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Steven Dakin, Marc Tibber, John Greenwood, Frederick Kingdom, Michael Morgan; The common perceptual metric for human discrimination of number and density. Journal of Vision 2011;11(11):1203. doi: 10.1167/11.11.1203.
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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 of objects (i.e. how closely spaced they are). If humans have a sense of “visual number” (as has been proposed) then it should operate independently of density perception. Here we show that it does not. We had subjects discriminate the density or numerosity of two patches that were mismatched in size and show that larger patches appear both denser and (somewhat) more numerous, and that size-mismatching elevates thresholds for discriminating number and (to a lesser degree) density. We propose that density and number are both initially encoded as the ratio of responses from a pair of filters tuned to low and high spatial frequencies, but that number-estimation requires that this measure be scaled by relative stimulus-size. This model explains the rather complex dependence of observers' accuracy and precision on patch-size variation, using a simple, biologically plausible common metric for number and density. Because this model does not have any explicit representation of “objects” it predicts that (for example) mismatching element size will drastically affect number and density discrimination, whereas contrast-mismatching will not (Tibber, Greenwood & Dakin, VSS 2011).
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