Perception of visual texture flows contributes to object segmentation, shape perception, and object recognition. To better understand the visual mechanisms underlying texture flow perception, we studied the factors limiting detection of simple forms of texture flows composed of local dot dipoles (Glass patterns) and related stimuli. To provide a benchmark for human performance, we derived an ideal observer for this task. We found that human detection thresholds were 8.0 times higher than ideal. We considered three factors that might account for this performance gap: (1) false matches between dipole dots (correspondence errors), (2) loss of sensitivity with increasing eccentricity, and (3) local orientation bandwidth. To estimate the effect of correspondence errors, we compared detection of Glass patterns with detection of matched line-segment stimuli, where no correspondence uncertainty exists. We found that eliminating correspondence errors reduced human thresholds by a factor of 1.8. We used a novel form of classification image analysis to directly estimate loss of sensitivity with eccentricity and local orientation bandwidth. Incorporating the eccentricity effects into the ideal observer model increased ideal thresholds by a factor of 2.9. Interestingly, estimated orientation bandwidth increased ideal thresholds by only 8%. Taking all three factors into account, human thresholds were only 58% higher than model thresholds. Our findings suggest that correspondence errors and eccentricity losses account for the great majority of the perceptual loss in the visual processing of Glass patterns.

^{2}against a background luminance of 60.3 cd/m

^{2}.

^{2}against a background luminance of 60.3 cd/m

^{2}. The lower luminance of line segments compared to dipoles (102 cd/m

^{2}) equalized the total contrast energy of the two patterns. Pilot experiments revealed that in fact detection does not improve with contrast.

^{2}on average) we used.

*N*= 200, the ideal observer only has to consider the number of elements at the signal orientation

*n*

_{0}. We make the further assumption that the ideal observer knows the signal level, i.e., the number

*n*

_{s}of elements deterministically set to the signal orientation on signal-present trials. (Note that the actual set of elements

*n*

_{0}at the signal orientation consists of the signal elements plus the noise elements that happen to be at the signal orientation. See Figures 3a–3c.) This assumption is not unreasonable, as human thresholds are the outcome of a sequence of QUEST trials in which the signal level

*n*

_{s}gradually converges to a fixed value. Thus, in principle, it is possible for the ideal observer to develop a good estimate of the signal level

*n*

_{s}at its threshold. (In the Prior knowledge of signal level section, we explore how weaker knowledge of the signal level affects our analysis.)

*n*

_{s}, we assume that the ideal observer knows that the prior odds of signal present/absent is 50/50 and that the objective is to maximize proportion correct, consistent with a uniform reward for hits and correct rejects or equivalently a uniform loss for false positives and misses. With this knowledge, and an observed number of elements

*n*

_{0}at the known signal orientation, the ideal observer reports the signal present if and only if this is the more likely event. Letting

*H*

_{0}and

*H*

_{1}denote the signal-absent and signal-present events, respectively, the decision rule can be written as

*n*

_{0}follows a binomial distribution:

*p*

_{ θ }= 1/

*n*

_{ θ }is the probability that a single noise element will have a particular orientation, given

*n*

_{ θ }discrete stimulus orientations (

*n*

_{ θ }= 6, 12, or 24 in our experiments). The decision rule in Equation 1 then determines the ideal criterion

*n*

_{0}′:

*n*

_{0}′ is the minimum number of elements in the signal orientation for which the likelihood ratio equals or exceeds 1.

*n*

_{0}′ as a function of signal level

*n*

_{s}, the proportion of correct responses

*p*

_{c}for every possible signal level

*n*

_{s}∈ [0, …,

*N*] (where

*N*= 200) can be computed as the average of the hit rate

*p*

_{Hit}and correct-reject rate

*p*

_{CR}:

*p*

_{c}(

*n*

_{s}) for every integer signal level

*n*

_{s}∈ [0, …,

*N*], we compute the ideal threshold

*n*

_{s}′ for 75% correct performance by linear interpolation.

Parameter | F | df | P-value |
---|---|---|---|

Main effects | |||

Observer (O) | 250 | 2 | <0.001 |

Signal orientation (SO) | 38.7 | 1 | <0.001 |

Stimulus type (ST) | 879 | 1 | <0.001 |

Number of orientations (NO) | 12.9 | 2 | <0.001 |

Two-way interactions | |||

O × SO | 7.06 | 2 | <0.001 |

O × ST | 7.65 | 2 | <0.001 |

O × NO | 2.10 | 4 | =0.080 |

SO × ST | 81.9 | 1 | <0.001 |

SO × NO | 10.5 | 2 | <0.001 |

ST × NO | 14.2 | 2 | <0.001 |

Three-way interactions | |||

O × SO × ST | 17.2 | 2 | <0.001 |

O × SO × NO | 2.13 | 4 | =0.077 |

O × ST × NO | 2.60 | 4 | <0.036 |

SO × ST × NO | 2.39 | 2 | =0.093 |

Four-way interactions | |||

O × SO × ST × NO | 4.83 | 4 | <0.001 |

Residual | 394 |

*p*< 0.05). Since several of these significant interactions involve the observer factor, we also conducted three-way (conditional) ANOVAs for each observer (Table 2) to assess the generalization of observed effects across individuals. While a complete account of all higher order interactions is beyond the scope of this paper, we shall attempt to shed light at least on the main effects and selected two-way interactions in the following.

Observer CFO | |||
---|---|---|---|

Parameter | F | df | P-value |

Main effects | |||

Signal orientation (SO) | 0.324 | 1 | =0.570 |

Stimulus type (ST) | 305 | 1 | <0.001 |

Number of orientations (NO) | 3.35 | 2 | <0.038 |

Two-way interactions | |||

SO × ST | 1.69 | 1 | =0.196 |

SO × NO | 0.252 | 2 | =0.778 |

ST × NO | 0.423 | 2 | =0.656 |

Three-way interactions | |||

SO × ST × NO | 0.354 | 2 | =0.703 |

Residual | 138 | ||

Observer DN | |||

Main effects | |||

Signal orientation (SO) | 44.6 | 1 | <0.001 |

Stimulus type (ST) | 313 | 1 | <0.001 |

Number of orientations (NO) | 13.6 | 2 | <0.001 |

Two-way interactions | |||

SO × ST | 27.7 | 1 | <0.001 |

SO × NO | 10.9 | 2 | <0.001 |

ST × NO | 11.7 | 2 | <0.001 |

Three-way interactions | |||

SO × ST × NO | 15.5 | 2 | <0.001 |

Residual | 108 | ||

Observer YM | |||

Main effects | |||

Signal orientation (SO) | 22.8 | 1 | <0.001 |

Stimulus type (ST) | 305 | 1 | <0.001 |

Number of orientations (NO) | 3.32 | 2 | <0.040 |

Two-way interactions | |||

SO × ST | 115 | 1 | <0.001 |

SO × NO | 7.45 | 2 | <0.001 |

ST × NO | 11.6 | 2 | <0.001 |

Three-way interactions | |||

SO × ST × NO | 0.284 | 2 | =0.753 |

Residual | 148 |

*p*< 0.001) for the two observers who show it and nowhere near significant (

*p*= 0.57) for the third observer. This finding suggests that the inverse oblique effect may only hold for a subset of the population, although clearly a larger sample size is required to substantiate this claim.

*p*< 0.04). Note that the decline in thresholds is predicted by the ideal observer model and reflects the decline in the number of noise elements at the signal orientation as the number of orientations in the stimulus increases.

*C*

_{r}at discrete eccentricity bins

*i*:

*n*

_{FA}is the number of false-alarm trials,

*n*

_{CR}is the number of correct-reject trials,

*N*

_{FA}(

*i*) is the number of elements falling in eccentricity bin

*i*over all false-alarm trials, and

*N*

_{CR}(

*i*) is the number of elements falling in eccentricity bin

*i*over all correct-reject trials.

*w*

_{ ij }is a weight assigned to the

*j*th element falling in bin

*i*:

*N*

_{ i }is the number of elements falling in bin

*i*over all (signal-absent) trials.

*r*increases as

*r*

^{2}, the bin width is made to decrease as

*r*is the eccentricity of an element, defined as the angular deviation of the element's midpoint from fixation, and

*r*

_{0}is the space constant for the falloff.

*C*

_{r}(

*r*) with eccentricity

*r,*a least-squares fit of our model

_{r}(

*r*) directly to this empirical sensitivity function

*C*

_{r}(

*r*) is sensitive to the choice of the bin width. We can eliminate this dependence by using one bin per oriented texture element, centring each on the eccentricity of that element, and making the bin widths so small that they each encompass only one point. It is important to note that there may be gaps and overlaps between the bins, but this is not a problem, since our goal is only to estimate the sensitivity at a discrete number of

*r*values, not to estimate a probability density. Under these conditions, the number of bins is equal to the total number of elements and the estimated sensitivity at the

*i*th bin is given by

*w*

_{ ij }defined as in Equations 9 and 10. Now a least-squares fit of the model

_{r}(

*r*) to the data can be computed by minimizing

*I*

_{FA}and

*I*

_{CR}are the index sets for all noise elements in the false-alarm and correct-reject trials, respectively. We minimize Equation 13 over

*A*

_{r}and

*r*

_{0}using a standard gradient descent method (MATLAB

*fminsearch*). We report only the results for

*r*

_{0}, since

*A*

_{r}is an arbitrary scaling constant.

*r*

_{0}as a function of our four factors: (1) observer (CFO, DN, or YM), (2) stimulus type (Glass patterns or line patterns), (3) signal orientation (horizontal or oblique), and (4) number of orientations (6, 12, or 24). We find significant main effects of observer and signal orientation as well as three significant two-way interactions (observer × signal orientation, observer × stimulus type, stimulus type × signal orientation) and one significant three-way interaction (observer × stimulus type × signal orientation). (Given only one estimate per condition, the fourth-order interaction is undefined.) As there is no significant effect of the number of orientations, either as a main effect or in interactions, we averaged the space constants over the three values of this factor (Figure 5b).

Parameter | F | df | P-value |
---|---|---|---|

Main effects | |||

Observer (O) | 36.6 | 2 | <0.003 |

Signal orientation (SO) | 22.6 | 1 | <0.009 |

Stimulus type (ST) | 0.228 | 1 | =0.658 |

Number of orientations (NO) | 1.62 | 2 | =0.306 |

Two-way interactions | |||

O × SO | 13.0 | 2 | <0.018 |

O × ST | 7.19 | 2 | <0.048 |

O × NO | 1.13 | 4 | =0.455 |

SO × ST | 10.8 | 1 | <0.031 |

SO × NO | 0.879 | 2 | =0.483 |

ST × NO | 0.472 | 2 | =0.654 |

Three-way interactions | |||

O × SO × ST | 11.2 | 2 | <0.024 |

O × SO × NO | 0.790 | 4 | =0.588 |

O × ST × NO | 3.19 | 4 | =0.143 |

SO × ST × NO | 0.046 | 2 | =0.956 |

Residual | 4 |

*I*

_{FA}(

*θ*) and

*I*

_{CR}(

*θ*) denote index sets for all noise elements at orientation

*θ*in false-alarm and correct-reject trials, respectively. Then, the empirical sensitivity function over orientation is

*r*

_{ i }is the eccentricity of noise element

*i, r*

_{0}is the space constant estimated separately for each observer and stimulus parameter (Eccentricity loss section), and

*n*

_{FA}and

*n*

_{CR}are the numbers of false-alarm and correct-reject trials, respectively. In this way, elements that are closer to fixation and have greater influence on detection are weighted more heavily in estimating the orientation bandwidth.

*σ*

_{ θ }provides an estimate of the orientation bandwidth for oriented texture detection. The difference between the mean

*μ*

_{ θ }and the signal orientation

*θ*

_{0}(i.e.,

*μ*

_{ θ }−

*θ*

_{0}) indicates observer bias. We compute error estimates for the orientation bandwidth

*σ*

_{ θ }as 68% confidence intervals based on 500 bootstrapped estimates.

*μ*

_{ θ }−

*θ*

_{0}and bandwidths

*σ*

_{ θ }of the Gaussian functions fit to the empirical sensitivity functions over orientation. The statistical results reported below are based on ANOVA models without interaction terms, all of which were found to be not significant (

*p*> 0.12) in the full models.

*F*(2,7) = 2.37,

*p*= 0.16, stimulus type,

*F*(1,7) = 0.632,

*p*= 0.45, or signal orientation,

*F*(1,7) = 1.30,

*p*= 0.29. Importantly, an additional

*t*-test reveals an average bias of −0.22 deg not significantly different from zero bias,

*t*(11) = 0.322,

*p*= 0.75 (two-tailed).

*F*(1,7) = 2.75,

*p*= 0.14. This finding suggests that superior performance for detection of line patterns derives mainly from elimination of correspondence errors, not a reduction in orientation loss. It also suggests that local orientation processing is similar for the two types of oriented elements, consistent with Dakin's (1997) psychophysical findings. Indeed, orientation-selective neurons in V1 and V2 have been shown to be sensitive to both Glass-pattern dipoles (Smith et al., 2002, 2007) and line textures (Kastner, De Weerd, & Ungerleider, 2000; Knierim & van Essen, 1992).

*F*(1,7) = 10.4,

*p*< 0.02 (Figures 6 and 7). Superficially at least, this may seem like a surprising result, given that larger orientation bandwidth should lead to reduced detection performance, yet we generally find that observers are better at detecting oblique patterns (Figure 4). Combined with our analysis of eccentricity loss (Figure 5b), our results suggest that the inverse oblique effect is due to broader spatial pooling for oblique patterns and not to superior orientation tuning. Our results thus explain how the classical oblique effect (Jastrow, 1892) and the inverse oblique effect for Glass patterns are not inconsistent. Whereas the first effect is due to weaker orientation tuning for oblique signals, the latter is due to broader spatial pooling for oblique signals.

*F*(2,7) = 9.54,

*p*< 0.02 (Figure 7). A Tukey's post hoc pairwise comparison test revealed that only YM differed in his orientation bandwidth from the other two observers (YM vs. CFO:

*p*< 0.01, YM vs. DN:

*p*< 0.05, CFO vs. DN:

*p*= 0.48). It is worth noting that YM also had a larger eccentricity loss (Figure 5b).

*σ*

_{ θ }= 11.1 deg, consistent with Maloney et al.'s (1987) estimate of 11 deg for Glass patterns. Our estimate is equivalent to 13.1 deg, half-width at half-height, within the range of orientation bandwidths (7–15 deg) estimated from grating stimuli (Blakemore & Nachmias, 1971; Campbell & Kulikowski, 1966; Phillips & Wilson, 1984; Snowden, 1992).

*w*

_{ j }

^{ e }were given by the exponential model estimated in the Classification image analysis section. Specifically, given a stimulus with

*n*

_{0}elements at the signal orientation, the modified ideal observer uses the decision variable:

*w*

_{ j }

^{ e }= exp(−

*r*

_{ j }/

*r*

_{0}), and

*r*

_{ j }is the eccentricity of the

*j*th element at the signal orientation.

_{0}′ of the modified ideal observer, we simulated 5000 signal-present trials for each possible value of the signal level

*n*

_{s}∈ [0, …,

*N*] and 5000 signal-absent trials, sampling the associated binomial distributions in each case (Equations 2 and 3) to determine values for

*n*

_{0}. Eccentricity values

*r*

_{ j }for each of the

*n*

_{0}elements at the signal orientation in a given trial were then simulated by sampling

*r*

_{ j }

^{2}from a uniform distribution [0,

*R*

^{2}], where

*R*= 7 deg is the radius of the stimulus. The weights

*w*

_{ j }

^{ e }and decision variable

_{0}were then computed based on Equation 16.

_{0}′(

*n*

_{s}) was determined by maximizing the proportion of correct responses for this signal level. The model threshold for 75% correct performance was then estimated by linear interpolation.

*N*= 200 elements in each stimulus were determined by setting

*n*

_{s}to the signal orientation and distributing the remaining

*N*−

*n*

_{s}according to a uniform distribution over all orientations to mimic the oriented textures generated for human observers (Stimuli section; Figure 3). The decision variable

_{0}was then determined as

*w*

_{ j }

^{ θ }were determined from the estimated Gaussian tuning function (Equation 15):

_{0}′(

*n*

_{s}) and threshold at 75% correct performance was then determined as in the Eccentricity loss section.

*σ*= 2.5 elements and mean

*μ*= (

*N*−

*n*

_{s}) /

*n*

_{ θ }for signal-present trials and

*μ*=

*N*/

*n*

_{ θ }for signal-absent trials, where

*N*= 200 is the

*expected*total number of elements in the stimulus,

*n*

_{s}is the number of signal elements, and

*n*

_{ θ }is the number of orientations (

*n*

_{ θ }= 24 in this case). The same three observers performed this experiment, and the other experimental setup was the same as previously described (Stimuli and Procedure sections). Note that the total number of elements in the stimulus now varies from trial to trial but is not predictive of the signal.

*F*(1,1) = 5.47,

*p*= 0.26; stimulus type:

*F*(1,1) = 0.027,

*p*= 0.90), and the interaction effect is also not significant (

*F*(1,1) = 2.20,

*p*= 0.38). Thus, orientation bandwidths estimated using white noise do not differ significantly from the original results based on fixing the total number of elements in the stimulus (Figure 10).

*n*

_{s}∈ [1, …,

*N*]. The expected signal level is then (

*N*+ 1) / 2 and the variance is

*N*= 200, the variance corresponds to a standard deviation of 57.7 elements. The resulting ideal observer model can be taken as a lower bound on the performance (or upper bound on the threshold) of an observer that perceives the stimulus perfectly.

*n*

_{0}(Equation 2) to marginalize over the unknown signal level

*n*

_{s}′:

*p*(

*n*

_{s}′) is the prior over

*n*

_{s}′. For the uniform prior (lower bound), this becomes

*n*

_{s}is the true signal level and

*σ*

_{ n s }

^{2}is the empirical variance in the signal level over a block.

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