Peter van der Helm: Symmetry processing

Peter A. van der Helm

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Holographic bootstrapping

Within the holographic approach, the detection of visual regularities is modeled by way of the holographic bootstrap model, which was introduced in Psychological Review 1999. This holographic detection model builds on Jenkins' (1983) and Wagemans et al.'s (1993) processing ideas, but it proposes propagation steps that agree with the construction of mental representations in the coding model of SIT. This also means that it honours the holographic difference in structure between symmetry and repetition (point versus block structure), and thereby, it provides a faithful process translation of the holographic detectability model introduced in Psychological Review 1996. The proposed propagation steps can be sketched as follows.


Jenkins postulated that the anchors for the detection process are parallel and midpoint-colinear virtual lines in symmetry, and parallel and equally long virtual lines in repetition. Later, Wagemans et al. developed a bootstrap model in which the detection process propagates by way of correlational quadrangles formed by two virtual lines, that is, trapezoids in symmetry and parallelograms in repetition. Neither Jenkins nor Wagemans et al. postulated further processing differences between symmetry and repetition, and resorted to a proximity effect to explain that symmetry is better detectable than repetition.

In the holographic bootstrap model, symmetry detection propagates just as in Wagemans et al.'s original bootstrap model: In parallel for all virtual lines found so far, the process searches for new virtual lines to form additional trapezoids, and so on. Repetition detection, however, propagates differently: The four elements that form a parallelogram are grouped into two blocks that give rise to one virtual line, after which a new virtual line is searched to form a new parallelogram, and so on.

Thus, the holographic model implies that the propagation spreads exponentially in symmetry but linearly in repetition.

As is sketched next, this model is supported by empirical evidence -- first, for the proposed anchors of the propagation, and second, for the proposed difference in propagation between symmetry and repetition.

Perspective effect

The evidence for the proposed anchors of the propagation in symmetry and repetition detection follows the development of the model:

First, by considering the effect of jitter in othrofrontally presented regularities, Jenkins (1983) found empirical evidence supporting the perceptual relevance of the first-order structures, that is, of the parallel and midpoint-colinear virtual lines in symmetry, and of the parallel and equally long virtual lines in repetition.
Second, Wagemans et al. (1993) considered the effect of affine skewing, which affects detectability even though the first-order structures are preserved; this yielded empirical evidence supporting the perceptual relevance of the second-order structures, that is, of the correlational quadrangles formed by two virtual lines (which are affected by skewing).
Third, van der Vloed et al. (2005) tested the perceptual relevance of both first-order and second-order structures in proper perspective views (which are ecologically more relevant than affine skewing), as depicted in the next figure.

A theoretical analysis showed that perspective distorts the retinal first-order and second-order structures of symmetry and repetition differently (and in another way than affine skewing does). Two experiments on this distortion difference between symmetry and repetition yielded evidence that, in perspective views, regularity detection in these kind of stimuli is not preceded by normalization but occurs directly on the basis of the retinal first-order and second-order structures. That is, the different degrees in which perspective distorts these structures in symmetry and repetition correspond to the different degrees in which it affects the detectability of symmetry and repetition.

Blob effect

Just as the holographic detectability model, the holographic detection model predicts that symmetry is better detectable than repetition, and that the detectability of symmetry depends hardly on the number of pattern elements whereas the detectability of repetition depends strongly on the number of pattern elements (see Symmetry perception).

In addition, the holographic propagation difference between symmetry and repetition leads to a remarkable further prediction which has been put forward in Psychological Review 1999 and which can be introduced as follows.


Suppose that, in a first stage, only the white areas of the stimulus are available to the regularity detection process. Then, at first, the propagation proceeds as usual (the structure detected so far is here indicated by the red dots).

Soon, however, the restriction to the white areas stops the exponentially spreading propagation in symmetry, whereas the linearly spreading propagation in repetition still can go on detecting structure in the white areas. Hence, symmetry is hindered by the split situation whereas repetition is not.

When, in a second stage, the split is lifted so that the entire stimulus becomes available, the propagation again proceeds as usual and symmetry restores its advantage over repetition.

To understand this effect of the split situation by way of an analogy, one may think of a slow car (repetition) for which it matters hardly whether or not there is much traffic on the road, versus a fast car (symmetry) for which it matters a lot. In the next figure, the split situation arises in the stimuli containing relatively coarse-scaled areas. These areas consist of salient blobs which attract attention and which, therefore, are probably processed before the rest of the pattern is processed.

Symmetry				Repetition

Hence, although the literature features the idea (no data) that such salient subpatterns might improve symmetry detection, the holographic bootstrap model predicts, as argued above, that such salient blobs hinder symmetry. Yet, this counter-intuitive holographic prediction was confirmed by Csathó et al. (2003) who also found that such blobs do not hinder repetition but, probably due to the Number effect, do help repetition -- see the sketch of their results below, in which "blob scale" refers to the scale of the blob areas (0 means no difference with respect to the surround; 1 and 2 refer to two degrees of coarser-scaled blob areas).

Notice that this finding is relevant to the question, in biology, of whether symmetry or size -- of sexual ornaments and other morphological traits -- is more relevant in mate selection. That is, a large local trait in a global symmetry may be salient as such but also seems to reduce the salience of the global symmetry. For further biologically relevant considerations, see Symmetry 2011 and The origin of visual regularities.