Symmetry perception

	Single regularities	Number effect	Noise and (a)symmetry effects	Weber's law?	Hierarchy effects	Multiple symmetry

(for further regularity phenomena, see Symmetry processing and Perceptual organization)

A key phenomenon regarding the detectability of visual regularities is that symmetry and Glass patterns are about equally good and better than repetition. The traditionally assumed transformational approach to visual regularity (Garner, 1974; Palmer, 1983) does not account for this phenomenon. As a consequence, in the literature, attempts to account for this phenomenon often capitalized on proximity as the allegedly decisive factor. For instance, in symmetry, corresponding elements near the axis of symmetry are close to one another and are therefore matched easily, whereas in repetition, corresponding elements are always one repeat apart. It is true that proximity may be relevant, but it specifies neither the structural anchors of regularity detection nor its processing mechanisms.

Against this background, the holographic approach to visual regularity was introduced in Psychological Review 1996. It builds on the formalization in Journal of Mathematical Psychology 1991, in which visual regularity is characterized as being transparent holographic regularity (for a mathematical synopsis, see The nature of visual regularities). This characterization implies that symmetry, repetition, and Glass patterns have different visual structures. That is, as illustrated in the next figure, the holographic approach specifies these regularities by identity relationships between substructures, which indicates that they have a point structure, a block structure, and a dipole structure, respectively. This specification is based on the fact that these regularities preserve their nature under growth, that is, under expansion by symmetry pairs, repeats, and dipoles, respectively (see also Evolutionary considerations).




This unique structural differentiation reverberates in all aspects of symmetry perception (see also Symmetry processing and Perceptual organization). It forms, in particular, the heart of a quantitative model of the detectability of single, perturbed, and combined regularities. In this model, the detectability of a regularity is quantified by the holographic weight of evidence (W) for the regularity in the stimulus. The next examples illustrate the explanatory power of this goodness measure.


Single regularities

The detectability of single regularities (as in the figure above) is quantified by W=E/n, where n is the total number of stimulus elements (the dots in the figure above) and E the number of holographic identity relationships (the links in the figure above) that constitute the regularity. In symmetry, for instance, E equals the number of symmetry pairs, so that a perfect symmetry gets a W-load of W=0.5. Furthermore, in Glass patterns, E equals the number of dipoles minus one, so that a perfect Glass pattern gets nearly the same W-load as perfect symmetry. In repetition, however, E equals the number of repeats minus one, so that a perfect repetition gets a much lower W-load which, moreover, depends heavily on the number of stimulus elements in each repeat.

In Psychological Review 1999, these holographic strength differences have been translated into differences in detection speed (see Symmetry processing). According to this process translation, detection propagates linearly for repetition but exponentially for symmetry and Glass patterns. Hence, both the quantitative weight-of-evidence model and the qualitative propagation model explain the key phenomenon that symmetry and Glass patterns are about equally good and better than repetition.


Number effect

Most empirical studies on visual regularity are about symmetry only. One of the strengths of the holographic approach, however, is that it predicts differences between regularities. For instance, both the weight-of-evidence model and the propagation model predict that the detectability of repetition but not of symmetry depends on the number of stimulus elements. The literature already contained evidence supporting the lack of a number effect in symmetry, and to complement this, Csathó et al. (2003) contrasted symmetry and repetition in stimuli as in the next figure -- to see if repetition improves as the number of stimulus elements decreases.




In these stimuli, the number of blobs decreases as the scale gets coarser. Csathó et al. found that this indeed hardly affects symmetry detection but also that it improves repetition detection -- thus confirming the prediction by the holographic approach.

Neuroscientifically, this might be understood as follows for these kind of stimuli. As the number of blobs decreases, the size of the blobs increases, so that they tend to activate a smaller number of larger receptive fields (RFs) in the brain. Fewer but larger RFs with identical responses can be used to detect repetition but not to detect symmetry which still has to be detected at a relatively fine scale. As argued in Psychological Review 2004, this suggests that the holographic number effect in terms of stimulus elements corresponds to a neural number effect in terms of RFs involved in the detection of regularity.

Note: Baylis and Driver (1994) also reported a number effect, but in stimuli featuring what we call antirepetition (see Antiregularity).

For another predicted and confirmed scale effect, see Blob effect.


Noise and (a)symmetry effects

Most regularities in the world are imperfect, so that it is ecologically relevant to assess how perturbed regularities are perceived. When a regularity is perturbed without evoking new spurious structures, the strength of the remaining regularity is again quantified as above. Thus, for instance, the W-load of a perturbed symmetry is W=R/n, where n is the total number of stimulus elements and R the number of still intact symmetry pairs. This implies that symmetry is predicted to degrade gracefully (i.e., deteriorates gradually as noise increases), as already had been found by Barlow & Reeves (1979). It also implies that, under varying n, symmetry is predicted to remain equally detectable as long as R/n remains constant.

Olivers, Chater, and Watson (2004) tested the latter prediction by means of starlike shapes with 6, 12, 24, and 48 contour elements, respectively. The next figure shows several of their stimuli in which, just for this illustration, the noise is marked by shading so that the nonshaded areas mark the remaining symmetry.



Olivers et al. found that symmetry was about equally detectable for 6, 12, and 24 elements, and better for 48 elements. This confirms the holographic prediction above: As shown by our re-analysis in Psychological Review 2004, their stimuli with 6, 12, and 24 elements all have a W-load of about 0.30, and their stimuli with 48 elements have a higher W-load of about 0.40.

Furthermore, perturbed Glass patterns are predicted to follow the formula W=(R-1)/n in which, this time, R is the number of still intact dipoles. This prediction finds support in the empirical study by Maloney et al. (1987) who, albeit by another reasoning, concluded that perturbed Glass patterns follow the formula R/n which, for large R, is virtually the same as the formula given by the holographic approach. Hence, also perturbed symmetry and perturbed Glass patterns are about equally good.

Whereas symmetry and Glass patterns degrade gracefully, it seems that repetition is destroyed perceptually much more easily. The holographic approach predicts that the detectability of perturbed repetition depends on the location of the perturbations, and some evidence for this can be found in Psychological Review 2004.

Moreover, in the literature, it had been noted that humans seem to overestimate the degree of symmetry in a pattern. Freyd and Tversky (1984) investigated this phenomenon by means of triadic comparisons as in the following figure.




Freyd and Tversky found that, when pedestal and targets had a relatively high level of symmetry, subjects tended to choose the more symmetrical target (a symmetry effect), but when pedestal and targets had a relatively low level of symmetry, subjects tended to choose the less symmetrical target (an asymmetry effect).

The holographic approach predicts that the overall level of symmetry indeed plays a role but it predicts moreover that, also at a constant overall level of symmetry, both symmetry effects and asymmetry effects may occur. According to the holographic approach, these effects are not caused by an incorrect assessment (overestimation or underestimation) of the amount of symmetry in a pattern, but by a correct assessment of the symmetry-to-noise ratio in a pattern. This has been confirmed by Csathó et al. (2004) who used the same triadic comparison paradigm but with more detailed stimulus conditions.


Weber's law?

The Weber-Fechner law, or Weber's law (as Fechner, 1860, coined it when elaborating Weber's, 1834, work), is a classical psychophysical law. It usually applies to first-order structures like length, weight, or pitch, and it holds that just noticeable differences in a signal vary in proportion to the strength of the signal. Symmetry, however, is a higher-order structure which relies on structural relationships between elements in a stimulus. A relevant question therefore is whether symmetry in the presence of noise also follows Weber's law.

The answer is: if one excludes extremely weak or strong symmetry signals (for which Weber's law is known to hold poorly) then one could maintain that Weber's law holds, but the holographic approach provides a law which, then, fits equally well, and which, moreover, accounts for the obvious floor and ceiling effects for extremely weak or strong symmetry signals. The holographic law deviates from Weber's law in that it implies that, in the middle range of noise proportions, the sensitivity to variations in regularity-to-noise ratio (which is the signal to be considered) is disproportionally higher than in both outer ranges. Details can be found in Attention, Perception, & Psychophysics 2010, but the following gives a gist.

Weber's law, on the one hand, is dp=k*dS/S, with signal strength S and percept strength p, and with constant k to be determined experimentally. Integration of this differential equation yields p=k*ln(S)+C, where ln is the natural logarithm, and C another constant to be determined experimentally. The holographic approach, on the other hand, implies that, for a noisy symmetry with R intact symmetry pairs and N noise elements, the total number of elements is n=2R+N, so that the holographic weight-of-evidence W=R/n can be rewritten into W=1/(2+N/R). This suggests that the regularity-to-noise ratio R/N defines the strength of the signal (S) to be considered, so that the holographic law is given by p=g/(2+1/S), with constant g to be determined experimentally. The following figure shows best fits of Weber's law p=k*ln(S)+C and the holographic law p=g/(2+1/S) to Barlow & Reeves' (1979) data.




Both laws yield a goodness-of-fit of R²=0.96, but notice that the holographic law does so more parsimoneously, that is, with only one free parameter (g) instead of two free parameters (k and C) as in Weber's law. Above all, notice that the qualitative S-shape in the data (revealing floor and ceiling effects) is captured by the holographic law, whereas it is not captured by Weber's law. In fact, the fits above were lenient to Weber's law: because the real first and last data points cannot be fitted in this log-scale, they were replaced by points obtained via linear interpolation between the real first and last pairs of data points -- choosing these points closer to the real first and last data points drops the fit for Weber's law below R²=0.70, whereas the fit for the holographic law remains R²=0.96.


Hierarchy effects

Corballis and Roldan (1974) found that repetition and symmetry are about equally well detectable if the repetition repeats and the symmetry halves contain additional regularity (they compared the repetition << with the symmetry <>). This agrees with the holographic approach in which the coding model of SIT is used to assess whether a global regularity and a local regularity form a hierarchically compatible combination. Due to the different holographic structures of repetition (block structure) and symmetry (point structure), hierarchically compatible local regularity counts in each repeat of a global repetition but it counts only once in a global symmetry.

Hence, as supported by Corballis and Roldan's finding, repetition is predicted to benefit more from local regularity than symmetry does. Furthermore, for the detectability of seven different combinations of a global and local symmetries, Nucci and Wagemans (2007) found a correlation of R²=0.88 with the predictions by the holographic approach.

Moreover, Olivers et al. (2004) investigated the detectability of symmetry in dipole stimuli which, schematically, looked like:

As shown by our re-analysis in Psychological Review 2004, their data agree well with the predictions for these cases by the holographic approach. The perceived pattern in stimulus A is predicted to be a gobal symmetry with a nested dipole structure; this nested dipole structure makes the symmetry better detectable than an otherwise-random symmetry (which would have W=0.50). Stimuli B and C exhibit a global Glass pattern but are, just as stimulus D, predicted to be perceived as a gobal symmetry with a nested Glass pattern; the orientational uniformity of the dipoles in the nested Glass pattern makes the symmetry even better detectable. Stimulus E does exhibit a symmetry but this is a broken symmetry (the dipoles are positioned symmetrically without being mirrored); the stimulus is actually predicted to be perceived as just a global Glass pattern, and the symmetry is therefore poorly detectable.


Multiple symmetry

Multiple symmetries are patterns with two or more global symmetry axes. A multiple symmetry is, in the holographic approach, also assessed as a hierarchical combination of single symmetries. Whereas the traditionally assumed transformational approach predicts that the goodness of multiple symmetry increases with the number of symmetry axes, the holographic approach predicts that, for instance, 3-fold symmetry is actually slightly worse than 2-fold symmetry (see next figure). This prediction agrees with the study by Wenderoth and Welsh (1998), who found that 3-fold symmetry is not better than 2-fold symmetry and even tends to be worse.


In the holographic approach, the usage of the coding model of SIT implies that the relative orientation of symmetry axes determines the extent to which superimposed single symmetries are hierarchical compatible. Treder et al. (2011) found that the relative orientation of symmetry axes is indeed perceptually relevant. That is, they probed noisy 2-fold symmetries constructed by superimposing two perfect 1-fold symmetries in orthogonal and nonorthogonal relative orientations, all else being equal. They found that performance (discrimination from random patterns) was better for orthogonal axes than for nonorthogonal axes. This suggests that, perceptually, a multiple symmetry is not processed as one regularity, but as consisting of separate single symmetries which, after having been detected, engage in an orientation-dependent interaction.

This is a fine example of the more general Gestalt motto that "the whole is something else than the sum of its parts". Furthermore, as argued in Symmetry 2011, it might well explain the preponderance of 3-fold and 5-fold symmetries in flowers as well as their absence in decorative art -- see also The distribution of multiple symmetries in nature and art.