Numerosity underestimation in sets with illusory contours

The interplay between spatial and non-spatial grouping cues over approximate number perception

Humans and animals share the cognitive ability to quickly extract approximate number information from sets. Main psychophysical models suggest that visual approximate numerosity relies on segmented units, which can be affected by Gestalt rules. Indeed, arrays containing spatial grouping cues, such as connectedness, closure, and even symmetry, are underestimated compared to ungrouped arrays with equal low-level features. Recent evidence suggests that non-spatial cues, such as color-similarity, also trigger numerosity underestimation. However, in natural vision, several grouping cues may coexist in the scene. Notably, conjunction of grouping cues (color and closure) reduces perceived numerosity following an additive rule. To test whether the conjunction-effect holds for other Gestalt cues, we investigated the effect of connectedness and symmetry over numerosity perception both in isolation and, critically, in conjunction with luminance similarity. Participants performed a comparison-task between a reference and a test stimulus varying in numerosity. In Experiment 1, test stimuli contained two isolated groupings (connectedness or luminance), a conjunction (connectedness and luminance), and a neutral condition (no groupings). Results show that point of subjective equality was higher in both isolated grouping conditions compared to the neutral condition. Furthermore, in the conjunction condition, the biases from isolated grouping cues added linearly, resulting in a numerosity underestimation equal to the sum of the isolated biases. In Experiment 2 we found that conjunction of symmetry and luminance followed the same additive rule. These findings strongly suggest that both spatial and non-spatial isolated cues affect numerosity perception. Crucially, we show that their conjunction effect extends to symmetry and connectedness.

The symmetry-induced numerosity illusion depends on visual attention

Symmetry is an important and strong cue we rely on to organize the visual world. Although it is at the basis of objects segmentation in a visual scene, it can sometimes bias our perception. When asked to discriminate numerical quantities between symmetric and asymmetric arrays, individuals tend to underestimate the number of items in the symmetric stimuli. The reason for this underestimation is currently unknown. In this study we investigated whether the symmetry-induced numerosity underestimation depends on perceptual grouping mechanisms by depriving attentional resources. Twenty-six adults judged the numerosity of dot arrays arranged symmetrically or randomly, while ignoring a visual distractor (single task) or while simultaneously judging its color and orientation (dual-task). Diverting attention to the concurrent color-orientation conjunction task halved the symmetry-induced numerosity underestimation. Taken together these results showed that the bias in numerosity perception of symmetric arrays depends-at least partially-on attentional resources and suggested that it might originate from the recruitment of attentional dependent incremental grouping mechanisms.

Visual number sense for real-world scenes shared by deep neural networks and humans

Recently, visual number sense has been identified from deep neural networks (DNNs). However, whether DNNs have the same capacity for real-world scenes, rather than the simple geometric figures that are often tested, is unclear. In this study, we explore the number perception of scenes using AlexNet and find that numerosity can be represented by the pattern of group activation of the category layer units. The global activation of these units increases with the number of objects in the scene, and the variations in their activation decrease accordingly. By decoding the numerosity from this pattern, we reveal that the embedding coefficient of a scene determines the likelihood of potential objects to contribute to numerical perception. This was demonstrated by the more optimized performance for pictures with relatively high embedding coefficients in both DNNs and humans. This study for the first time shows that a distinct feature in visual environments, revealed by DNNs, can modulate human perception, supported by a group-coding mechanism.

Attention drives human numerosity-selective responses

Numerosity, the set size of a group of items, helps guide behavior and decisions. Previous studies have shown that neural populations respond selectively to numerosities. How numerosity is extracted from the visual scene is a longstanding debate, often contrasting low-level visual with high-level cognitive processes. Here, we investigate how attention influences numerosity-selective responses. The stimuli consisted of black and white dots within the same display. Participants' attention was focused on either black or white dots, while we systematically changed the numerosity of black, white, and total dots. Using 7 T fMRI, we show that the numerosity-tuned neural populations respond only when attention is focused on their preferred numerosity, irrespective of the unattended or total numerosities. Without attention, responses to preferred numerosity are suppressed. Unlike traditional effects of attention in the visual cortex, where attention enhances already existing responses, these results suggest that attention is required to drive numerosity-selective responses.

Numerosity tuning in human association cortices and local image contrast representations in early visual cortex

Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus’s retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity’s straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing.

The authors show that spatial frequency domain Fourier power closely but nonlinearly follows numerosity, simplifying computing numerosity from early visual responses. Monotonic early visual cortex and neural network responses follow Fourier power, while later tuned responses follow numerosity.

Nonsymbolic numerosity in sets with illusory-contours exploits a context-sensitive, but contrast-insensitive, visual boundary formation process

The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by illusory contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2, or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.

Numbers, numerosities, and new directions

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

Dogs (canis familiaris) underestimate the quantity of connected items: first demonstration of susceptibility to the connectedness illusion in non-human animals

In humans, numerical estimation is affected by perceptual biases, such as those originating from the spatial arrangement of elements. Different animal species can also make relative quantity judgements. This includes dogs, who have been proposed as a good model for comparative neuroscience. However, dogs do not show the same perceptual biases observed in humans. Thus, the exact perceptual/cognitive mechanisms underlying quantity estimations in dogs and their degree of similarity with humans are still a matter of debate. Here we explored whether dogs are susceptible to the connectedness illusion, an illusion based on the tendency to underestimate the quantity of interconnected items. Dogs were first trained to choose the larger of two food arrays. Then, they were presented with two arrays containing the same quantity of food, of which one had items interconnected by lines. Dogs significantly selected the array with unconnected items, suggesting that, like in humans, connectedness determines underestimation biases, possibly disrupting the perceptual system's ability to segment the display into discrete objects. The similarity in dogs' and humans' susceptibility to the connectedness, but not to other numerical illusions, suggests that different mechanisms are involved in the estimation of quantity of stimuli with different characteristics.

The dynamics of grouping-induced biases in apparent numerosity revealed by a continuous tracking technique

Connecting pairs of items causes robust underestimation of the numerosity of an ensemble, presumably by invoking grouping mechanisms. Here we asked whether this underestimation in numerosity judgments could be revealed and further explored by continuous tracking, a newly developed technique that allows for fast and efficient data acquisition and monitors the dynamics of the responses. Participants continuously reproduced the perceived numerosity of a cloud of dots by moving a cursor along a number line, while the number of dots and the proportion connected by lines varied over time following two independent random walks. The technique was robust and efficient, and correlated well with results obtained with a standard psychophysics task. Connecting objects with lines caused an underestimation of approximately 15% during tracking, agreeing with previous studies. The response to the lines was slower than the response to the physical numerosity, with a delay of approximately 150 ms, suggesting that this extra time is necessary for processing the grouping effect.

The Grouping-Induced Numerosity Illusion Is Attention-Dependent

Perceptual grouping and visual attention are two mechanisms that help to segregate visual input into meaningful objects. Here we report how perceptual grouping, which affects perceived numerosity, is reduced when visual attention is engaged in a concurrent visual task. We asked participants to judge the numerosity of clouds of dot-pairs connected by thin lines, known to cause underestimation of numerosity, while simultaneously performing a color conjunction task. Diverting attention to the concomitant visual distractor significantly reduced the grouping-induced numerosity biases. Moreover, while the magnitude of the illusion under free viewing covaried strongly with AQ-defined autistic traits, under conditions of divided attention the relationship was much reduced. These results suggest that divided attention modulates the perceptual grouping of elements by connectedness and that it is independent of the perceptual style of participants.

Number is not just an illusion: Discrete numerosity is encoded independently from perceived size

While seminal theories suggest that nonsymbolic visual numerosity is mainly extracted from segmented items, more recent views advocate that numerosity cannot be processed independently of nonnumeric continuous features confounded with the numerical set (i.e., such as the density, the convex hull, etc.). To disentangle these accounts, here we employed two different visual illusions presented in isolation or in a merged condition (e.g., combining the effects of the two illusions). In particular, in a number comparison task, we concurrently manipulated both the perceived object segmentation by connecting items with Kanizsa-like illusory lines, and the perceived convex-hull/density of the set by embedding the stimuli in a Ponzo illusion context, keeping constant other low-level features. In Experiment 1, the two illusions were manipulated in a compatible direction (i.e., both triggering numerical underestimation), whereas in Experiment 2 they were manipulated in an incompatible direction (i.e., with the Ponzo illusion triggering numerical overestimation and the Kanizsa illusion numerical underestimation). Results from psychometric functions showed that, in the merged condition, the biases of each illusion summated (i.e., largest underestimation as compared with the conditions in which illusions were presented in isolation) in Experiment 1, while they averaged and competed against each other in Experiment 2. These findings suggest that discrete nonsymbolic numerosity can be extracted independently from continuous magnitudes. They also point to the need of more comprehensive theoretical views accounting for the operations by which both discrete elements and continuous variables are computed and integrated by the visual system.

Dogs (Canis lupus familiaris) are susceptible to the Kanizsa's triangle illusion

The ability to complete partially missing contours is widespread across the animal kingdom, but whether this extends to dogs is still unknown. To address this gap in knowledge, we assessed dogs' susceptibility to one of the most common contour illusions, the Kanizsa's triangle. Six dogs were trained to discriminate a triangle from other geometrical figures using a two-alternative conditioned discrimination task. Once the learning criterion was reached, dogs were presented with the Kanizsa's triangle and a control stimulus, where inducers were rotated around their centre, so as to disrupt what would be perceived as a triangle by a human observer. As a group, dogs chose the illusory triangle significantly more often than control stimuli. At the individual level, susceptibility to the illusion was shown by five out of six dogs. This is the first study where dogs as a group show susceptibility to a visual illusion in the same manner as humans. Moreover, the analyses revealed a negative effect of age on susceptibility, an effect that was also found in humans. Altogether, this suggests that the underling perceptual mechanisms are similar between dogs and humans, and in sharp contrast with other categories of visual illusions to which the susceptibility of dogs has been previously assessed.

Numerosity representation in a deep convolutional neural network

Enumerating objects in the environment (i.e., “number sense”) is crucial for survival in many animal species, and foundational for the construction of more abstract and complex mathematical knowledge in humans. Perhaps surprisingly, deep convolutional neural networks (DCNNs) spontaneously emerge a similar number sense even without any explicit training for numerosity estimation. However, little is known about how the number sense emerges, and the extent to which it is comparable with human number sense. Here, we examined whether the numerosity underestimation effect, a phenomenon indicating that numerosity perception acts upon the perceptual number rather than the physical number, can be observed in DCNNs. In a typical DCNN, AlexNet, we found that number-selective units at late layers operated on the perceptual number, like humans do. More importantly, this perceptual number sense did not emerge abruptly, rather developed progressively along the hierarchy in the DCNN, shifting from the physical number sense at early layers to perceptual number sense at late layers. Our finding hence provides important implications for the neural implementation of number sense in the human brain and advocates future research to determine whether the representation of numerosity also develops gradually along the human visual stream from physical number to perceptual number.

Grouping-Induced Numerosity Biases Vary with Autistic-Like Personality Traits

Individuals with autism spectrum disorder are thought to have a more local than global perceptual style. We used a novel paradigm to investigate how grouping-induced response biases in numerosity judgments depend on autistic-like personality traits in neurotypical adults. Participants judged the numerosity of clouds of dot-pairs connected by thin lines, known to cause underestimation of numerosity. The underestimation bias correlated strongly with autism-spectrum quotient (r = 0.72, Bayes factor > 100), being weaker for participants with high autistic traits. As connecting dots probably activates global grouping mechanisms, causing dot-pairs to be processed as an integrated whole rather than as individual dots, the results suggest that these grouping mechanisms may be weaker in individuals self-reporting high levels of autistic-like traits.

The Number Sense Represents (Rational) Numbers

On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique-the arguments from congruency, confounds, and imprecision-and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as "numerosities" or "quanticals," as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

The ratio effect in visual numerosity comparisons is preserved despite spatial frequency equalisation

How non-symbolic numerosity is visually extracted remains a matter of intense debate. Most evidence suggests that numerosity is directly extracted on individual objects following Weber's law, at least for a moderate numerical range. Alternative accounts propose that, whatever the range, numerosity is indirectly derived from summary texture-statistics of the raw image such as spatial frequency (SF). Here, to disentangle these accounts, we tested whether the well-known behavioural signature of numerosity encoding (ratio effect) is preserved despite the equalisation of the SF content. In Experiment 1, participants had to select the numerically larger of two briefly presented moderate-range numerical sets (i.e., 8-18 dots) carefully matched for SF; the ratio between numerosities was manipulated by levels of increasing difficulty (e.g., 0.66, 0.75, 0.8). In Experiment 2, participants performed the same task, but they were presented with both the original and SF equalised stimuli. In both experiments, the results clearly showed a ratio-dependence of the performance: numerosity discrimination became harder and slower as the ratio between numerosities increased. Moreover, this effect was found to be independent of the stimulus type, although the overall performance was better with the original rather than the SF equalised stimuli (Experiment 2). Taken together, these findings indicate that the power spectrum per se cannot explain the main behavioural signature of Weber-like encoding of numerosities (the ratio effect), at least over the tested numerical range, partially challenging alternative indirect accounts of numerosity processing.

Similarly oriented objects appear more numerous

Several non-numerical factors influence the numerical estimation of visual arrays, including the spacing of items and whether they are arranged randomly or symmetrically. Here we report a novel numerosity illusion we term the coherence illusion. When items in an array have a coherent orientation (all pointing in the same direction) they seem to be more numerous than when items are oriented randomly. Participants show parametric effects of orientation coherence in three distinct numerical judgment tasks. These findings are not predicted by any current model of numerical estimation. We discuss array entropy as a possible framework for explaining both the coherence illusion and the previously reported regular-random illusion.

Does the standard search task predict performance in related tasks for Kanizsa-style illusory contours?

It is often assumed that results from standard visual search tasks will be replicated in related tasks but his idea is rarely tested. In a conceptual replication of Li, Cave, and Wolfe (2008), we investigated the attentional demands of Kanizsa-style illusory contours using orientation-based search, comparing performance for items defined by real- as compared to illusory contours. After confirming the initial findings in standard search, we tested the same manipulation in multiple-target search, Thornton and Gilden's (2007) hybrid standard/multiple-target search, and simple- and selective enumeration. The RT slope differences between real- and illusory contours did not replicate in Thornton and Gilden's task, though they did in multiple-target search and selective enumeration. In fact, absolute differences between real- and illusory contours in RT costs per distractor were 2 - 6 times larger than in standard search. To determine whether performance differences between real and illusory contours originated from shape-definition (necessary for distinguishing target shapes from distractors) or unit formation (grouping disconnected parts to define an item/unit), simple and selective enumeration were compared. The differences between real- and illusory-contours only emerged in selective enumeration (enumerating targets among distractors), which suggests the discrepancies between conditions originate from shape definition rather than unit formation processes. There was no evidence of subitizing in selective enumeration for illusory contour figures, but contrary to attention-based theories of enumeration, there was no subitizing for the real-contour controls either. This study contributes to research on illusory contours but it is especially important to the study of search and enumeration.

Similarity Grouping as Feature-Based Selection

Across the natural world as well as the artificial worlds of maps, diagrams, and data visualizations, feature similarity (e.g., color and shape) links spatially separate areas into sets. Despite a century of study, it is yet unclear what mechanism underlies this gestalt similarity grouping. One recent proposal is that similarity grouping-for example, seeing a red, vertical, or square group-is just global selection of those features. Although parsimonious, this account makes the counterintuitive prediction that similarity grouping is strictly serial: A green group cannot be constructed at the same time as a red group. We tested this prediction with a novel measure-a grouping illusion within number-estimation tasks that should work only if participants simultaneously construct groups-and found the strongest evidence yet in favor of serial feature-based attention ( Ns = 14, 12, and 12 for Experiment 1, Experiment 2, and Experiment 3, respectively).

When the weaker conquer: A contrast-dependent illusion of visual numerosity

Humans and many nonhuman species have the capacity to perceive the approximate numerosity of a large set of elements. In the visual domain, various factors have been found to affect perceived numerosity, including, as studied here, the effect of luminance contrast. We report a new numerosity illusion in which low-contrast elements appear to be more numerous than high-contrast ones when intermixed in the same display, an effect we have called "the weak conquer the strong." The illusion occurred for both positive and negative contrast disks of the same sign; for example, white disks appeared less numerous than light gray disks when presented on a dark gray field (all positive contrasts). The illusion grew in size as the contrast difference between the two sets of disks became smaller but did not hold for disks with opposite signs, such as white and dark gray disks on a light gray field. The illusion was also eliminated when the disks were segregated spatially, in which case the high-contrast (white) disks and low-contrast (gray) disks appeared equally numerous. The illusion is due to the loss of some high-contrast elements; e.g., the perceived numerosity of the white disks was decreased whereas that of gray ones was preserved as shown by matching to isolated sets of white or gray disks. We speculate that failure in segregation coupled with error in contrast normalization may account for the numerosity illusion.

Approximate number sense theory or approximate theory of magnitude?

Leibovich et al. argue that the evidence in favor of a perceptual mechanism devoted to the extraction of numerosity from visual collections is unsatisfactory and propose to replace it with an unspecific mechanism capturing approximate magnitudes from continuous dimensions. We argue that their representation of the evidence is incomplete and that their theoretical proposal is too vague to be useful.

Numerosity but not texture-density discrimination correlates with math ability in children

Considerable recent work suggests that mathematical abilities in children correlate with the ability to estimate numerosity. Does math correlate only with numerosity estimation, or also with other similar tasks? We measured discrimination thresholds of school-age (6- to 12.5-years-old) children in 3 tasks: numerosity of patterns of relatively sparse, segregatable items (24 dots); numerosity of very dense textured patterns (250 dots); and discrimination of direction of motion. Thresholds in all tasks improved with age, but at different rates, implying the action of different mechanisms: In particular, in young children, thresholds were lower for sparse than textured patterns (the opposite of adults), suggesting earlier maturation of numerosity mechanisms. Importantly, numerosity thresholds for sparse stimuli correlated strongly with math skills, even after controlling for the influence of age, gender and nonverbal IQ. However, neither motion-direction discrimination nor numerosity discrimination of texture patterns showed a significant correlation with math abilities. These results provide further evidence that numerosity and texture-density are perceived by independent neural mechanisms, which develop at different rates; and importantly, only numerosity mechanisms are related to math. As developmental dyscalculia is characterized by a profound deficit in discriminating numerosity, it is fundamental to understand the mechanism behind the discrimination. (PsycINFO Database Record