**We habitually move our eyes when we enumerate sets of objects. It remains unclear whether saccades are directed for numerosity processing as distinct from object-oriented visual processing (e.g., object saliency, scanning heuristics). Here we investigated the extent to which enumeration eye movements are contingent upon the location of objects in an array, and whether fixation patterns vary with enumeration demands. Twenty adults enumerated random dot arrays twice: first to report the set cardinality and second to judge the perceived number of subsets. We manipulated the spatial location of dots by presenting arrays at 0°, 90°, 180°, and 270° orientations. Participants required a similar time to enumerate the set or the perceived number of subsets in the same array. Fixation patterns were systematically shifted in the direction of array rotation, and distributed across similar locations when the same array was shown on multiple occasions. We modeled fixation patterns and dot saliency using a simple filtering model and show participants judged groups of dots in close proximity (2°–2.5° visual angle) as distinct subsets. Modeling results are consistent with the suggestion that enumeration involves visual grouping mechanisms based on object saliency, and specific enumeration demands affect spatial distribution of fixations. Our findings highlight the importance of set computation, rather than object processing per se, for models of numerosity processing.**

*M*= 20.15 years,

*SD*= 3.34). All had normal or corrected-to-normal vision. The study adheres to the Declaration of Helsinki and was approved by the author's University's Human Research Ethics Committee.

*n*= 1–12 by dividing a circular region at the center of the display (diameter subtending 15° visual angle) into

*n*equally-sized segments. The distance of each dot from the center of the display was sampled from an inverse cumulative distribution (i.e., square-root of the radius) to ensure equally probable density across the entire circular region. The angular position of each dot was Gaussian distributed from the center of each segment to avoid regular radial patterns. This sampling procedure was repeated until all pairwise distances between dots were greater than a visual degree to prevent occlusion and reduce the likelihood of crowding. Two unique dot arrays were constructed for each set size.

^{2}(

*xy*= 0.299, 0.338).

*N*

_{i}observers with

*M*

_{k}fixations at

*x*spatial coordinates, a probability density for the distribution of fixations can be defined for an individual (Equation 1) and the entire sample (Equation 2):

*δ*value represents Kronecker delta in Equation 1 (i.e.,

*δ*(

*x*) = 1 for each fixated coordinate, otherwise

*δ*(

*x*) = 0), and was scaled by fixation duration in all analyses to quantify processing time. A saliency map

*S*is computed by pointwise multiplication of the fixation probability density map

*f*with an isotropic bidimensional Gaussian kernel

*G*(see Equation 3):

*σ*represents the standard deviation of the Gaussian kernel, which was set to approximately one degree visual angle as an estimate of central fovea acuity. Saliency map intensity values were normalized within conditions for comparison. Image saliency maps for each dot array were calculated using the same logic. Each array was first converted into a 768 × 768 grayscale image before being multiplied pointwise by a Gaussian kernel with

*σ*= 1° (see Equation 3) to produce a smoothed saliency map.

*P*and

*Q*, the Jensen-Shannon divergence measure was calculated (

*D*, derived from the Kullback-Leibler divergence

_{JS}*D*; see Equation 4): where

_{KL}*D*(

_{KL}*P*‖

*Q*) = ∑

*(*

_{i}P*i*)log (

*P*(

*i*)/

*Q*(

*i*)) and

*M*= 1/2 (

*P*+

*Q*).

*D*). The JS-distance is preferred over the KL-divergence because of it metric properties that help with interpreting similarity: Comparisons between fixation maps are symmetric and bounded between [0–1], with values closer to zero representing lower similarity in the dispersion of fixations (and vice versa).

_{JS}*R*

^{2}); in other words, over half of the variance in RTs was common despite enumeration of different aspects of the same array. Importantly, overall accuracy for the Dot Enumeration task was high (mean = 0.943, range = 0.702 – 1.00) and participants gave responses to most trials in the Subset Enumeration task (mean = 0.989, range = 0.952 – 1.00). Trials were discarded from analyses if a response was made in less than 200 ms or more than 10,000 ms from trial onset (2.64% and 2.93% of Dot Enumeration and Subset Enumeration trials, respectively).

*p*< 0.001), while the number of fixations was also moderated correlated with the perceived number of subsets the Subset Enumeration task (Kendall rank correlation τ = 0.436,

*p*< 0.001). Outliers were removed before calculating these correlations (i.e., trials with a fixation count three times greater than the median absolute deviation value; 62.92% and 71.04% of Dot Enumeration and Subset Enumeration trials were analyzed, respectively).

*M*(

*SD*) = 5.86 (5.04), compared to judging the number of sets perceived,

*M*(

*SD*) = 4.67 (4.41), paired-sample

*t*test (

*df*= 1816) = 11.779,

*p*< 0.001. There was a moderate correlation between the number of saccades made on trials where the same dot arrays were shown across the two tasks (Kendall rank correlation τ = 0.531,

*p*< 0.001). Mean fixation durations per trial were longer on average for Subset Enumeration,

*M*(

*SD*) = 282.07 (116.24) ms, than Dot Enumeration,

*M*(

*SD*) = 272.87 (91.41) ms, paired-sample

*t*test (

*df*= 1580) = −2.751,

*p*= 0.006. However, there was a low correlation in mean fixation durations across tasks (Kendall rank correlation τ = 0.155,

*p*< 0.001, adjusted-

*R*

^{2}= 0.057). Fixations with duration less than 100 ms, and more than 1,100 ms were excluded from this analysis (7.35% and 7.72% of fixations for Dot Enumeration and Subset Enumeration tasks, respectively).

^{2}(1) = 19.6,

*p*< 0.001, and Subset Enumeration, χ

^{2}(1) =13.2,

*p*< 0.001.

*SE*) = 0.006 (0.002),

*t*(94) = 3.061,

*p*= 0.003, showing a high degree of consistency in scan patterns across participants, irrespective of task.

*SE*) = 0.038 (0.002),

*t*(86) = 21.462,

*p*< 0.001, and Subset Enumeration, closed circles: mean slope (

*SE*) = 0.024 (0.002),

*t*(86) = 13.623,

*p*< 0.001. The exception was for set size one, which showed a relatively high overlap between fixations and dot locations indicating that participants tended to look directly at the single dot in those arrays, so was not included in calculating slope coefficients.

*σ*) specifies the region over which saliency is integrated; for instance, a width of 1°–2° visual angle is consistent with the size of foveal vision, while ≤ 8° visual angle typically demarcates the central visual field from peripheral vision (Strasburger et al., 2011). Filter widths of

*σ*= 1°–5° (steps of 0.5°) were used that correspond to the spatial resolution of the eye tracker (lower limit) and the central third of the display area (upper limit). The assumption underlying this modeling is that saccades are directed towards salient regions across central and peripheral vision (Schütz, Trommershäuser, & Gegenfurtner, 2012).

*SD*= 0.985, range = 1–6). Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis tests were conducted to compare the empirical response distribution and each predicted filter distribution (see Figure 5F). Since the empirical distribution of perceived number of subsets was positively skewed, we computed one-sided KS-test statistics to identify the filter size at which the predicted filter distribution was larger than the empirical distribution. A filter width of 2.5° was the smallest distribution of scores not significantly larger than the empirical response distribution (1°–2°:

*K-S*= 0, all

*p*s > 0.05) and (2.5°:

*K-S*= 0.123; 3°:

*K-S*= 0.330; 3.5°:

*K-S*= 0.456; 4°:

*K-S*= 0.623; 4.5°:

*K-S*= 0.707; 5°:

*K-S*= 0.790, all

*p*s < 0.001). These results show dots located within 2.5° (≤ 2.5 cm) from each other were perceived as belonging to the same set.

*k*means) provides a less plausible explanation since these models require knowledge of precise dot coordinates for computation to converge on an optimal solution. Moreover, given it is unlikely that object coordinates would be known prior to initiating saccades, an unintuitive prediction of these models would be to maintain central fixation to locate individual dots.

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