Dot lattices are very simple multi-stable images where the dots can be perceived as being grouped in different ways. The probabilities of grouping along different orientations as dependent on inter-dot distances along these orientations can be predicted by a simple quantitative model. L. Bleumers, P. De Graef, K. Verfaillie, and J. Wagemans (2008) found that for peripheral presentation, this model should be combined with random guesses on a proportion of trials. The present study shows that the probability of random responses decreases with decreasing ambiguity of lattices and is different for bi-stable and tri-stable lattices. With central presentation, similar effects can be produced by adding positional noise to the dots. The results suggest that different levels of internal positional noise might explain the differences between peripheral and central proximity grouping.

*a*and

*b*are inter-dot distances and

*k*is the free distance sensitivity parameter (attraction constant).

*b*/

*a*(a measure of dominance of the orientation

**a**) was varied from 1 to 1.5 in 0.125 steps (see Figure 2). The absolute orientation of the lattices was random.

*b*/

*a*ratio were varied within blocks.

**a**orientation of a lattice in each trial (the center of the category comprising

**a**orientation was assigned orientation zero). The data are pooled across absolute orientations. For better reliability, I also averaged over negative and positive (clockwise and anticlockwise) orientations relative to the

**a**orientation. Thus, a data set for five

*b*/

*a*ratios and two lattice types has 40 degrees of freedom.

*p*

_{ v }′ is proportion of responses based on dominant grouping orientation

**v**∈ {

**a**,

**b**,

**c**, …},

*φ*(

*θ*,

*θ*

_{ v },

*σ*) is Gaussian distribution with mean

*θ*

_{ v }and standard deviation (

*SD*)

*σ*,

*p*

_{r}is proportion of random guesses (distributed uniformly across 180 deg—the full range of orientations). When using PDL,

*p*

_{ v }′ =

*p*(

**v**)(1 −

*p*

_{r}), where

*p*(

**v**) is proportion of responses

**v**predicted by that model.

*SD*small relative to the period (as in this study), there is practically no need to wrap a distribution over more than a single cycle; only periodic boundary conditions must be taken into account.)

*O*

_{ ij }is observed and

*E*

_{ ij }is the expected frequency in cell

*i*,

*j*.

*p*

_{ v }′) of responses based on each of the dominant orientations of a lattice, the proportion of random guesses (

*p*

_{r}), and

*SD*of orientation errors (

*σ*) that accounted for the observed response distributions best. Because of the small number of trials per condition, I used only the 2 and 3 most dominant orientations for rectangular and triangular lattices, respectively.

*p*(

**v**)) along four dominant orientations, combined with Gaussian orientation errors and implemented within the present framework with 8 response alternatives. The model has two free parameters: distance sensitivity parameter of PDL (

*k*) and

*SD*of orientation errors (

*σ*).

*SD*of orientation errors, and proportion of random guessing (

*p*

_{r}).

*b*/

*a*ratios. The model has 12 parameters in total: distance sensitivity,

*SD*of orientation errors, and 10 proportions of random guessing.

*G*-tests based on chi-square distribution and Akaike's information criterion with small sample correction AIC

_{C}= −2 ln(

*L*) +

*L*is the maximum of the likelihood function of the model,

*k*is number of free parameters, and

*n*is number of independent data points (Burnham & Anderson, 2002).

*b*/

*a*ratio.

*b*/

*a*ratio affect the important aspects of the response distributions, I used simple atheoretical modeling. From the observed response distributions, I estimated the: (1) proportions of responses based on supposedly salient orientations within a lattice (

**a**,

**b**,

**c**), (2)

*SD*of a perceived orientation (

*σ*), and (3) proportion of random responses (

*p*

_{r}).

*SD*of orientation errors was assumed to be invariant across the

*b*/

*a*ratios; the proportions of response categories were estimated separately for each condition.

*b*/

*a*ratio. There was also a decrease of random responses with increasing stimulus size (not shown in the figure).

*b*/

*a*ratio but depends on lattice type as well.

*SD*of orientation errors was slightly larger for peripheral presentation (CM, 12.4 deg; EP, 10.9 deg) as compared with central presentation (CM, 10.7 deg; EP, 7.9 deg). There were no systematic differences in this parameter across different sizes or lattice types.

*SD*was adjusted together with other free parameters). The fits are given in Table 1, and the optimal values of parameters are in Table 2.

Observer | PDL | PDL + fixed proportion random | PDL + varied proportion random |
---|---|---|---|

EP | 66.4 (p < 0.01) | 66.4 (p < 0.01) | 30.4 (n.s.) |

DI | 114.2 (p < 0.001) | 112.4 (p < 0.001) | 41.0 (n.s.) |

Parameters | Observer | PDL | PDL + fixed proportion random | PDL + varied proportion random |
---|---|---|---|---|

Distance sensitivity (k) | EP | 6.9 | 7.0 | 8.3 |

DI | 9.3 | 10.5 | 23.8 | |

SD of orientation errors (deg) | EP | 12.9 | 12.9 | 10.9 |

DI | 17.3 | 16.9 | 13.4 |

*b*/

*a*ratio produced an acceptable fit (no statistically significant difference between the model and data). Still, the tests reported in Table 1 do not imply that the third model fits the data significantly better than two others. As the models are nested, the differences in fit can be directly tested using the differences in the values of

*G*-statistics. These direct pairwise tests too showed highly significant (

*p*< 10

^{−4}) superiority of the third model over two others for both observers. In addition, I calculated an information criterion (AIC

_{C}) that takes into account both fit and number of parameters (the smallest value corresponds to the best model). This criterion (Table 3) also favors Model 3.

Observer | PDL | PDL + fixed proportion random | PDL + varied proportion random |
---|---|---|---|

EP | 3771 | 3776 | 3766 |

DI | 3767 | 3770 | 3725 |

*b*/

*a*ratio must be considerably larger with triangular as compared with rectangular lattices.

*b*/

*a*ratio and type of lattice. It seems that with peripheral presentation, grouping performance is limited by some kind of noise that masks relatively weak and ambiguous orientation signals while the

*b*/

*a*ratio is close to 1. With increasing

*b*/

*a*ratios, not only the relative but also absolute strength of a dominant orientation signal probably increases, making the grouping along that orientation visible within the noise. The characteristic differences between the two lattice types suggest that a simple positional noise might play a role in these results. A close look at the lattices used in this study reveals that regardless of the same minimal inter-dot distance and the same

*b*/

*a*ratio, the triangular lattices have smaller average distances between the dots. Consequently, adding the same 2D noise to the positions of the dots must distort this lattice more severely. In addition, several studies have reported spatial uncertainty as an important property of peripheral vision and measured internal positional noise at different eccentricities (e.g., Hess & Dakin, 1999; Hess & Hayes, 1994; Levi, Klein, & Aitsebaomo, 1985).

*SD*3 pixels) was added to the

*x*and

*y*coordinates of each dot. Examples of stimuli are given in Figure 7. Two observers participated in the experiment (600 trials). Figure 7 depicts the proportions of random responses, estimated in the same way as in Experiment 2. The pattern of results is really similar to those of Experiment 2. Both observers exhibit a decreasing proportion of random responses with increasing

*b*/

*a*ratios and a higher proportion of random responses with triangular lattices. Thus, the results show that adding positional noise to the dots of a centrally presented lattice produces the same effects as presenting a noiseless lattice in the visual periphery. This is strong support for the hypothesis that internal positional noise in the visual periphery might be the main cause of the central–peripheral differences in proximity grouping.

*b*/

*a*ratios, with and without positional noise (300 simulated trials per condition). For a better comparability with the experimental data, I convolved the response distributions with a typical distribution of human orientation errors (

*SD*10 deg) and presented the results in 8 orientation categories.

*p*> 0.9). With positional noise, the estimated proportions of random responses follow the same pattern as observed in the experiments with human observers exposed to the similar noisy stimuli, or when noiseless stimuli are presented in the visual periphery. Thus, the simulation provides an additional support to the idea that positional noise can produce the results that are observed in experiments with peripheral presentation of dot lattices.

*b*/

*a*ratio on the probability of random responses in that study. They varied

*b*/

*a*ratio over a relatively small range (from 1 to 1.26). Their lattices were somewhat larger (relative to eccentricity) and exposure duration was longer, which should reduce the total amount of random responses. They did not use tri-stable lattices that produced the largest proportion of random responses in the present study. In addition, note that only using response alternatives far from dominant grouping orientations (as in the present study) allows a robust estimation of the probability of random responses. However, it is not impossible that their procedure really produced stimulus-independent lapses of attention on some proportion of trials. (I reanalyzed their results and found a significant decrease of random responses with increasing

*b*/

*a*ratios at 15-deg eccentricity in their Experiment 1 but not in Experiment 2.)

*b*/

*a*ratio and lattice type (Kubovy et al., 1998). In addition, I have found that using variable

*SD*of orientation errors could not fit the present data as well as variable proportions of random responses.