The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art

A role of rectangularity in perceiving a 3D shape of an object

Rectangularity and perpendicularity of contours are important properties of 3D shape for the visual system and the visual system can use them asa prioriconstraints for perceivingshape veridically. The presentarticle provides a comprehensive review ofpriorstudiesofthe perception of rectangularity and perpendicularity anditdiscussestheir effects on3D shape perception from both theoretical and empiricalapproaches. It has been shown that the visual system is biased to perceive a rectangular 3D shape from a 2D image. We thought that this bias might be attributable to the likelihood of a rectangular interpretation but this hypothesis is not supported by the results of our psychophysical experiment. Note that the perception ofa rectangular shape cannot be explained solely on the basis of geometry. A rectangular shape is perceived from an image that is inconsistent with a rectangular interpretation. To address thisissue, we developed a computational model that can recover a rectangular shape from an image of a parallelopiped. The model allows the recovered shape to be slightly inconsistent so that the recovered shape satisfies the a priori constraints of maximum compactness and minimal surface area. This model captures someof thephenomenaassociated withthe perception of the rectangular shape that were reported inpriorstudies. This finding suggests that rectangularity works for shape perception by incorporatingitwith someadditionalconstraints.

The brain does not process horizontal reflection when attending to vertical reflection, and vice versa

Previous work has found that feature attention can modulate electrophysiological responses to visual symmetry. In the current study, participants observed spatially overlapping clouds of black and white dots. They discriminated vertical symmetry from asymmetry in the target dots (e.g., black or white) and ignored the regularity of the distractor dots (e.g., white or black). We measured an electroencephalography component called the sustained posterior negativity (SPN), which is known to be generated by visual symmetry. There were five conditions with different combinations of target and distractor regularity. As well as replicating previous results, we found that an orthogonal axes of reflection in the distractor dots had no effect on SPN amplitude. We conclude that the visual system can processes reflectional symmetry in independent axis-orientation specific channels.

Electrophysiological response to visual symmetry: Effects of the number of symmetry axes

The symmetry axes of a stimulus are a critical determinant of visual perception. Although much is known about the effects of a single symmetry axis on perception, the effects of multiple symmetry axes are still poorly understood. Here, we investigated the influence of the number of symmetry axes on brain activity using event-related potentials (ERPs). Our results showed that altering the number of symmetry axes affects both the amplitude and the latency of ERPs. Specifically, the amplitude of ERP components increased as the number of symmetry axes increased, starting at the N1 (165-175 ms) component and lasting until the P2 (230-250 ms) component in the bilateral posterior areas and until the N2 (340 ms) component in the frontal-central areas. Importantly, the latency of ERP components was reduced when the number of symmetry axes increased, starting at the N1 in the right posterior area and lasting until the P2 component in the bilateral posterior areas. The temporal and spatial differences in these effects imply that activity related to symmetry axes gradually changes throughout the ventral visual streams in the human brain.

No effect of multi-axis dot pattern symmetry on subjective duration

Previous work has shown that symmetrical stimuli are judged as lasting longer than asymmetrical ones, even when actual duration is matched. This effect has been replicated with different methods and stimuli types. We aimed to a) replicate the effect of symmetry on subjective duration, and b) assess whether it was further modulated by the number of symmetrical axes. There was no evidence for either effect. This null result cannot be explained by reduced statistical power or enhanced floor or ceiling effects. There is no obvious stimulus-based explanation either. However, we are mindful of the reproducibility crisis and file drawer problems in psychology. Other symmetry and time perception researchers should be aware of this null result. One possibility is that the effect of symmetry on subjective duration is limited to very specific experimental paradigms.

The neural basis of visual symmetry and its role in mid- and high-level visual processing

Symmetry is an important and prominent feature of the visual world. It has been studied as a basis for image segmentation and perceptual organization, but it also plays a role in higher level processes, such as face and object perception. Over the past decade, there has been progress in the study of the neural mechanisms of symmetry perception in humans and other animals. There is extended activity in the ventral stream, including the lateral occipital complex (LOC) and VO1; this activity starts in V3 and it occurs independently of the task (automatic response). Additionally, when the task requires processing of symmetry, the activation may emerge for objects that are symmetrical, even though they do not project a symmetrical image. There is also some evidence of hemispheric lateralization, especially for the LOC. We review the studies on the cortical basis of visual symmetry processing and its links to encoding of other aspects of the visual world, such as faces and objects.

Visual cortex activation predicts visual preference: Evidence from Britain and Egypt

The term “Perceptual goodness” refers to the strength, obviousness, or salience of a visual configuration. Recent work has found strong agreement between theoretical, neural, and behavioural measures of perceptual goodness across a wide range of different symmetrical visual patterns. We used these pattern types again to explore the relationship between perceptual goodness and aesthetic preference. A group of 50 U.K. participants rated the patterns on a 0 to 100 scale. Preference ratings positively correlated with four overlapping measures of perceptual goodness. We then replicated this finding in Egypt, suggesting that our results reflect universal aspects of human preference. The third experiment provided consistent results with a different stimulus set. We conclude that symmetry is an aesthetic primitive that is attractive because of the way it is processed by the visual system.

Searching for Radial Symmetry

Symmetry is ubiquitous in the natural world. Numerous investigations, dating back over one hundred years, have explored the visual processing of symmetry. However, these studies have been concerned with mirror symmetry, overlooking radial (or rotational) symmetry, which is also prevalent in nature. Using a visual search paradigm, which approximates the everyday task of searching for an object embedded in background clutter, we have measured how quickly and how accurately human observers detect radially symmetric dot patterns. Performance was compared with mirror symmetry. We found that with orders of radial symmetry greater than 5, radial symmetry can be detected more easily than mirror symmetry, revealing for the first time that radial symmetry is a salient property of objects for human vision.

A group theoretical model of symmetry cognition

We report on two experiments, published originally in Japanese, on judged goodness and simplicity of dot patterns with reflectional and rotational symmetries (with 1-4 reflection axes and repeats, respectively) under free-viewing tasks. We found that (a) both goodness and simplicity increase monotonously with the number of transformations under which a pattern is invariant; (b) stimulus outlines, such as squares and hexagons, affect both goodness and simplicity; and (c) factors such as contrast polarity and collinearity affect simplicity rather than goodness. The employed free-viewing tasks contrast with detection tasks involving short presentation times, and based on behavioural and neurophysiological evidence, we conclude that this transformational approach captures late rather than early aspects of visio-cognitive processing of visual regularities.

An Electrophysiological Index of Perceptual Goodness

A traditional line of work starting with the Gestalt school has shown that patterns vary in strength and salience; a difference in “Perceptual goodness.” The Holographic weight of evidence model quantifies goodness of visual regularities. The key formula states that W = E/N, where E is number of holographic identities in a pattern and N is number of elements. We tested whether W predicts the amplitude of the neural response to regularity in an extrastriate symmetry-sensitive network. We recorded an Event Related Potential (ERP) generated by symmetry called the Sustained Posterior Negativity (SPN). First, we reanalyzed the published work and found that W explained most variance in SPN amplitude. Then in four new studies, we confirmed specific predictions of the holographic model regarding 1) the differential effects of numerosity on reflection and repetition, 2) the similarity between reflection and Glass patterns, 3) multiple symmetries, and 4) symmetry and anti-symmetry. In all cases, the holographic approach predicted SPN amplitude remarkably well; particularly in an early window around 300–400 ms post stimulus onset. Although the holographic model was not conceived as a model of neural processing, it captures many details of the brain response to symmetry.

The accentuation principle of figure-ground segregation and the downbeat illusion

Pinna and Sirigu (2011) demonstrated a new principle of grouping, called the accentuation principle, stating that, all else being equal, elements tend to group in the same oriented direction of the discontinuous element placed within a whole set of continuous/homogeneous components. The discontinuous element behaves like an accent, i.e. a visual emphasis within the wholeness of components as shown in the next section. In this work, the accentuation principle has been extended to new visual domains. In particular, it is shown how this principle affects shape perception. Moreover several visual object attributes are also highlighted, among which orientation, spatial position, inner dynamics and apparent motion that determine the so-called organic segmentation and furthermore tend to induce figure-ground segregation. On the basis of the results of experimental phenomenology, the accentuation can be considered as a complex principle ruling grouping, figure-ground segregation, shape and meaning formation. Through a new musical illusion of downbeat, it is also demonstrated that this principle influences perceptual organization not only in space but also in time and, thus, in both visual and musical domains. This illusion can be heard in eight measures of Pagodes, a solo piano music by Claude Debussy (1862-1918), where a strong physical-perceptual discrepancy in terms of upbeats and downbeats inversion is strongly perceived in both staves.

The Role of Visual Eccentricity on Preference for Abstract Symmetry

This study tested preference for abstract patterns, comparing random patterns to a two-fold bilateral symmetry. Stimuli were presented at random locations in the periphery. Preference for bilateral symmetry has been extensively studied in central vision, but evaluation at different locations had not been systematically investigated. Patterns were presented for 200 ms within a large circular region. On each trial participant changed fixation and were instructed to select any location. Eccentricity values were calculated a posteriori as the distance between ocular coordinates at pattern onset and coordinates for the centre of the pattern. Experiment 1 consisted of two Tasks. In Task 1, participants detected pattern regularity as fast as possible. In Task 2 they evaluated their liking for the pattern on a Likert-scale. Results from Task 1 revealed that with our parameters eccentricity did not affect symmetry detection. However, in Task 2, eccentricity predicted more negative evaluation of symmetry, but not random patterns. In Experiment 2 participants were either presented with symmetry or random patterns. Regularity was task-irrelevant in this task. Participants discriminated the proportion of black/white dots within the pattern and then evaluated their liking for the pattern. Even when only one type of regularity was presented and regularity was task-irrelevant, preference evaluation for symmetry decreased with increasing eccentricity, whereas eccentricity did not affect the evaluation of random patterns. We conclude that symmetry appreciation is higher for foveal presentation in a way not fully accounted for by sensitivity.

The role of constant curvature in 2-D contour shape representations

In early cortex, visual information is encoded by retinotopic orientation-selective units. Higher-level representations of abstract properties, such as shape, require encodings that are invariant to changes in size, position, and orientation. Within the domain of open, 2-D contours, we consider how an economical representation that supports viewpoint-invariant shape comparisons can be derived from early encodings. We explore the idea that 2-D contour shapes are encoded as joined segments of constant curvature. We report three experiments in which participants compared sequentially presented 2-D contour shapes comprised of constant curvature (CC) or non-constant curvature (NCC) segments. We show that, when shapes are compared across viewpoint or for a retention interval of 1000 ms, performance is better for CC shapes. Similar recognition performance is observed for both shape types, however, if they are compared at the same viewpoint and the retention interval is reduced to 500 ms. These findings are consistent with a symbolic encoding of 2-D contour shapes into CC parts when the retention intervals over which shapes must be stored exceed the duration of initial, transient, visual representations.

Interactions between constituent single symmetries in multiple symmetry

As a rule, the discriminability of multiple symmetries from random patterns increases with the number of symmetry axes, but this number does not seem to be the only determinant. In particular, multiple symmetries with orthogonal axes seem better discriminable than multiple symmetries with nonorthogonal axes. In six experiments on imperfect two-fold symmetry, we investigated whether this is due to extra structure in the form of so-called correlation rectangles, which arise only in the case of orthogonal axes, or to the relative orientation of the axes as such. The results suggest that correlation rectangles are not perceptually relevant and that the percept of a multiple symmetry results from an orientation-dependent interaction between the constituent single symmetries. The results can be accounted for by a model involving the analysis of symmetry at all orientations, smoothing (averaging over neighboring orientations), and extraction of peaks.

Bayesian confusions surrounding simplicity and likelihood in perceptual organization

In the study of perceptual organization, the Occamian simplicity principle (which promotes efficiency) and the Helmholtzian likelihood principle (which promotes veridicality) have been claimed to be equivalent. Proposed models of these principles may well yield similar outcomes (especially in everyday situations), but as argued here, claims that the principles are equivalent confused subjective probabilities (which are used in Bayesian models of the Occamian simplicity principle) and objective probabilities (which are needed in Bayesian models of the Helmholtzian likelihood principle). Furthermore, Occamian counterparts of Bayesian priors and conditionals have led to another confusion, which seems to have been triggered by a dual role of regularity in perception. This confusion is discussed by contrasting complete and incomplete Occamian approaches to perceptual organization.