**The ability of human participants to integrate fragmented stimulus elements into perceived coherent contours (amidst a field of distracter elements) has been intensively studied across a large number of contour element parameters, ranging from luminance contrast and chromaticity to motion and stereo. The evidence suggests that contour integration performance depends on the low-level Fourier properties of the stimuli. Thus, to understand contour integration, it would be advantageous to understand the properties of the low-level filters that the visual system uses to process contour stimuli. We addressed this issue by examining the role of stimulus element orientation bandwidth in contour integration, a previously unexplored area. We carried out three psychophysical experiments, and then simulated all of the experiments using a recently developed two-stage filter-overlap model whereby the contour grouping occurs by virtue of the overlap between the filter responses to different elements. The first stage of the model responds to the elements, while the second stage integrates the responses along the contour. We found that the first stage had to be fairly broadly tuned for orientation to account for our results. The model showed a very good fit to a large data set with relatively few free parameters, suggesting that this class of model may have an important role to play in helping us to better understand the mechanisms of contour integration.**

Parameter | Value (degrees visual angle) | Rationale |

σ_{u}_{,1} | 0.0629 | Fit to the data |

σ_{u}_{,1}/λ_{1} | 0.2906 | σ/_{u}λ = 0.2906 is the median value of this ratio from Jones and Palmer's (1987) physiological data set. |

σ_{v}_{,1}/σ_{u}_{,1} | 1.0714 | Aspect ratio σ/_{v}σ = 1.0714 is the smallest value from Jones and Palmer's (1987) data set, giving the largest physiologically plausible orientation bandwidth for the first-stage filter (full-width at half-height was 70.5°). The need for a wide orientation bandwidth for the first-stage filter was indicated by the relatively mild effect of the notch in Experiment 2._{u} |

σ_{u}_{,2} | 0.5032 | Fit to the data |

σ_{u}_{,2}/λ_{2} | 0.33964 | σ/_{u}λ = 0.33964 is the value of this ratio for Jones and Palmer's (1987) cell with the highest ratio of σ/_{v}λ. May and Hess (2008) argued that a high value of σ/_{v}λ would benefit contour integration. |

σ_{v}_{,2}/σ_{u}_{,2} | 2.407 | Aspect ratio σ/_{v}σ = 2.407 is the aspect ratio of Jones and Palmer's (1987) cell with the highest ratio of _{u}σ/_{v}λ, as described above. The orientation bandwidth (full-width at half-height) was 26.5°. |

^{2}, frame rate of 120 Hz, and resolution of 1024 × 768 pixels. The stimuli were generated using C routines called from MATLAB (The MathWorks, Inc., Natick, MA) version 7.0 and were linearly scaled to fit the range 0–255 and stored in an 8-bit frame store on the VSG card and subsequently scaled to the correct contrast and gamma corrected by mapping the 8-bit values onto 15-bit values. An analogue input to the monitor was generated from these 15-bit values using two 8-bit digital-to-analogue converters in the VSG card. Participants viewed the display binocularly from a distance of 60 cm.

^{1}Both participants had normal or corrected-to-normal vision. All experiments conformed to the ethical standards of the Federal Code of Regulations Title 45 (Public Welfare) and Department of Health and Human Services, Part 46 (Protection of Human Subjects). Institutional Review Board-approved (McGill University) informed written consent was obtained.

*f*

_{0}= 4.5 cycles per degree (cpd), and had a standard deviation of

*σ*= 2.25 cpd. Thus, if points in Fourier space are defined in Polar coordinates (

_{f}*f*,

*θ*), where

*f*is spatial frequency and

*θ*is angle (in degrees) from the horizontal axis in Fourier space, then the spatial frequency filter had the form

*θ*

_{0}is the orientation of the element from vertical (in stimulus space), and

*σ*, which we call the

_{θ}*orientation bandwidth SD*, controls the element's orientation bandwidth. The filter described in Equation 2 had to contain two wedges to maintain a complex-conjugate relationship between points on opposite sides of Fourier space, so that the image obtained after inverse-Fourier transforming did not contain any imaginary components.

*n*(

*f*,

*θ*) is the noise value at position (

*f*,

*θ*) in Fourier space, then the filtered noise was given by

*n*(

*f*,

*θ*) ×

*w*(

_{f}*f*,

*θ*) ×

*w*(

_{θ}*f*,

*θ*). The combined orientation and spatial frequency filter has the property that the spatial frequency bandwidth is constant across orientation, and the orientation bandwidth is constant across spatial frequency. The direct current (DC) component was then set to zero, and the filtered noise was inverse-Fourier transformed, and windowed with a circular Gaussian spatial envelope with standard deviation 0.17° visual angle. Finally, the element contrast was scaled to give a root-mean-square (RMS) contrast of 0.12, and was set to possess a mean pixel luminance that matched the background.

^{2}Part of an example contour is represented schematically in Figure 2. The contour was constructed along an invisible backbone of nine line segments, joined end to end. A contour element was placed at the center of each segment. The absolute difference in orientation between adjacent segments is referred to as the

*path angle*,

*α*(which controls contour curvature), and was varied systematically in our experiments. In order to reduce any chance alignments of three or more elements along the contour paths, we added ±10° of random (uniform distribution) path angle jitter, Δ

*α*, to all contour elements in all experiments of the current study (e.g., Hansen & Hess, 2006). Contour elements were further randomly jittered by ±5 pixels along the contour path (i.e., Δ

*d*). Finally, the

*element angle*(i.e., the difference in orientation between a given path segment and the central orientation of a given element) was varied by adding a value, Δ

*θ*, sampled from a zero-mean Gaussian distribution whose standard deviation (the angle

*SD*) was varied systematically in the experiment. Experiment 1 was designed to test the effects of varying the parameters of (a) orientation bandwidth

*SD*of the stimulus field elements, (b) element angle

*SD*, and (c) the path angle on the ability of humans to detect contours made up of orientation-filtered random noise elements. All three parameters were varied in a factorial design. In each trial, the stimulus element orientation bandwidth,

*σ*, was 2°, 10°, 20°, 30°, or 40°. For stimuli containing contours, the angle

_{θ}*SD*was 0°, 10°, 20°, 30°, or 40°, and the path angle was 0°, 10°, 20°, or 30°. Refer to Figure 3 for stimulus examples.

*w*in Equation 2 a narrower wedge shape,

_{θ}*w*

_{notch}, given by where

*σ*

_{notch}, which we call the notch

*SD*, controls the width of the notch. The orientation bandwidth

*SD*,

*σ*, was set to 25° for Experiment 2, and

_{θ}*σ*

_{notch}took values of 2°, 6°, or 12°. As for Experiment 1 stimuli, the angle

*SD*could take values of 0°, 10°, 20°, 30°, or 40°, and the path angle took values of 0°, 10°, 20°, or 30°. See Figure 4 for stimulus examples. All three variables were varied in a factorial design.

*SD*= 0°) since it is with that particular parameter setting that performance is at its highest (therefore allowing a full range of performance modulation).

*SD*, and path angle) were blocked by noise element orientation bandwidth and angle

*SD*; within each block the path angle of the contour varied between one of the four values mentioned above. Each block had 20 trials per path angle, resulting in 80 trials per block. Each block was repeated five times, resulting in 100 trials per level of path angle for each noise element orientation bandwidth and element angle (total number of trials for Experiment 1 was 10,000).

*SD*, angle

*SD*, and path angle) were blocked by notch

*SD*and element angle

*SD*; within each block, path angle varied between one of the four curvatures mentioned above. Each block had 20 trials for each level of path angle, resulting in 80 trials per block. Each block was repeated five times, resulting in 100 trials per level of curvature for each noise element orientation notch bandwidth and angle

*SD*(total number of trials for Experiment 2 was 6,000).

*λ*, the carrier wavelength;

*σ*, the envelope standard deviation in a direction perpendicular to the bars of the carrier (i.e., the receptive field width);

_{u}*σ*, the envelope standard deviation in a direction parallel to the bars of the carrier (i.e., the receptive field length). These three parameters (

_{v}*λ*,

*σ*,

_{u}*σ*) can alternatively be presented as (

_{v}*σ*,

_{u}*σ*/

_{u}*λ*,

*σ*/

_{v}*σ*). The

_{u}*σ*parameter describes the overall scale (i.e., size) of the filter kernel, while the

_{u}*σ*/

_{u}*λ*and

*σ*/

_{v}*σ*parameters describe the

_{u}*shape*of the kernel. To constrain the fit, we set the two shape parameters,

*σ*/

_{u}*λ*and

*σ*/

_{v}*σ*, at each filter stage to sensible values based on physiological findings and the demands of the contour integration task, and fit only the scale parameter,

_{u}*σ*, of each stage, and the threshold. The filter parameter values that we used, along with the rationale, are given in Table 1, with the extra subscript “1” or “2” indicating first- or second-stage filter kernel. The fitting process itself was a manual procedure, guided by intuition. Simulating the experiments with just one set of parameter values was immensely time-consuming, requiring many computers running in parallel for many months, so it was not possible to fit the parameters using standard fitting methods. Instead, we simulated some selected conditions with a few sets of Gabor filter parameters, and then, having found the set of filter parameters that fit best to the selected conditions, we proceeded to simulate every condition of every experiment with these filter parameters. The only parameter that was fit in the conventional sense was the threshold since, having obtained the filter outputs, we could easily set the threshold at many different levels and map out the contours for each threshold level. The threshold is expressed in standard deviations above the mean second-stage filter response across all orientation channels.

_{u}*X*+ 1,

*N*−

*X*+ 1), where

*N*is the total number of trials on that condition, and

*X*is the observed number of correct trials (Nicholson, 1985). The lower confidence limit is the 0.025 quantile of this distribution, and the upper limit is the 0.975 quantile: where

*B*

^{−1}is the inverse cumulative distribution function of the beta distribution. These can be calculated in MATLAB using the

*betainv*function in the Statistics Toolbox.

*path angle*) exhibited by the contour in a field of distracter elements, and (b) the contour element-to-path orientation alignment, i.e., the amount of noise in the relative orientation alignment of the elements making up the contour to that contour's path trajectory (referred to as

*angle SD*). There were two reasons for these other manipulations. Firstly, they would enable us to see whether the influence of orientation bandwidth interacted with the other stimulus properties; secondly, the angle

*SD*provided a manipulation that could be compared in size with the orientation bandwidth manipulation, in the same units.

*SD*, whereas in Figure 6, each curve plots the performance as a function of angle

*SD*. It is clear that performance drops much faster with increasing angle

*SD*(Figure 6) than with increasing orientation bandwidth

*SD*(Figure 5). This is a meaningful comparison because both angle

*SD*and orientation bandwidth

*SD*are expressed in the same units. To quantify this comparison, we calculated the drop in proportion correct as orientation bandwidth

*SD*increased from 10° to 40°. Table 2 shows the mean drop across subjects for each path angle and each angle

*SD*≥ 10°. We also calculated the drop in proportion correct as angle

*SD*increased from 10° to 40°. Table 3 shows the mean drop across subjects for each path angle and each orientation bandwidth

*SD*≥ 10°. By comparing corresponding cells in the two tables, we can see how much more performance was impaired by the increase in angle

*SD*than by the increase in orientation bandwidth

*SD*(we omitted the smallest orientation bandwidth

*SD*and angle

*SD*from this analysis, as they were not identical, being equal to 0° and 2°, respectively, so these conditions were not directly comparable). Table 4 shows the result of subtracting each cell of Table 2 from the corresponding cell of Table 3. In every case, the difference was positive, indicating that angle

*SD*had a greater effect. We subjected the data in Table 4 to a one-sample

*t*test against a hypothesized mean of zero, and found a highly significant difference,

*t*(15) = 8.34,

*p*= 5.15 × 10

^{−7}. This confirms that when the orientation bandwidth

*SD*is held at

*σ*while the angle

*SD*increases from 10° to 40°, the drop in performance is much greater than when the angle

*SD*is held at

*σ*while the orientation bandwidth

*SD*increases from 10° to 40°.

Path angle 0° | Path angle 10° | Path angle 20° | Path angle 30° | |

Angle SD 10° | 0.17 | 0.195 | 0.29 | 0.18 |

Angle SD 20° | 0.1325 | 0.08625 | 0.135 | −0.01 |

Angle SD 30° | −0.005 | 0 | −0.045 | 0.045 |

Angle SD 40° | 0.08 | 0.015 | −0.105 | 0.1 |

Path angle 0° | Path angle 10° | Path angle 20° | Path angle 30° | |

Bandwidth SD 10° | 0.365 | 0.375 | 0.44 | 0.205 |

Bandwidth SD 20° | 0.47 | 0.33 | 0.285 | 0.215 |

Bandwidth SD 30° | 0.265 | 0.3 | 0.26 | 0.15 |

Bandwidth SD 40° | 0.275 | 0.195 | 0.045 | 0.125 |

Path angle 0° | Path angle 10° | Path angle 20° | Path angle 30° | |

Angle & bandwidth SD 10° | 0.195 | 0.18 | 0.15 | 0.025 |

Angle & bandwidth SD 20° | 0.3375 | 0.24375 | 0.15 | 0.225 |

Angle & bandwidth SD 30° | 0.27 | 0.3 | 0.305 | 0.105 |

Angle & bandwidth SD 40° | 0.195 | 0.18 | 0.15 | 0.025 |

*SD*controls the spread of contrast energy across orientation for an individual element, the angle

*SD*controls how much the peak of this distribution deviates from the element's contour path segment. It is clear that for fairly straight paths, performance depends critically on alignment of the peak of the energy distribution with the path segment, and is relatively insensitive to the spread of this distribution across orientation. This is still true to some extent for the larger path angles (indicated by the positive values in every cell of Table 4), but the effects of angle

*SD*and element orientation bandwidth

*SD*are more similar to each other, both converging at chance performance. It therefore seems as though contour integration performance is more narrowly tuned for orientation bandwidth

*SD*for larger path angles, as long as the angle

*SD*is small enough to avoid floor effects (i.e., angle

*SD*≥ 20). To verify this quantitatively, the participant-averaged performance curves from Figure 5 for the 0° and 10° path angle conditions (angle

*SD*= 20 to avoid ceiling effects) were best fit by a truncated Gaussian function [–90 90] with a bandwidth (half-width at half-height) of ∼40°. Using the fitting procedure described above, the participant-averaged performance curves for the 20° path angle conditions yielded a bandwidth (half-width at half-height) of ∼25°. This difference in estimated bandwidth might seem to imply a difference between the underlying mechanisms for curved and straight contours, but in fact we show later that a single contour integration mechanism applied to all stimuli can account for this effect.

*SD*). On the other hand, a filter broadly tuned for orientation would bridge the notch, resulting in a unimodal distribution of energy centered on the element orientation, so a contour integration mechanism that was broadly tuned for element orientation should still perform reasonably well in the presence of a notch, as long as the angle

*SD*and path angles were low.

*SD*s of 0° and 10°, respectively. As argued above, this suggests that the elements are processed with a mechanism that is not simply narrowly tuned for orientation; otherwise one would expect to find chance (or near chance) performance at the 6° and 12° notch conditions. We return to this notion in the modeling section (Part II).

*SD*s for two different orientation notch filter bandwidth

*SD*s. Dakin and Hess (1998) employed a similar paradigm to assess the spatial frequency tuning of contour integration performance. Thus, the idea here was to provide an alternative paradigm to that explored in Experiment 2. However, it is worth noting that the spatial frequency alternation of contour elements in Dakin and Hess (1998) does not cleanly map onto a paradigm that alternates the orientation bandwidth of contour elements. Specifically, mechanisms tuned for spatial frequency will respond to some spatial frequencies but not others. Thus, if the spatial frequencies of the alternating elements differ by more than the tuning width of the contour integration mechanism, then contour integration will break down (leaving the critical spatial frequency difference as a measure of the tuning width of the contour integration mechanism). However, it's not clear whether such a breakdown process could be expected for alternating orientation bandwidths. Therefore, the primary motivation for conducting Experiment 3 was to provide additional data to test the two-stage filter-overlap contour integration model in order to develop a more robust estimate of orientation bandwidth of the front-end processes underlying the contour integration mechanism.

*SD*alternating between 2° and 30°, or alternating between 2° and 40°, with data replotted from Experiment 1 for comparison.

*SD*alternating between 0° (i.e., no notch) and 6°, or alternating between 0° and 12°, with data replotted from Experiment 2 for comparison. Interestingly, performance in both notch conditions (i.e., 0°–6° or 0°–12°) is much higher in the alternating no-notch and notch bandwidths than in the notch conditions measured in Experiment 2.

*SD*or notch width, showed a substantial effect of the threshold. This shows that, for more ordinary contours, which do not fluctuate wildly in their properties from element to element, May and Hess's (2008) contour integration algorithm is quite robust to changes in the threshold level, so the threshold level does not have to be set to a precise value for the mechanism to work well. This suggests that the model is a robust algorithm that could be implemented in imperfect biological hardware.

*SD*s above the mean second-stage filter response. Figures 10 through 13 plot the performance of the model with this threshold level, and all the filter parameters fixed at the values given in Table 1. This model, with a single set of parameters across all conditions of all experiments, gives a surprisingly good fit to the whole data set: Mean absolute difference in proportion correct between the model and the averaged human data was only 0.0438 (this is the value plotted with the dashed line for threshold 2.6 in Figure 9). Thus, the data for all the different contours in our experiment can be accounted for by a single contour integration mechanism. Given the limited search of parameter space that we could feasibly conduct, it is almost certain that we missed a set of parameters that would have fit the data even better, but the fits that we obtained with the parameter values in Table 1 are very satisfactory.

*r*, between these two datasets is 0.934, so the model explains 87.4% of the variance in the psychophysical data (this is 100 ×

*r*

^{2}). The

*t*statistic corresponding to

*r*= 0.934 is given by

*t*= 48.2 (

*df*= 174), which is so high that

*p*evaluates to zero in MATLAB, suggesting that

*p*is too small to represent using a 64-bit floating point number.

BCH | RFH | S1 | S2 | |

BCH | — | 0.0585 | 0.0597 | 0.0539 |

RFH | — | — | 0.0567 | 0.0625 |

S1 | — | — | — | 0.0482 |

S2 | — | — | — | — |

BCH | RFH | S1 | S2 | |

BCH | — | 0.889 | 0.892 | 0.905 |

RFH | — | — | 0.892 | 0.874 |

S1 | — | — | — | 0.923 |

S2 | — | — | — | — |

*SD*was large. We took the conditions from the top-right panel of Figure 11 (notch

*SD*12°, path angle 0°) and simulated these conditions with first-stage filters set to have a narrower orientation bandwidth. We refer to this new parameterization as the “narrowband model”; the parameterization shown in Table 1 (and plotted in all the other figures) is referred to as the “broadband model.” All the filter parameters in the narrowband model were as in Table 1, except for the first-stage aspect ratio,

*σ*

_{v,1}/

*σ*

_{u,1}which was set to 2. The higher aspect ratio elongated the Gabor envelope, resulting in an orientation bandwidth (full-width at half-height) of 37°. We varied the threshold from 1.7 to 3.6 in steps of 0.1, and selected the threshold that gave the lowest MAD between the narrowband model and the human data (averaged across subjects) from the five conditions in the top-right panel of Figure 11. The best-fitting threshold for the narrowband model was 1.8. Figure 16a plots the mean of the human data from the top-right panel of Figure 11, along with the performance of the narrowband model with a threshold of 1.8. For the smallest two angle

*SD*s (i.e., those with substantially above-chance human performance), the confidence intervals for the narrowband model and human data do not overlap. As a comparison, Figure 16b replots the broadband model data from the top-right panel of Figure 11, along with the mean human data from that panel: The confidence intervals of the broadband model and human data overlap considerably. In summary, we have shown that the model with the broadband first-stage filter is able to perform as well as humans when the notch

*SD*was large, whereas the model with the more narrowband first-stage filter cannot reach human performance. It should be noted that this comparison was greatly biased in favor of the narrowband model, because the threshold in this model was fitted just to the data in Figure 16, while the broadband model had its threshold set to give the best fit across all 176 conditions in the study. Yet, despite having this massive advantage, the narrowband model was unable to reach human performance levels on these conditions. We conclude that a broadband first-stage filter is necessary for the model to be able to integrate contours that contain a 12° orientation notch.

*SD*for small path angles, and more narrowly tuned for larger path angles. Experiment 2 showed that notch-filtering the stimulus elements with a notch centered on the element orientation impaired performance to some extent, but did not completely disrupt the contour integration process. The latter finding suggests that, in contour integration, the stimulus is initially processed with filters that have sufficiently wide orientation bandwidth to be able to bridge the notch. This notion was supported by the results of Experiment 3. As expected, element-to-path orientation misalignment (i.e., increased angle

*SD*) had a detrimental impact on integration performance, regardless of element orientation bandwidth or path angle.

*shapes*of the Gabor kernels (i.e., shapes of the receptive field profiles) were set to values constrained by taking into account both physiological data and the task requirements. The two parameters that determined the overall scales (i.e., sizes) of the first- and second-stage filter kernels (

*σ*

_{u}_{,1}and

*σ*

_{u}_{,2}) were fit to the data, as was the threshold.

*σ*

_{u}_{,1}and

*σ*

_{u}_{,2}, as free parameters in the current study. Since all conditions of all the experiments in this study had the same element spatial frequency and element spacing, all the conditions required the same filter scale parameters, so these parameters could be kept constant across all conditions. Keeping the element spatial frequency and element spacing constant, we varied several key stimulus parameters that should reveal characteristics of the contour integration mechanisms: These stimulus parameters were path angle, angle

*SD*, element orientation bandwidth, and element orientation notch width. Each stimulus parameter was varied over a very wide range. Across this large range of conditions, 100 (Experiment 1) + 60 (Experiment 2) + 8 (Experiment 3a) + 8 (Experiment 3b) = 176 conditions, a single set of model parameters fit well to all the data.

*integration*process. It is possible that increasing the association field width would cause more excessive grouping between distracter elements and contour elements than would be caused by increasing the front-end tuning width.

*afferent input*) at each spatial position was a probability distribution giving the relative probability of an edge as a function of orientation. Schinkel et al. assumed that this probability distribution could be identified with the cortical orientation tuning function. However, we argue that this is a false analogy. The afferent input to their model is essentially a posterior probability distribution that would arise from decoding the local orientation from the front-end neurons. The width of this posterior distribution does depend on neuronal tuning width, but it also depends on the number of neurons, the maximum spike rate of the neurons, the spontaneous firing rate, the Fano factor (ratio of neuronal spike variance to mean rate), and the stimulus itself. Any of these factors can trade off with each other in their contribution to decoding precision (see May & Solomon, 2014). If all we know is the posterior distribution, we cannot determine the cortical tuning width. Therefore, it cannot be correct to identify the afferent input to Schinkel et al.'s model with cortical tuning functions. We conclude that, although Schinkel et al.'s (2005) model is very interesting, it does not demonstrate conclusively that increasing the association field tuning can have similar effects to increasing the front-end filter tuning width, and further work will be needed to examine this hypothesis.

*f*, you can create a multidimensional space (

*x*,

*y*,

*f*), where

*x*and

*y*are spatial coordinates, and then plot the image in this space. Then image elements with similar attributes and spatial positions will be nearby and, if you blur the representation, they join up. As noted earlier, a quantitative comparison of the performance of this class of model with that of human subjects has so far been lacking. Our quantitative evaluation of the model showed a very good fit to a large data set with relatively few free parameters, suggesting that this class of model may have an important role to play in helping us to understand the mechanisms of contour integration, and perceptual grouping more generally.

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*Vision Research*^{1}Given the extensively systematic nature of the current study (e.g., 176 conditions in total), it is necessary to have experienced psychophysical observers so that they did not become better psychophysical observers during the course of the study. Further, the current study is completely exploratory, thus the participants had no expectations regarding ad hoc hypotheses and can both be considered naive to the outcome of the study.

^{2}Note that the relative contour and background elements were randomly positioned within each 16 × 16 grid cell (which is standard practice), and thus their distances from one another were also randomly distributed, and did not lead to any reliable detection cue when contour elements were present (note that contour detection performance in several of the conditions described later was at chance). Thus, performance cannot have been mediated by differences in density around the contours.