Discrimination of numerical proportions: A comparison of binomial and Gaussian models (original) (raw)
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A texture-processing model of the 'visual sense of number
Proceedings. Biological sciences / The Royal Society, 2014
It has been suggested that numerosity is an elementary quality of perception, similar to colour. If so (and despite considerable investigation), its mechanism remains unknown. Here, we show that observers require on average a massive difference of approximately 40% to detect a change in the number of objects that vary irrelevantly in blur, contrast and spatial separation, and that some naive observers require even more than this. We suggest that relative numerosity is a type of texture discrimination and that a simple model computing the contrast energy at fine spatial scales in the image can perform at least as well as human observers. Like some human observers, this mechanism finds it harder to discriminate relative numerosity in two patterns with different degrees of blur, but it still outpaces the human. We propose energy discrimination as a benchmark model against which more complex models and new data can be tested.
Modeling the approximate number system to quantify the contribution of visual stimulus features
Cognition, 2015
The approximate number system (ANS) subserves estimation of the number of items in a set. Typically, ANS function is assessed by requiring participants to compare the number of dots in two arrays. Accuracy is determined by the numerical ratio of the sets being compared, and each participant's Weber fraction (w) provides a quantitative index of ANS acuity. When making numerical comparisons, however, performance is also influenced by non-numerical features of the stimuli, such as the size and spacing of dots. Current models of numerosity comparison do not account for these effects and consequently lead to different estimates of w depending on the methods used to control for non-numerical features. Here we proffer a new model that teases apart the effects of ANS acuity from the effects of non-numerical stimulus features. The result is an estimate of w that is a more theoretically valid representation of numerical acuity and novel terms that denote the degree to which a participant&...
Numerosity Estimation in Visual Stimuli in the Absence of Luminance-Based Cues
PLoS ONE, 2011
Background: Numerosity estimation is a basic preverbal ability that humans share with many animal species and that is believed to be foundational of numeracy skills. It is notoriously difficult, however, to establish whether numerosity estimation is based on numerosity itself, or on one or more non-numerical cues like-in visual stimuli-spatial extent and density. Frequently, different non-numerical cues are held constant on different trials. This strategy, however, still allows numerosity estimation to be based on a combination of non-numerical cues rather than on any particular one by itself.
Mechanisms for perception of numerosity or texture-density are governed by crowding-like effects
Journal of Vision, 2015
We have recently provided evidence that the perception of number and texture density is mediated by two independent mechanisms: numerosity mechanisms at relatively low numbers, obeying Weber's law, and texture-density mechanisms at higher numerosities, following a square root law. In this study we investigated whether the switch between the two mechanisms depends on the capacity to segregate individual dots, and therefore follows similar laws to those governing visual crowding. We measured numerosity discrimination for a wide range of numerosities at three eccentricities. We found that the point where the numerosity regime (Weber's law) gave way to the density regime (square root law) depended on eccentricity. In central vision, the regime changed at 2.3 dots/8 2 , while at 158 eccentricity, it changed at 0.5 dots/8 2 , three times less dense. As a consequence, thresholds for low numerosities increased with eccentricity, while at higher numerosities thresholds remained constant. We further showed that like crowding, the regime change was independent of dot size, depending on distance between dot centers, not distance between dot edges or ink coverage. Performance was not affected by stimulus contrast or blur, indicating that the transition does not depend on low-level stimulus properties. Our results reinforce the notion that numerosity and texture are mediated by two distinct processes, depending on whether the individual elements are perceptually segregable. Which mechanism is engaged follows laws that determine crowding.
Perception of visual numerosity in humans and machines
2019
Numerosity perception is foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representation; others argue that visual numerosity is estimated using continuous magnitudes, such as density or area, which usually co-vary with number. Here we reconcile these contrasting perspectives by testing deep networks on the same numerosity comparison task that was administered to humans, using a stimulus space that allows to measure the contribution of non-numerical features. Our model accurately simulated the psychophysics of numerosity perception and the associated developmental changes: discrimination was driven by numerosity information, but non-numerical features had a significant impact, especially early during development. Representational similarity analysis further highlighted that both numerosity and continuous magnitudes were spontaneously encoded even wh...
While observers are adept at judging the density of elements (e.g., in a random-dot image), it has recently been proposed that they also have an independent visual sense of number. To test the independence of number and density discrimination, we examined the effects of manipulating stimulus structure (patch size, element size, contrast, and contrast-polarity) and available attentional resources on both judgments. Five observers made a series of two-alternative, forced-choice discriminations based on the relative numerosity/density of two simultaneously presented patches containing 16-1024 Gaussian blobs. Mismatches of patch size and element size (across reference and test) led to bias and reduced sensitivity in both tasks, whereas manipulations of contrast and contrast-polarity had varied effects on observers, implying differing strategies. Nonetheless, the effects reported were consistent across density and number judgments, the only exception being when luminance cues were made available. Finally, density and number judgment were similarly impaired by attentional load in a dual-task experiment. These results are consistent with a common underlying metric to density and number judgments, with the caveat that additional cues may be exploited when they are available.
Grouping effects in numerosity perception under prolonged viewing conditions
PLOS ONE, 2019
Humans can estimate numerosities-such as the number sheep in a flock-without deliberate counting. A number of biases have been identified in these estimates, which seem primarily rooted in the spatial organization of objects (grouping, symmetry, etc). Most previous studies on the number sense used static stimuli with extremely brief exposure times. However, outside the laboratory, visual scenes are often dynamic and freely viewed for prolonged durations (e.g., a flock of moving sheep). The purpose of the present study is to examine grouping-induced numerosity biases in stimuli that more closely mimic these conditions. To this end, we designed two experiments with limited-dot-lifetime displays (LDDs), in which each dot is visible for a brief period of time and replaced by a new dot elsewhere after its disappearance. The dynamic nature of LDDs prevents subjects from counting even when they are free-viewing a stimulus under prolonged presentation. Subjects estimated the number of dots in arrays that were presented either as a single group or were segregated into two groups by spatial clustering, dot size, dot color, or dot motion. Grouping by color and motion reduced perceived numerosity compared to viewing them as a single group. Moreover, the grouping effect sizes between these two features were correlated, which suggests that the effects may share a common, feature-invariant mechanism. Finally, we find that dot size and total stimulus area directly affect perceived numerosity, which makes it difficult to draw reliable conclusions about grouping effects induced by spatial clustering and dot size. Our results provide new insights into biases in numerosity estimation and they demonstrate that the use of LDDs is an effective method to study the human number sense under prolonged viewing.
Psychophysical evidence for the number sense
Philosophical Transactions of the Royal Society B: Biological Sciences, 2017
It is now clear that most animals, including humans, possess an ability to rapidly estimate number. Some have questioned whether this ability arises from dedicated numerosity mechanisms, or is derived indirectly from judgements of density or other attributes. We describe a series of psychophysical experiments, largely using adaptation techniques, which demonstrate clearly the existence of a number sense in humans. The number sense is truly general, extending over space, time and sensory modality, and is closely linked with action. We further show that when multiple cues are present, numerosity emerges as the natural dimension for discrimination. However, when element density increases past a certain level, the elements become too crowded to parse, and the scene is perceived as a texture rather than array of elements. The two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws, and show different dependencies on eccentricity, luminanc...
A common visual metric for approximate number and density
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.