The computation of relative numerosity, size and density

Phenomenology, Quantity, and Numerosity

There are many situations in which we interact with collections of objects, from a crowd of people to a bowl of blackberries. There is an experience of the quantity of these items, although not a precise number, and we have this impression quickly and effortlessly. It can be described as an expressive property of the whole. In the literature, the study of this sense of numerosity has a long history, which is reviewed here with examples. I argue that numerosity is a direct perceptual experience, and that all experiences of numerosity, not only estimations, are affected by perceptual organisation.

Distinct Mechanisms in Number Comparison of Random and Regular Dots: An ERP Study

Numerosity comparison for regular patterns shows different features compared with that for random ones in previous studies, suggesting an underlying mechanism distinct from numerosity. In this study, we went further to compare the event-related potentials (ERP) components in numerosity processing of random and regular patterns, which are identical in all aspects of texture features except for the distribution. ERP components were recorded and analyzed while participants compared which of the two successively presented sets was more numerous. P2p amplitude was revealed to be significantly weaker for regular patterns compared with that for random patterns over right occipital-parietal cites, whereas no difference was found for P1 or N1 components. The difference in P2p amplitude, which is consistent with the behavior dissociation revealed in our previous studies, suggests that regular distribution can trigger distinct processing in numeral comparison tasks. Processing of continuous magnitudes or configuration cannot explain the decrease in P2p amplitude for regular distributed patterns. Therefore, this study further supports that P2p is mediated by numerosity processing.

The channel for detecting contrast modulation also responds to density modulation (or vice versa)

In textures composed of black and white dots, we modulated dot density and/or dot contrast in one direction of visual space. Just as Mulligan and MacLeod (Vision Research 28 (1988) 503-519) found a strong reciprocity between density and luminance for dots viewed against a darker background, we found a strong reciprocity between density and contrast: detection thresholds for in-phase modulations of density and contrast were 30% - 55% lower than detection thresholds for density and contrast modulations that were 180° out of phase. These findings support the existence of at least one psychophysical channel that is excited by both density modulations and contrast modulations. A good, quantitative fit to our data can be obtained with a two-channel model.

Proximity model of perceived numerosity

The occupancy model (OM) was proposed to explain how the spatial arrangement of dots in sparse random patterns affects their perceived numerosity. The model's central thesis maintained that each dot seemingly fills or occupies its surrounding area within a fixed radius r_o and the total area collectively occupied by all the dots determines their apparent number. Because the perceptual system is not adapted for the precise estimation of area, it looks likely that the OM is just a convenient computational algorithm that does not necessarily correspond to the processes that actually take place in the perceptual system. As an alternative, the proximity model (PM) was proposed, which instead relies on a binomial function with the probability β characterizing the perceptual salience with which each element can be registered by the perceptual system. It was also assumed that the magnitude of β is proportional to the distance between a dot and its nearest neighbor. A simulation experiment demonstrated that the occupancy area computed according to the OM can almost perfectly be replicated by the mean nearest neighbor distance. It was concluded that proximity between elements is a critical factor in determining their perceived numerosity, but the exact algorithm that is used for the measure of proximities is yet to be established.

Shape facilitates number: brain potentials and microstates reveal the interplay between shape and numerosity in human vision

Recognition of simple shapes and numerosity estimation for small quantities are often studied independently of each other, but we know that these processes are both rapid and accurate, suggesting that they may be mediated by common neural mechanisms. Here we address this issue by examining how spatial configuration, shape complexity, and luminance polarity of elements affect numerosity estimation. We directly compared the Event Related Potential (ERP) time-course for numerosity estimation under shape and random configurations and found a larger N2 component for shape over lateral-occipital electrodes (250–400 ms), which also increased with higher numbers. We identified a Left Mid Frontal (LMF; 400–650 ms) component over left-lateralised medial frontal sites that specifically separated low and high numbers of elements, irrespective of their spatial configuration. Different luminance-polarities increased N2 amplitude only, suggesting that shape but not numerosity is selective to polarity. Functional microstates confined numerosity to a strict topographic distribution occurring within the LMF time-window, while a microstate responding only to shape-configuration was evidenced earlier, in the N2 time-window. We conclude that shape-coding precedes numerosity estimation, which can be improved when the number of elements and shape vertices are matched. Thus, numerosity estimation around the subitizing range is facilitated by a shape-template matching process.

Models for discriminating image blur from loss of contrast

Observers can discriminate between blurry and low-contrast images (Morgan, 2017). Wang and Simoncelli (2004) demonstrated that a code for blur is inherent to the phase relationships between localized pattern detectors of different scales. To test whether human observers actually use local phase coherence when discriminating between image blur and loss of contrast, we compared phase-scrambled chessboards with unscrambled chessboards. Although both stimuli had identical amplitude spectra, local phase coherence was disrupted by phase-scrambling. Human observers were required to concurrently detect and identify (as contrast or blur) image manipulations in the 2 × 2 forced-choice paradigm (Nachmias & Weber, 1975; Watson & Robson, 1981) traditionally considered to be a litmus test for “labelled lines” (i.e. detection mechanisms that can be distinguished on the basis of their preferred stimuli). Phase scrambling reduced some observers’ ability to discriminate between blur and a reduction in contrast. However, none of our observers produced data consistent with Watson and Robson's most stringent test for labeled lines, regardless whether phases were scrambled or not. Models of performance fit significantly better when (a) the blur detector also responded to contrast modulations, (b) the contrast detector also responded to blur modulations, or (c) noise in the two detectors was anticorrelated.

Grouping effects in numerosity perception under prolonged viewing conditions

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.

Regular Distribution Inhibits Generic Numerosity Processing

This study investigated the role of pattern regularity in approximate numerical processing. Experiment 1 demonstrated that the change in stimulus size has a distinct effect on the adaptation aftereffect for random and regular patterns. For regular patterns, adapting to large patterns and being tested with small patterns caused stronger aftereffects than the reverse treatment, in which the participants adapted to small patterns and were tested with large patterns. For random patterns, this effect was absent. Experiment 2 revealed a distinct connectedness effect on the numerosity processing of random and regular patterns. For random patterns, reference stimuli were perceived to contain fewer items when the dots were connected by lines than when they were not connected, and the number of items in the connected reference was further underestimated when the participants adapted to unconnected patterns with the same number of dots. For regular patterns, this effect was absent. Distinct mechanisms were thus suggested for the numerosity coding of random and regular patterns. For random patterns, the change in primary texture features would be abstracted from numerosity processing, while connectedness could affect this coding by affecting the processing of numerical unit individuation. For regular patterns, generic numerosity processing is inhibited, and numerical judgments appear to be inferred from the visual processing results of texture features such as dot size or the distance between adjacent dots.

Psychophysical evidence for the number sense

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, luminance levels and effects of perceptual grouping. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

This article is part of the discussion meeting issue ‘The origins of numerical abilities’.

Simultaneous density contrast and binocular integration

Most research on texture density has utilized textures rendered as two-dimensional (2D) planar surfaces, consistent with the conventional definition of density as the number of texture elements per unit area. How the brain represents texture density information in the three-dimensional (3D) world is not yet clear. Here we tested whether binocular information affects density processing using simultaneous density contrast (SDC), in which the perceived density of a texture region is changed by a surround of different density. We considered the effect on SDC of two types of binocular information: the stereoscopic depth relationships and the interocular relationships between the center and surround textures. Observers compared the perceived density of two random dot patterns, one with a surround (test stimulus) and one without (match), using a 2AFC staircase procedure. In Experiment 1 we manipulated the stereo-depth of the surround plane systematically from near to far, relative to the center plane. SDC was reduced when the difference in stereo-depth between test center and surround increased. In Experiment 2 we spread the surround dots randomly across a stereo-depth volume from small to large volume sizes, and found that SDC was slightly reduced with volume size. The decrease of SDC in both experiments was observed with dense surrounds only, but not with sparse surrounds. In the last experiment we presented center and surround in the same depth plane but dichopticly, monopticly, and binocularly. A strong interocular transfer of SDC was found in the dichoptic condition. Together these results show that texture density processing is sensitive to binocularity.