In this paper, we examine the mechanisms underlying the perceptual integration of two types of contour: *snakes* (composed of Gabor elements parallel to the path of the contour) and *ladders* (with elements perpendicular to the path). We varied the element separation and carrier wavelength. Increasing the element separation impaired detection of snakes but did not affect ladders; at high separations, snakes and ladders were closely matched in difficulty. One subject showed no effect of carrier wavelength, and the other showed a decline in performance as the wavelength increased. We discuss how these results might be accommodated by association field models. We also present a new model in which the linkage results from overlap in the filter responses to adjacent elements. We show that, if 1st-order filters are used, the model's performance on widely spaced snake contours deteriorates greatly as the carrier wavelength of the elements decreases, in contrast to our psychophysical results. To integrate widely spaced contours with short carrier wavelengths, the model requires a 2nd-order process, in which a nonlinearity intervenes between small-scale 1st-stage filters and large-scale 2nd-stage filters. This model detects snakes when the 1st and 2nd stage filters have the same orientation, and detects ladders when they are orthogonal.

*snakes*and

*ladders*.

*c*is the carrier, and

*w*is the envelope, as defined in Equations 2 and 3, respectively:

*L*is the luminance at position (

*x, y*), measured from the center of the Gabor patch;

*L*

_{0}is the mean (background) luminance of 52 cd/m

^{2};

*C*is the Michelson contrast, which was set to 0.9;

*λ*is the wavelength of the Gabor carrier;

*σ*controls the “width” of the Gaussian envelope; and

*θ*is the orientation of the element from vertical.

*h*/√2 from its nearest neighbor, where

*h*is the height (and width) of a grid square. This ensured that there was never more than a slight overlap between the elements. Overlapping values were simply added (note that this addition occurred before the contrast function,

*cw,*was converted to luminance using Equation 1). For stimuli containing a contour, the contour was first positioned randomly within the grid, and then each remaining empty grid square was filled with one element with random orientation and random position within the grid square, as with the no-contour stimulus.

*element separation, s*. For snake contours, the element was oriented parallel to the segment; for ladders, the element was orthogonal to the segment.

*path angle*. The sign of this difference was random for each pair of adjacent segments. For each pair, the path angle was jittered by adding a random value uniformly distributed between ±10°.

*s*/ (1 + √2), which ensured that the mean separation between adjacent distractors was close to the element separation,

*s*(Beaudot & Mullen, 2003). For more details of the contour generation algorithm, see Beaudot and Mullen (2003).

Parameter | Value |
---|---|

Grid size (in terms of number of cells) | 10 × 10 |

Element contrast | 0.9 |

Carrier spatial frequency (c/deg) | 5.19, 3.67, 2.59, 1.83 |

Carrier wavelength, λ (deg visual angle) | 0.193, 0.273, 0.385, 0.545 |

σ (deg visual angle) | 0.136 |

λ/ σ | 1.41, 2, 2.83, 4 |

Separation, s (deg visual angle) | 1.09, 1.54, 2.18, 3.08 |

Width of a single square within the grid, for each separation value (deg visual angle) | 0.903 1.28 1.81 2.55 |

s/ λ | 5.66 4.00 2.83 2.00 8.00 5.66 4.00 2.83 11.3 8.00 5.66 4.00 16.0 11.3 8.00 5.66 |

s/ σ | 8, 11.3, 16, 22.6 |

Path angle | 0°, 10°, 20°, 30°, 40° |

Path angle jitter | Uniform probability between ±10° |

Orientation jitter | None |

Separation jitter | None |

Stimulus duration | 500 ms |

Inter-stimulus interval duration | 1000 ms |

^{2}). Subjects were allowed to move their eyes during the presentation of the stimuli. On each trial, the subject had to indicate, using a button box, which interval contained the contour. After each trial, the subject received auditory feedback to indicate whether the response was correct or incorrect.

*α*

_{1}, was chosen such that the type I error probability across all four correlations,

*α*

_{ n}, was equal to 0.05. If there are

*n*correlations, then

*n*= 4,

*α*

_{1}= 0.0127. For the correlations of performance against separation, we took each set of four scores for a particular contour type and wavelength (i.e., each set of connected data points in Figure 5) and divided each score within the set of four by the mean of the set. This normalization reduced any variance due to differences in performance across wavelength, leading to a purer measure of the effect of separation. Similarly, for the correlations of performance against wavelength, we took each set of four scores for a particular contour type and separation (i.e., each set of connected data points in Figure 6) and divided each score within the set of four by the mean of the set.

Subject | Independent variable | Contour type | Pearson correlation of normalized performance vs. independent variable | Pearson correlation of unnormalized performance vs. independent variable |
---|---|---|---|---|

BCH | Separation | Snake | r = −0.904 ( p = 1.5 × 10 ^{−6})* | r = −0.866 ( p = 1.4 × 10 ^{−5})* |

Separation | Ladder | r = −0.554 ( p = 0.026) | r = −0.529 ( p = 0.035) | |

Wavelength | Snake | r = −0.572 ( p = 0.021) | r = −0.271 ( p = 0.31) | |

Wavelength | Ladder | r = 0.0988 ( p = 0.72) | r = 0.0770 ( p = 0.78) | |

KAM | Separation | Snake | r = −0.915 ( p = 6.8 × 10 ^{−7})* | r = −0.837 ( p = 5.2 × 10 ^{−5})* |

Separation | Ladder | r = −0.0146 ( p = 0.96) | r = −0.0140 ( p = 0.96) | |

Wavelength | Snake | r = −0.801 ( p = 1.9 × 10 ^{−4})* | r = −0.384 ( p = 0.14) | |

Wavelength | Ladder | r = −0.818 ( p = 1.1 × 10 ^{−4})* | r = −0.814 ( p = 1.2 × 10 ^{−4})* |

*λ*) for snake stimuli, while keeping the Gabor envelope size (

*σ*) constant. Their four longest-wavelength stimuli had the same ratios

*λ*/

*σ*as the stimuli in our four different wavelength conditions (i.e., our Gabor micro-patches were scaled copies of theirs). They found that, over this range, the carrier wavelength had little effect, although performance was impaired when the carrier frequency was very high (>8 c/deg). They speculated that this might be because the critical variable for determining contour detectability is separation expressed in units of

*λ*. However, our results, plotted in Figure 7, rule out this idea.

*association field*that links features together. There have been few attempts to implement an association field model that can integrate both snake and ladder contours. One such model, which accounted for a wide range of psychophysical data, was described by Yen and Finkel (1996, 1997, 1998). Their model contained two sets of facilitatory links, one favoring co-axial (snake) configurations and the other favoring trans-axial (ladder) configurations. Within each set, the association strength fell as Gaussian functions of element separation and deviation from co-circularity. Mutually facilitated units developed synchronized temporal oscillations, and the model grouped temporally synchronized units into a single contour.

*crowding,*whereby identification of a peripherally located target letter is disrupted by the presence of flanking letters.

^{1}The lack of an effect of feature width on linking performance shown by BCH fits well with the proposal that contour integration and crowding are mediated by similar mechanisms. A key characteristic of crowding is that the critical target-flanker spacing necessary for crowding to occur is independent of the size of the target or flanker (Pelli et al., 2004). This suggests that the range of integration/association field sizes available is not determined by the size or scale of the features to be integrated.

*filter-overlap*models, the linkage occurs purely because the filter responses to adjacent elements overlap. Hess and Dakin (1997, 1999) implemented a filter-overlap model that linked filter responses within individual orientation channels. The output of each filter was thresholded, giving rise to zero-bounded regions (ZBRs), and the model performed the contour integration task by looking for the longest ZBR. Hess and Dakin showed that the model was much worse at detecting curved contours than human subjects viewing the stimuli in the fovea.

*λ*is the carrier wavelength;

*x*and

*y*represent the horizontal and vertical displacement from the center of the kernel; and

*u*and

*v*represent displacement perpendicular to and parallel to the bars of the sinusoidal carrier, respectively. If the carrier has an orientation of

*θ*from vertical, then

*u*and

*v*are given by

*σ*

_{ u}and

*σ*

_{ v}are the standard deviations of the envelope perpendicular to and along the bars of the carrier, respectively.

*σ*

_{ v}/

*λ*to approximately the highest physiologically plausible value. To arrive at this value, we examined the data of Jones and Palmer (1987), who fitted a Gabor model to the receptive fields of simple cells in the cat's striate cortex. We selected the cell with the highest ratio

*σ*

_{v}/

*λ*and, for this cell (labeled 0811 by Jones & Palmer, 1987), we noted the values of the ratios

*σ*

_{v}/

*σ*

_{u}and

*σ*

_{u}/

*λ,*which were 2.41 and 0.340, respectively

^{2}. We fixed the corresponding ratios in our filter kernels at these values so that, apart from the orientation,

*λ*was the only free parameter of the kernel.

*θ,*from 0° to 172.5°, differing in steps of 7.5°. Examples of the filtered images are shown in Figure 8. A threshold was set at 2.2 standard deviations above the mean of the filtered image values across all orientations (the expected value of this mean was zero). Values above the threshold were set to 1; all other values were set to 0. This created a set of ZBRs in each orientation channel, also shown in Figure 8.

*σ*

_{ v}(filter kernel length) to kernel carrier wavelength was set at the highest physiologically plausible value, to maximize the model's ability to integrate the contour when the element separation was much larger than the element carrier wavelength. But even though we maximized this ratio, Figure 12 shows that, when the element wavelength is set at the smallest value used in the experiment (0.19°), the model can only integrate the contour with the smallest separation, and for that, requires a filter with a carrier wavelength 1.5 octaves above that of the elements. If the filter kernel size is increased by another half-octave, then its carrier wavelength is so large that it cannot respond significantly to the elements, and the result is largely noise.

- increasing the separation between the elements had a disruptive effect on the detection of snakes but had no effect on ladders, so that as separation increased, performance on the two contour types converged;
- in most cases, performance was largely a function of absolute separation rather than separation expressed as a multiple of the carrier wavelength of the stimulus elements; and
- increasing the carrier wavelength had no effect on one subject and caused a decline in performance for the other.

*x*is the distance between them. Suppose, for a particular contour element, there is another contour element (the “signal”) at a distance

*s*from the element, and a distractor element (the “noise”) at a distance

*n*. Then the signal-to-noise ratio, SNR

_{1}, for that pair of elements (assuming equal association strength in other respects due to equal levels of co-circularity, etc.) will be given by

*m*> 1, then the new signal-to-noise ratio, SNR

_{2}, is given by

*m*> 1, SNR

_{2}/SNR

_{1}> 1 if

*n*>

*s,*and SNR

_{2}/SNR

_{1}< 1 if

*n*<

*s*. Thus, increasing the element separation causes the signal-to-noise ratio to increase for distractor elements further away than the neighboring contour element and causes it to decrease for distractor elements closer than the neighboring contour element. The overall effect of increasing the separation on the signal-to-noise ratio will depend on the distribution of distractor positions, and the relative strengths of the inputs from the different elements due to other factors, such as co-circularity.

^{2}In Jones and Palmer's Gabor receptive field model (as in ours),

*σ*

_{ u}and

*σ*

_{ v}are the standard deviations of the envelope along its minor and major axes, respectively. A difference is that, in our model, the bars of the carrier were always parallel to the major axis of the envelope, whereas Jones and Palmer allowed the carrier to be oriented differently: For cell 0811, the carrier was oriented 5° from the major axis of the envelope, so our filter kernels differed slightly from scaled versions of the Gabor function that Jones and Palmer fitted to this cell's receptive field.