The role of visual information in numerosity estimation

Mainstream theory suggests that the approximate number system supports our non-symbolic number abilities (e.g. estimating or comparing different sets of items). It is argued that this system can extract number independently of the visual cues present in the stimulus (diameter, aggregate surface, etc.). However, in a recent report we argue that this might not be the case. We showed that participants combined information from different visual cues to derive their answers. While numerosity comparison requires a rough comparison of two sets of items (smaller versus larger), numerosity estimation requires a more precise mechanism. It could therefore be that numerosity estimation, in contrast to numerosity comparison, might rely on the approximate number system. To test this hypothesis, we conducted a numerosity estimation experiment. We controlled for the visual cues according to current standards: each single visual property was not informative about numerosity. Nevertheless, the results reveal that participants were influenced by the visual properties of the dot arrays. They gave a larger estimate when the dot arrays consisted of dots with, on average, a smaller diameter, aggregate surface or density but a larger convex hull. The reliance on visual cues to estimate numerosity suggests that the existence of an approximate number system that can extract numerosity independently of the visual cues is unlikely. Instead, we propose that humans estimate numerosity by weighing the different visual cues present in the stimuli.     

Evolutionary foundations of number: spontaneous representation of numerical magnitudes by cotton-top tamarins

Although animals of many species have been shown to discriminate between visual-spatial arrays or auditory-temporal sequences based on numerosity, most of the evidence for numerosity discrimination derives from experiments involving extensive laboratory training. Under these conditions, animals' discrimination of two numerosities depends on their ratio and is independent of their absolute value. It is an open question whether any untrained non-human animal spontaneously represents number in this way, as do human children and adults. We present the results of familiarization-discrimination experiments on cotton-top tamarin monkeys (Saguinus oedipus) that provide evidence for numerosity discrimination in the absence of training. Presented with auditory stimuli (speech syllables) controlled for the continuous variables of sequence duration, item duration, inter-stimulus interval and overall energy, tamarins readily discriminated sequences of 4 versus 8, 4 versus 6, and 8 versus 12 syllables. By contrast, tamarins failed to discriminate sequences of 4 versus 5 and 8 versus 10 syllables, providing evidence that their numerosity discrimination is approximate and shows the ratio signature of numerosity discrimination in humans and trained non-human animals. These results provide strong support for the hypothesis that representations of large, approximate numerosity are evolutionarily ancient and spontaneously available to non-human animals.     

Bears "Count" Too: Quantity Estimation and Comparison in Black Bears (Ursus Americanus)

Studies of bear cognition are notably missing from the comparative record despite bears' large relative brain size and interesting status as generalist carnivores facing complex foraging challenges, but lacking complex social structures. We investigated the numerical abilities of three American black bears (Ursus Americanus) by presenting discrimination tasks on a touch-screen computer. One bear chose the larger of two arrays of dot stimuli, while two bears chose the smaller array of dots. On some trials the relative number of dots was congruent with the relative total area of the two arrays. On other trials number of dots was incongruent with area. All of the bears were above chance on trials of both types with static dots. Despite encountering greater difficulty with dots that moved within the arrays, one bear was able to discriminate numerically larger arrays of moving dots, and a subset of moving dots from within the larger array, even when area and number were incongruent. Thus, although the bears used area as a cue to guide responding, they were also able to use number as a cue. The pattern of performance was similar to that found previously with monkeys, and suggests that bears may also show other forms of sophisticated quantitative abilities.     

A texture-processing model of the ‘visual sense of number’

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.     

Perception of shape-from-motion in macaque monkeys and humans

Motion is one of the most efficient cues for shape perception. We conducted behavioral experiments to examine how monkeys perceive shapes defined by motion cues and whether they perceive them as humans do. We trained monkeys to perform a shape discrimination task in which shapes were defined by the motion of random dots. Effects of dot density and dot speed on the shape perception of monkeys were examined. Human subjects were also tested using the same paradigm and the test results were compared with those of monkeys. In both monkeys and humans, correct performance rates declined when density or speed of random dots was reduced. Both of them tended to confuse the same combinations of shapes frequently. These results suggest that monkeys and humans perceive shapes defined by motion cues in a similar manner and probably have common neural mechanisms to perceive them. Electronic Publication     

On visual orientation of dot patterns

Two-dimensional normally distributed random dot patterns were used in two experiments on visual orientation estimation. In the first experiment the patterns differed in their sample correlation and in dot number. In the second one the number of dots was maintained constant but the patterns were generated as a superposition of two normally distributed orthogonal sets composed of different number of dots. In both experiments the estimated orientation depended on stimuli correlation-with increasing correlation the estimated orientation gets closer to the orientation of the least square distance axis of the pattern. Even at very low unsignificant correlations there still remained a hint about stimulus orientation which was not estimated at random. Equalizing consecutively the number of dots in the two orthogonal dot patterns during the second experiment did not result in chance performance either. The bimodal angular distributions of the obtained responses permitted to approach the problem of orientation ambiguity. The results are discussed in terms of optimization processes taking place in the visual system.     

On the estimation of average treatment effects with right-censored time to event outcome and competing risks

We are interested in the estimation of average treatment effects based on right-censored data of an observational study. We focus on causal inference of differences between t-year absolute event risks in a situation with competing risks. We derive doubly robust estimation equations and implement estimators for the nuisance parameters based on working regression models for the outcome, censoring, and treatment distribution conditional on auxiliary baseline covariates. We use the functional delta method to show that these estimators are regular asymptotically linear estimators and estimate their variances based on estimates of their influence functions. In empirical studies, we assess the robustness of the estimators and the coverage of confidence intervals. The methods are further illustrated using data from a Danish registry study.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Cross- and auto-correlation in early vision

Neurons that respond selectively to the orientation of visual stimuli were discovered in V1 more than 50 years ago, but it is still not fully understood how or why this is brought about. We report experiments planned to show whether human observers use cross-correlation or auto-correlation to detect oriented streaks in arrays of randomly positioned dots, expecting that this would help us to understand what David Marr called the 'computational goal' of V1. The streaks were generated by two different methods: either by sinusoidal spatial modulation of the local mean dot density, or by introducing coherent pairs of dots to create moiré patterns, as Leon Glass did. A wide range of dot numbers was used in the randomly positioned arrays, because dot density affects cross- and auto-correlation differently, enabling us to infer which method was used. This difference stems from the fact that the cross-correlation task is limited by random fluctuations in the local mean density of individual dots in the noisy array, whereas the auto-correlation task is limited by fluctuations in the numbers of randomly occurring spurious pairs having the same separation and orientation as the deliberately introduced coherent pairs. After developing a new method using graded dot luminances, we were able to extend the range of dot densities that could be used by a large factor, and convincing results were obtained indicating that the streaks generated by amplitude modulation were discriminated by cross-correlation, while those generated as moiré patterns were discriminated by auto-correlation. Though our current results only apply to orientation selectivity, it is important to know that early vision can do more than simple filtering, for evaluating auto-correlations opens the way to more interesting possibilities, such as the detection of symmetries and suspicious coincidences.     

The influence of two-dimensional stimulus properties in a reconstruction task with minimal information stimuli

Effects of viewpoint that are typically found in shape recognition tasks have been interpreted in the past as strong evidence against approaches based on the use of geometric invariants by the human visual system. At first sight, the use of such invariants would indeed predict shape recognition to be viewpoint independent. It has been argued before, however, that the visual estimation of invariants need not itself be invariant. Changes of viewpoint introduce changes in 2D stimulus features. The latter may make the estimation of invariants by the visual system more or less difficult. The present study explores how such 2D features influence estimation distributions in a reconstruction paradigm. Subjects were shown four coplanar dots. They were asked to estimate the position of one of the dots relative to the three others on the basis of a slanted version of the configuration. An estimation distribution was obtained for every position of the dot to be estimated. Biases in these distributions indicated that subjects performed the task by using 2D reference axes that were unambiguously identifiable in both the images of the dot pattern.     

Models for the perception of orientation in random dot patterns

The aim of this paper is to summarize and to compare some known mathematical models of orientation perception in random dot patterns and to propose new solutions of this question. The model adequacy is judged from the previously obtained experimental results. Apart from the models based on some simple function of the coordinates of dots forming a pattern, also models derived from the so-called image function of the pattern are analysed. The latter ones were found more flexible to render the different features of experimentally obtained data, mainly in the case of orientation ambiguity for some special patterns. The stochastic variants of the deterministic models are introduced.     

A generalized sense of number

Much evidence has accumulated to suggest that many animals, including young human infants, possess an abstract sense of approximate quantity, a number sense. Most research has concentrated on apparent numerosity of spatial arrays of dots or other objects, but a truly abstract sense of number should be capable of encoding the numerosity of any set of discrete elements, however displayed and in whatever sensory modality. Here, we use the psychophysical technique of adaptation to study the sense of number for serially presented items. We show that numerosity of both auditory and visual sequences is greatly affected by prior adaptation to slow or rapid sequences of events. The adaptation to visual stimuli was spatially selective (in external, not retinal coordinates), pointing to a sensory rather than cognitive process. However, adaptation generalized across modalities, from auditory to visual and vice versa. Adaptation also generalized across formats: adapting to sequential streams of flashes affected the perceived numerosity of spatial arrays. All these results point to a perceptual system that transcends vision and audition to encode an abstract sense of number in space and in time.     

Distance Estimation Is Influenced by Encoding Conditions

Background

It is well established that foveating a behaviorally relevant part of the visual field improves localization performance as compared to the situation where the gaze is directed elsewhere. Reduced localization performance in the peripheral encoding conditions has been attributed to an eccentricity-dependent increase in positional uncertainty. It is not known, however, whether and how the foveal and peripheral encoding conditions can influence spatial interval estimation. In this study we compare observers'' estimates of a distance between two co-planar dots in the condition where they foveate the two sample dots and where they fixate a central dot while viewing the sample dots peripherally.

Methodology/Principal Findings

Observers were required to reproduce, after a short delay, a distance between two sample dots based on a stationary reference dot and a movable mouse pointer. When both sample dots are foveated, we find that the distance estimation error is small but consistently increases with the dots-separation size. In comparison, distance judgment in peripheral encoding condition is significantly overestimated for smaller separations and becomes similar to the performance in foveal trials for distances from 10 to 16 degrees.

Conclusions/Significance

Although we find improved accuracy of distance estimation in the foveal condition, the fact that the difference is related to the reduction of the estimation bias present in the peripheral conditon, challenges the simple account of reducing the eccentricity-dependent positional uncertainty. Contrary to this, we present evidence for an explanation in terms of neuronal populations activated by the two sample dots and their inhibitory interactions under different visual encoding conditions. We support our claims with simulations that take into account receptive fields size differences between the two encoding conditions.     

Numerosity estimation in visual stimuli in the absence of luminance-based cues

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.

Methodology/Principal Findings

Here we introduce a novel method, based on second-order (contrast-based) visual motion, to create stimuli that exclude all first-order (luminance-based) cues to numerosity. We show that numerosities can be estimated almost as well in second-order motion as in first-order motion.

Conclusions/Significance

The results show that numerosity estimation need not be based on first-order spatial filtering, first-order density perception, or any other processing of luminance-based cues to numerosity. Our method can be used as an effective tool to control non-numerical variables in studies of numerosity estimation.     

Sequential mapping of weighting functions for visual location.

The effect upon perceived location of adding an extra dot offset from the centre of a cluster of pseudorandom dots was investigated using a vernier acuity task. With this technique, weighting functions showing the extent to which the added dot pulls the apparent location of the entire cluster can be defined as a function of distance from the centre of the cluster. When dot density within the cluster is high, the weighting functions approximate to what would be expected on the basis of centroid alignment. With low dot densities, it appears that performance is determined by aligning the outermost dots within each cluster. The peak amplitudes of these weighting functions are proportional to the square root of dot density within the clusters. The results are consistent with the view that each vernier element is localised in an orthoaxial direction prior to discrimination of the vernier offset.     

Estimation in Bayesian disease mapping

Recent work on Bayesian inference of disease mapping models discusses the advantages of the fully Bayesian (FB) approach over its empirical Bayes (EB) counterpart, suggesting that FB posterior standard deviations of small-area relative risks are more reflective of the uncertainty associated with the relative risk estimation than counterparts based on EB inference, since the latter fail to account for the variability in the estimation of the hyperparameters. In this article, an EB bootstrap methodology for relative risk inference with accurate parametric EB confidence intervals is developed, illustrated, and contrasted with the hyperprior Bayes. We elucidate the close connection between the EB bootstrap methodology and hyperprior Bayes, present a comparison between FB inference via hybrid Markov chain Monte Carlo and EB inference via penalized quasi-likelihood, and illustrate the ability of parametric bootstrap procedures to adjust for the undercoverage in the "naive" EB interval estimates. We discuss the important roles that FB and EB methods play in risk inference, map interpretation, and real-life applications. The work is motivated by a recent analysis of small-area infant mortality rates in the province of British Columbia in Canada.     

Comparison of Bayesian methods for flexible modeling of spatial risk surfaces in disease mapping

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)‐based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF‐based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.     

Symbolic grouping versus simple cell models

The apparent line-like structure in dot patterns derives substantially from the orientation defined by pairings of adjacent dots. Two alternative models have been proposed for making these pairings, one in which the individual dots are treated as discrete grouping tokens, and the second in which the pairing orientation derives from spatial summation by simple cell receptive fields. Contradictory evidence has been found both directly in support of, and directly against, both models. Much of the debate about these two models has hinged on the degree of linearity of summation expected in the simple cell model. Recent neurophysiological evidence changes the balance of the debate, invalidating certain earlier arguments based on linearity and providing a model way of showing that simple cells do indeed play a major, but not necessarily exclusive, role in dot groupings.     

Effects of Awareness on Numerosity Adaptation

Numerosity perception is a process involving several stages of visual processing. This study investigated whether distinct mechanisms exist in numerosity adaptation under different awareness conditions to characterize how numerosity perception occurs at each stage. The status of awareness was controlled by masking conditions, in which monoptic and dichoptic masking were proposed to influence different levels of processing. Numerosity adaptation showed significant aftereffects when the participants were aware (monoptic masking) and unaware (dichoptic masking) of adaptors. The interocular transfer for numerosity adaptation was distinct under the different awareness conditions. Adaptation was primarily binocular when participants were aware of stimuli and was purely monocular when participants were unaware of adaptors. Moreover, numerosity adaptation was significantly reduced when the adaptor dots were clustered into chunks with awareness, whereas clustering had no effect on unaware adaptation. These results show that distinct mechanisms exist in numerosity processing under different awareness conditions. It is suggested that awareness is crucial to numerosity cognition. With awareness, grouping (by clustering) influences numerosity coding through altered object representations, which involves higher-level cognitive processing.