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 having a transparent holographic nature (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
Wload 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
Wload as perfect symmetry. In repetition, however,
E equals the number of repeats minus one, so that a perfect repetition
gets a much lower
Wload 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 weightofevidence 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
weightofevidence 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
Wload
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
reanalysis in
Psychological
Review 2004, their stimuli with 6, 12, and 24
elements
all have a
Wload of about
0.30, and their stimuli with 48
elements have a higher
Wload of about
0.40.
Furthermore, perturbed Glass patterns are predicted to follow the
formula
W=(R1)/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.


Triadic
comparison of a pedestal imperfect symmetry (at the top) with a
slightly more symmetrical target (bottomleft) and a slightly less
symmetrical target (bottomright).
Task: Judge which of the two targets is more
similar to the pedestal.

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 symmetrytonoise 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 WeberFechner 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 firstorder 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
higherorder 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 regularitytonoise 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 d
p=
k*d
S/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=2
R+N, so that the
holographic
weightofevidence
W=R/n
can be rewritten into
W=1/(2+
N/R). This
suggests that the regularitytonoise 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.
Weber's
law 

Holographic
law 






Both laws yield a goodnessoffit of
R^{2}=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 Sshape 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 logscale, 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
R^{2}=0.70,
whereas the fit for the holographic law remains
R^{2}=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
R^{2}=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: