 SEM. (b) Polar plots of mean detection thresholds for each observer for all conditions.
Figure 2
 
(a) Detection thresholds for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is represented by symbol color and shape. Light and dark colors represent thresholds collected with different levels of polar angle uncertainty. Error bars represent ±1 SEM. (b) Polar plots of mean detection thresholds for each observer for all conditions.
Figure 3
 
Average detection thresholds grouped according to visual quadrant. Each visual quadrant contains thresholds from six separate polar angle conditions. Light and dark bars represent conditions where polar angle was fixed and varied across trials, respectively. Angular frequency is represented by the color of each bar (DoG 3: blue; DoG 6: red), and errors bars represent ±1 SEM.
Figure 3
 
Average detection thresholds grouped according to visual quadrant. Each visual quadrant contains thresholds from six separate polar angle conditions. Light and dark bars represent conditions where polar angle was fixed and varied across trials, respectively. Angular frequency is represented by the color of each bar (DoG 3: blue; DoG 6: red), and errors bars represent ±1 SEM.
Figure 4
 
Thresholds measured at oblique and cardinal orientations in Experiments 1 and 2. Average thresholds are shown individual observers as well as averaged across observers. Error bars represent ±1 SEM.
Figure 4
 
Thresholds measured at oblique and cardinal orientations in Experiments 1 and 2. Average thresholds are shown individual observers as well as averaged across observers. Error bars represent ±1 SEM.
Figure 5
 
Detection accuracy plotted against log-transformed modulation amplitude for conditions in which the location of curvature deformations was either (a) blocked or (b) interleaved across trials. In both figures, data were collapsed across observers and irrelevant experimental variables, and the number of trials for each point is represented by symbol size. The black lines are the best-fitting SDT–PS model for each condition with Q = 1, while the dashed red lines are the best fitting SDT–PS model for each condition with Q = 24. The horizontal dotted lines represent chance performance.
Figure 5
 
Detection accuracy plotted against log-transformed modulation amplitude for conditions in which the location of curvature deformations was either (a) blocked or (b) interleaved across trials. In both figures, data were collapsed across observers and irrelevant experimental variables, and the number of trials for each point is represented by symbol size. The black lines are the best-fitting SDT–PS model for each condition with Q = 1, while the dashed red lines are the best fitting SDT–PS model for each condition with Q = 24. The horizontal dotted lines represent chance performance.
Figure 6
 
Detection thresholds measured in Experiment 2 for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is coded for by line color and symbol shape. Error bars represent ±1 SEM.
Figure 6
 
Detection thresholds measured in Experiment 2 for three observers plotted as a function of polar angle of peak positive curvature. The angular frequency of the DoG contour is coded for by line color and symbol shape. Error bars represent ±1 SEM.
Figure 7
 
Polar angle identification results from Experiment 2 on trials in which the curvature was near detection threshold. (a) Mean accuracy of angle identification judgements for each angular frequency and each polar angle. The dashed lines indicate chance performance. (b) Mean absolute error in angle identification responses (in degrees of polar angle) plotted as a function of DoG angular frequency and polar angle. Dashed lines represent the predicted error based on guessing. Error bars represent ±1 SEM.
Figure 7
 
Polar angle identification results from Experiment 2 on trials in which the curvature was near detection threshold. (a) Mean accuracy of angle identification judgements for each angular frequency and each polar angle. The dashed lines indicate chance performance. (b) Mean absolute error in angle identification responses (in degrees of polar angle) plotted as a function of DoG angular frequency and polar angle. Dashed lines represent the predicted error based on guessing. Error bars represent ±1 SEM.
Figure 8
 
(a) Detection accuracy plotted against log-transformed modulation amplitude, separately for trials on which identification was correct or incorrect. (b) Identification accuracy plotted separately for trials on which detection was correct or incorrect. In both figures, the data were collapsed across conditions and observers, and the number of trials for each point is represented by symbol size. The green lines are the best-fitting (weighted least-squares) Weibull function fit to data from correct (a) identification or (b) detection responses; the black lines are the Weibull functions fit to all of the data. The horizontal dashed lines represent chance performance, and the dotted lines indicate points on the psychometric functions where d′ equals 2, which corresponds to 92% and 71% accuracy in the detection and identification tasks, respectively.
Figure 8
 
(a) Detection accuracy plotted against log-transformed modulation amplitude, separately for trials on which identification was correct or incorrect. (b) Identification accuracy plotted separately for trials on which detection was correct or incorrect. In both figures, the data were collapsed across conditions and observers, and the number of trials for each point is represented by symbol size. The green lines are the best-fitting (weighted least-squares) Weibull function fit to data from correct (a) identification or (b) detection responses; the black lines are the Weibull functions fit to all of the data. The horizontal dashed lines represent chance performance, and the dotted lines indicate points on the psychometric functions where d′ equals 2, which corresponds to 92% and 71% accuracy in the detection and identification tasks, respectively.
Table 1
 
Jeffreys-Zellner-Siow (JZS) Bayes factors (BF) quantifying the strength of evidence that thresholds differed between cardinal and oblique orientations for each observer in Experiments 1 and 2.
Table 1
 
Jeffreys-Zellner-Siow (JZS) Bayes factors (BF) quantifying the strength of evidence that thresholds differed between cardinal and oblique orientations for each observer in Experiments 1 and 2.
Table 2
 
Parameter values of probability summation models fit to data using different number of monitored channels (Q) for the blocked and interleaved conditions in Experiment 1. Values for gain(g) and tau(τ) were estimated separately for each condition prior to generating function fits for models manipulating parameter Q.
Table 2
 
Parameter values of probability summation models fit to data using different number of monitored channels (Q) for the blocked and interleaved conditions in Experiment 1. Values for gain(g) and tau(τ) were estimated separately for each condition prior to generating function fits for models manipulating parameter Q.
Table 3
 
Contingency tables for detection and identification judgements for near-threshold trials (between staircase reversals 8 and 16) in Experiment 2.
Table 3
 
Contingency tables for detection and identification judgements for near-threshold trials (between staircase reversals 8 and 16) in Experiment 2.
Table 4
 
Best-fitting values of α and β in Eq. 4 for each subject and task in Experiment 2. The fits were performed on data that were combined across stimulus conditions. The value of the lapse rate (γ) was fixed at 0.01, and the guessing rate (g) was 0.5 and 0.125 for the detection and identification tasks, respectively.
Table 4
 
Best-fitting values of α and β in Eq. 4 for each subject and task in Experiment 2. The fits were performed on data that were combined across stimulus conditions. The value of the lapse rate (γ) was fixed at 0.01, and the guessing rate (g) was 0.5 and 0.125 for the detection and identification tasks, respectively.
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