Differences between target and flanker orientations become exaggerated in the tilt illusion. However, small differences sometimes go unnoticed. This small-angle assimilation shares many similarities with other types of visual crowding but is typically found only with small and/or hard-to-see stimuli. In Experiment 1, we investigated the effect of stimulus visibility on orientation bias using relatively large stimuli. The introduction of visual noise increased the perceived similarity of target and flanker orientations at retinal eccentricities of 4° and 10°; however, small-angle assimilation was found only at 10°. The effects of eccentricity were reduced in Experiment 2, when our stimuli were “M-scaled” for equal cortical coverage. Further support for a cortical substrate was obtained in Experiment 3, in which the effects of target–flanker separation were measured. When biases from all three experiments are expressed as a fraction of the inducing flankers' angle, and plotted as a function of the approximate cortical separation between the target and its closest flanker, they form a curve like the cross-section of half a Mexican hat. We conclude that the tilt illusion and small-angle assimilation reflect opponent influences on orientation perception. The strength of each influence increases with cortical proximity and stimulus visibility, but the one responsible for assimilation has a lesser extent.

^{1}

*σ*equals 0.28°) centered on a white stripe. The center-to-center spacing of the target with each flanker was 1.75°. All four flankers had the same tilt with respect to horizontal. It was selected at random from the set {−22°, −10°, −5°, 5°, 10°, 22°}, where negative angles indicate anti-clockwise tilts. When noise was present, it was added to the target and flankers. Each of its 192 × 192 pixels was drawn from a normal distribution, fixed at 32% r.m.s. contrast. To discourage eye movements, the target, flankers, and noise (when present) were displayed simultaneously for 170 ms. In the Peripheral conditions, these stimuli appeared randomly on the left or right, 10° of visual angle away from fixation. In the parafoveal conditions, the retinal eccentricity was 4°, and in the foveal conditions, it was 0. There were six conditions in total: three retinal eccentricities with noise and the same eccentricities without noise. Different flanker tilts were randomly interleaved in blocks of 240 trials within each condition.

*w*is eccentricity in degrees, and

*M*(

*w*) is the cortical magnification factor as a function of eccentricity. Note that with this scaling procedure, spatial frequency as well as size changes with eccentricity, but bandwidth remains constant. We opted to at least approximately equate Experiment 2's parafoveal stimuli with those of Experiment 1. Thus, for all stimuli, we used a center-to-center spacing of 3.5

*λ*, where

*λ*denotes the wavelength of the grating. In the non-foveal conditions, the viewing distance remained 57 cm, but the fixation point was moved 3 cm from the left edge of the monitor, and the stimuli were only presented to the right of fixation. At 4° eccentricity, target and flankers had a spatial frequency of 1.96 c/deg. Their Gaussian windows had a standard deviation of 0.25°. Target–flanker separation was 1.8°. For the stimuli presented at 10° eccentricity, M-scaling resulted in a spatial frequency of 0.94 c/deg, Gaussian windows with standard deviation of 0.53°, and a target–flanker separation of 3.84°. To obtain the requisite spatial frequency at the fovea, we had to increase the viewing distance to 192 cm. This resulted in a spatial frequency of 12 c/deg, Gaussian windows with a standard deviation of 0.04°, and a target–flanker separation of 0.29°.

*θ*to the proportion of clockwise responses:

*μ*, and threshold is its standard deviation

*σ*(Solomon et al., 2004). In this paper, we discuss only biases.

^{2}

Fovea | Parafovea | Periphery | |
---|---|---|---|

Noise absent E1 | 0.03% | 1.64% | 1.94% |

Noise present E1 | 1.17% | 1.64% | 2.68% |

Noise absent E2 | 1.02% | 1.11% | 1.99% |

Noise present E2 | 2.60% | 1.71% | 1.12% |

Noise absent E3 | 1.20% | 0.76% |

*per se*, but rather whether they were assimilative or repulsive. Therefore, we flipped the sign of the biases that we measured with anti-clockwise flankers and pooled them with the biases that we measured with the corresponding clockwise flankers. The first four plots show the means of each observer's biases. The final plot shows the means of these means, after weighting the latter by the reciprocal of their standard errors.

^{3}In some cases (NG ± 5, IM ± 5, HLW ± 5, and HLW ± 10), the rightmost point of the V falls above the dashed line, indicating small-angle assimilation. In no cases, however, does the leftmost point fall above the dashed line. Thus, simply moving a stimulus into the periphery can change repulsion (i.e., the tilt illusion) into small-angle assimilation (and thus crowding). A similar finding was reported in the motion domain by Murakami and Shimojo (1993), who reported a switch from induced motion to motion capture that depended on viewing eccentricity.

^{4}. This can be seen even more clearly in Figure 5, where differential bias (i.e., bias in noise minus bias without noise) is plotted as a function of eccentricity. The addition of noise seems to have had little effect on the appearance of foveated stimuli and stimuli surrounded by ±5° flankers, but when the same stimuli were viewed at 4° and 10° eccentricities, biases increased. This indicates that the target looked more like its flankers when noise was present. In particular, the peripheral stimuli that produced moderate small-angle assimilation in the absence of noise produce quite marked small-angle assimilation here, where their visibility has been reduced by the addition of random texture.

^{5}, these data do not exhibit the non-monotonicity seen in the results of Experiment 1. Indeed, these data do not contain any suggestion of assimilation. It is worth recalling that the stimuli presented at 4° eccentricity were very similar in the two experiments, while the foveal stimuli were much smaller and the peripheral stimuli were much larger in Experiment 2 than in Experiment 1. It appears that the result of our attempt to compensate for cortical coverage by M-scaling the stimuli is a severely attenuated effect of retinal eccentricity.

*M*(

*w*) is the cortical magnification factor (in millimeters) as a function of eccentricity

*w*(in degrees), and

*c*is a constant. If Equation 3 is to hold for all

*w*> 0, then

*M*(

*w*) has to be logarithmic.

^{7}

*μ*, defined empirically in Equation 2) reflects the difference between an assimilative force and a repulsive force,

^{8}each of which varies in proportion to the difference between target and flank orientations Δ

*θ*, and a decaying function of the cortical distance between the target and its nearest flanker. In its most successful form, our model can be expressed as

*D*is the cortical distance and

*M*

_{ass},

*σ*

_{ass},

*M*

_{rep}, and

*λ*

_{rep}are potentially free parameters, but see below.

*D*= 0). Gaussian functions change most rapidly away from their peak (i.e., when

*D*> 0).

*w*

_{target}and its most peripheral flanker

*w*

_{flank}to approximate these distances:

*θ*is the (unsigned) orientation difference between the target and its flanks, and

*k*is a small, positive constant. Inclusion of this factor was motivated by the existence of orientation columns: In small regions of visual cortex, the proximity of any two neurons tends to increase with the similarity of their orientation preferences (Hubel & Wiesel, 1974). From a functional standpoint, inclusion of this term effectively reduces the influence of more oblique flankers on orientation bias.

*M*

_{ass},

*σ*

_{ass},

*M*

_{rep}, and

*λ*

_{rep}. The first of these parameters has a further constraint, stemming from the fact that assimilation can be no greater than 100% of the difference between target and flank orientations, i.e.,

*M*

_{ass}≤ 1. One final constraint concerns the constant

*k*. While it does not seem unreasonable to imagine any of the other four parameters changing with signal strength, it does not really make sense that the distance between orientation columns would also change. Therefore, when finding the best possible fit of the most general, yet sensible form of this model to our data, we allowed 9 parameters to vary freely: one value for

*k*, plus 2 values for each of the other four parameters (one for noisy stimuli, the other for noise-free stimuli).

*χ*

_{(1)}

^{2}(−2 ln Λ) < 0.02] to the more general (9-parameter) model. In one of these inferior fits, we fixed

*k*= 0. In two others, we either forced the high-SNR and low-SNR values of

*M*

_{ass}to be the same or we forced the high-SNR and low-SNR values of

*σ*

_{ass}to be the same. On the other hand, two nested models were not significantly inferior. (Comparison of the generalized likelihood ratio with chi-square suggests 1 −

*χ*

_{(1)}

^{2}(−2 ln Λ) > 0.25.) In one of these models, there was only one value for

*M*

_{rep}(i.e., the same for both signal-to-noise ratios). In the other, there was only one value for

*λ*

_{rep}. Either of these models would have been suitable for illustration in Figure 10. We selected the latter, simply because it produced a slightly better fit. When both

*M*

_{rep}and

*λ*

_{rep}were forced to remain invariant with SNR, the fit was significantly inferior.

*x*

_{ i,1},

*x*

_{ i,2}, …,

*x*

_{ i,Ni }} is a sample from Gaussian distribution

*i*, then we can estimate the likelihood of all measurements

*λ*, given any predicted set of predicted values {

*p*

_{1},

*p*

_{2}, …,

*p*

_{ M }}:

*φ*denotes the standard normal probability density function. The log likelihood is thus

*p*

_{ i }:

*T*=

*N*

_{ i }

*SSE*

_{ i }is minimized.

*λ*

_{0}and sup

*λ*denote the maximum likelihoods for any two nested models. To determine whether the more general model fits significantly better, we can apply the chi-square test to their generalized likelihood ratio (see Mood, Graybill, & Boes, 1974, p. 440):

*T*is the difference between the two models' total numbers of squared standard errors when maximum likelihood fit to all of the measurements.

^{1}It should be noted that the converse is not true: a loss of orientation acuity does not necessarily imply small-angle assimilation (Solomon & Morgan, 2009).

^{2}Responses based on flanker appearance should produce assimilative biases. We found strong assimilation only with noise in Experiment 1's peripheral condition. This condition did produce the highest lapse rate, but further analysis of these errors reveals that only 53% of these lapses were in the same direction as the flank tilt. Since there was no consistent direction in these errors, we can be confident the assimilation reported below actually stems from the appearance of the target.

^{8}We must be very careful to discriminate between small-angle assimilation, which is a name given to measured biases (Howard, 1982), and the visual process that causes it. We refer to the latter as an “assimilative force.” This appellation is consistent not only with compulsory pooling, but other potential mechanisms as well. Although the term “feature repulsion” is similarly agnostic with regard to mechanism, we prefer “repulsive force,” because it more obviously works in opposition to the assimilative force.

*Callithrix jacchus*). Journal of Comparative Neurology, 372, 264–282. [CrossRef]