A common visual metric for approximate number and density

Abstract

There is considerable interest in how humans estimate the number of objects in a scene in the context of an extensive literature on how we estimate the density (i.e., spacing) of objects. Here, we show that our sense of number and our sense of density are intertwined. Presented with two patches, observers found it more difficult to spot differences in either density or numerosity when those patches were mismatched in overall size, and their errors were consistent with larger patches appearing both denser and more numerous. We propose that density is estimated using the relative response of mechanisms tuned to low and high spatial frequencies (SFs), because energy at high SFs is largely determined by the number of objects, whereas low SF energy depends more on the area occupied by elements. This measure is biased by overall stimulus size in the same way as human observers, and by estimating number using the same measure scaled by relative stimulus size, we can explain all of our results. This model is a simple, biologically plausible common metric for perceptual number and density.psychophysics | vision | texture | numerical cognition

The effects of sampling and internal noise on the representation of ensemble average size

“…In contrast, work from the ensemble feature literature has often taken a sample to be one of the segmentable objects within the ensemble and has implemented sampling without replacement (e.g., Haberman & Whitney, 2010;. We believe that extracting ensemble features may not rely on segmenting individual items, but rather may rely on mechanisms more similar to texture processing-a suggestion that has also been made elsewhere (Balas, Nakano, & Rosenholtz, 2009;Dakin, Tibber, Greenwood, Kingdom, & Morgan, 2011;Freeman & Simoncelli, 2011;Haberman & Whitney, 2010;Parkes et al, 2001). …”

Topology-defined units in numerosity perception

“…Importantly, the fMRI adaptation results revealed the neural basis of the behavioral observation and provided converging evidence supporting the topological account. Because the intraparietal sulcus is not directly sensitive to low-level image features, these results have the additional advantage of being free from lowlevel and nontopological feature confounds (2,20,21).…”

“…It might be argued that, in the numerosity comparison task performed in the above experiments, the comparison judgment could be confounded by other perceptual attributes of the displays, such as density or area, and other local features rather than being based on the actual representation of dot numerosity (17)(18)(19)(20)(21)(22)(23)(24). We, therefore, used, instead of the comparison task, a subjective estimation task, which required observers to report the number of dots directly (19).…”

Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence

“…Because single neurons cannot account for complex behaviors across a broad range of stimuli, a challenge for future research will be to determine how populations of neuronal firing support aspects of shared and distinct processing for numerical and nonnumerical magnitudes. One class of models in vision suggests that shared representations emerge via relative outputs from spatial filters tuned to high vs. low frequencies (51,52), whereas another class of models, which emphasizes distinct representations, suggests that summation processes invoke additional mechanisms that normalize computations across irrelevant cues (53)(54)(55). Although extant models have typically considered shared and distinct representations separately, findings such as those from the present study point to the need for integrative neuro-computational models that simulate how shared and distinct representations might coexist as part of a general magnitude system.…”