# Behind the Looking-Glass: A Review on Human Symmetry Perception

## Abstract

**:**

**Contents**

1 | Introduction | 1511 |
---|---|---|

1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1511 | |

1.2 The Symmetry of Nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1511 | |

1.3 The Nature of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1513 | |

2 | The Role of Symmetry Processing in Perceptual Organization | 1514 |

2.1 Object Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1514 | |

2.2 Automatic Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1516 | |

2.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1516 | |

3 | The Relationship between Symmetry Processing and Cognition | 1517 |

3.1 Attention and Awareness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1517 | |

3.2 Cuing, Expectancy and Voluntary Control. . . . . . . . . . . . . . . . . . . . . . . . . . . | 1519 | |

4 | Functional Characteristics | 1519 |

4.1 Modus Operandi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1519 | |

4.2 Temporal Efficiency and Noise-Resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . | 1521 | |

4.3 Orientation and Location of the Symmetry Axis. . . . . . . . . . . . . . . . . . . . . . . | 1521 | |

4.4 Information Integration and Scale Invariance. . . . . . . . . . . . . . . . . . . . . . . . . | 1522 | |

4.5 Multiple Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1523 | |

4.6 Symmetry versus Antisymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1524 | |

4.7 Symmetry Detection in Complex Biological Stimuli. . . . . . . . . . . . . . . . . . . . | 1526 | |

4.8 Recovery of 3D Structure from Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1526 | |

5 | Neural Implementation | 1527 |

6 | Models of Symmetry Processing | 1528 |

6.1 Representational Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1529 | |

6.2 Process Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1530 | |

6.3 Spatial Filtering Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1532 | |

6.4 Artificial Neural Network Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1534 | |

6.5 Neural Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 1535 | |

7 | Discussion and Conclusions | 1535 |

## 1. Introduction

#### 1.1. Motivation

#### 1.2. The Symmetry of Nature

#### 1.3. The Nature of Symmetry

## 2. The Role of Symmetry Processing in Perceptual Organization

#### 2.1. Object Formation

#### 2.2. Automatic Processing

#### 2.3. Conclusion

## 3. The Relationship between Symmetry Processing and Cognition

#### 3.1. Attention and Awareness

#### 3.2. Cuing, Expectancy and Voluntary Control

## 4. Functional Characteristics

#### 4.1. Modus Operandi

#### 4.2. Temporal Efficiency and Noise-Resistance

#### 4.3. Orientation and Location of the Symmetry Axis

#### 4.4. Information Integration and Scale Invariance

#### 4.5. Multiple Symmetry

#### 4.6. Symmetry versus Antisymmetry

#### 4.7. Symmetry Detection in Complex Biological Stimuli

#### 4.8. Recovery of 3D Structure from Symmetry

## 5. Neural Implementation

## 6. Models of Symmetry Processing

#### 6.1. Representational Models

#### 6.2. Process Models

#### 6.3. Spatial Filtering Models

#### 6.4. Artificial Neural Network Models

#### 6.5. Neural Models

## 7. Discussion and Conclusions

- symmetry detection is quick, sensitive to deviations from perfect symmetry, and robust to noise
- symmetry detection operates on 2D projections of (possibly 3D) objects, but disparity cues are incorporated when they have been processed
- symmetry detection operates automatically and it is involved in object formation
- symmetry detection is affected by higher-level cognition
- the salience of symmetry varies with the orientation of symmetry axis, with the most salient axes being, in order of salience, vertical, horizontal, left/right oblique axes
- generally, the salience of symmetry increases with the number of symmetry axes
- symmetry detection is most efficient when the symmetry axis is foveated, but performance can be equated across stimulus eccentricities by appropriate up-scaling with eccentricity
- the uptake of symmetry information is limited but it is scale invariant
- neurally, symmetry processing is supported by a widespread network of extrastriate visual areas, including V3A, V7, and LOC

## Acknowledgements

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**Figure 1.**Symmetry in the natural and in the artificial environment. (a) Bilateral symmetry of a human, a dog, and a frog. In flowers, one often encounters symmetries with more than one symmetry plane; only one plane is depicted here. (b) Bilateral symmetry in objects used by humans (chair and bicycle), symmetry in architecture (Maya pyramid; only one symmetry plane shown), and the symmetric arrangement of an interior consisting of multiple objects.

**Figure 2.**Emergence of symmetry. (a) A random blob pattern. (b) If the left half of that pattern is reflected about the vertical midline, the percept becomes perceptually organized. (c) This can be strengthened by adding a horizontal symmetry axis, creating a 2-fold symmetry. (d,e) Different perspective views of the same symmetry. For illustrative purposes, solid lines connect some of the symmetry pairs. In the frontoparallel view, lines are parallel and their midpoints lie on the symmetry axis. These characteristics have been conceived as ’anchors’ for symmetry detection. If the pattern is slanted by 40°, the 2D projection yields lines that are not exactly parallel any more but converge towards a vanishing point.

**Figure 3.**Perceptual interpretations of the symmetric structure-from-motion (SFM) stimulus in [31]. (a) Sketch of the physical stimulus, a dot pattern symmetric about the vertical midline. As the arrows indicate, symmetric dots move in opposite directions with the same velocity. (b) Classical SFM interpretations, a clockwise or counterclockwise rotating cylinder. (c) Novel SFM interpretations, two symmetric surfaces. At the midline, symmetric elements meet and they can be perceived as crossing by each other without physical contact (crossing surfaces) or as colliding and then bouncing off each other in the opposite direction (colliding surfaces). With the symmetric dot stimulus, all of these four interpretations can be perceived in an alternating fashion.

**Figure 4.**(a–d) shows the stimuli used in [30], kindly provided by the first author, and (e–h) gives a schematic sketch of the stimuli used in [38]. (a) Frontoparallel dot pattern. (b) The same dot pattern rotated about the vertical midline (y-axis). The retinal symmetry is distorted because the virtual lines connecting dot pairs are neither midpoint collinear nor parallel. (c) Blob pattern rotated about the vertical midline. (d) Blob pattern rotated about the horizontal midline (x-axis). In this pattern, both midpoint collinearity and orientational uniformity of the virtual lines connecting symmetry pairs are preserved. (e–h) Schematic overview of the stereoscopic manipulations used in [38]. Each panel shows a view of two depth planes that, for illustrative purposes, have been tilted backwards. Also for illustrative purposes, the planes are half-transparent (they were fully transparent in the experiment) so that dots on the lower plane covered by the upper plane appear greyish. (a) Baseline stimulus. A perfect symmetry is shown on the first (upper) depth plane, with no information in the other depth plane. (b) Starting from the same pattern, symmetry is spread across two depth planes such that each symmetry half resides on a different depth plane. (c) Starting from the pattern in (a), symmetry is spread across two depth planes such that the pattern on each individual plane looks random. (d) Starting from the pattern in (a), symmetry is spread across two depth planes such that symmetric relationships are preserved within depth planes.

**Figure 5.**Sensitivity to symmetry as a function of the orientation of the symmetry axis. Data points (black circles) are taken from Experiment 1 reported in [75]. The dotted horizontal line gives the sensitivity to horizontal symmetry, the grey curve is a cubic spline interpolation of the data points.

**Figure 6.**Position effects of symmetry information in a dot pattern. The pattern is split into three symmetric pairs of vertical stripes, with one pair containing symmetrically positioned dots and the other two pairs containing random dots. (a) Symmetry is centered around the symmetry axis. (b) Symmetry is confined to the second and fifth stripe. (c) Symmetry is confined to the outermost stripes.

**Figure 7.**Symmetry and antisymmetry in contour stimuli. The red contours in (a) form a symmetry because contour polarities are matched, that is, convexities/concavities in one contour correspond to convexities/concavities in the reflected contour. In contrast, the red contours in (b) form an antisymmetry because contour polarities are opposite, that is, convexities in the left contour match with concavities in the right contour and vice versa [48].

**Figure 8.**(a) Antisymmetric dot patterns with 32, 128, and 256 dots (left to right). Symmetric dots have opposite contrast polarity. (b) Second-order information. For each of the images in (a), dot density is estimated using Gaussian kernels. For the 32 dots pattern, there are substantial local variations in density, and these variations are symmetric about the vertical symmetry axis. As the number of dots increases, the density distribution becomes increasingly homogenous and, therefore, less informative regarding the presence of symmetry. (c) Random dot patterns with 32, 128, and 256 dots (left to right). (d) Analogous to (b), dot density is estimated for the random dot patterns. Again, homogeneity increases with the number of dots. For 256 dots, the density distribution for the random patterns looks very similar to the density distribution for the symmetry.

**Figure 9.**The transformational approach conceptualizes different types of symmetries associated with different invariance transformations. Symmetry halves get a block structure, as indicated by the dashed boxes. (a) Translational symmetry, obtained by translation along the x- or y-axis. (b) Reflection symmetry, obtained by 3D rotation around the vertical dashed line (indicating the symmetry axis). (c) Rotational symmetry, obtained by a rotation of 180°.

**Figure 10.**In the holographic approach, each substructure of a regularity is composed of that same regularity. (a) Repetition is characterized by relationships between repeats. A repetition grows by the addition of repeats, which is why it has a block structure. (b) Symmetry is characterized by relationships between symmetric elements and symmetry grows pair-by-pair. Consequently, symmetry has a point structure. (c) Glass patterns are characterized by relationships between equal pairs of elements (dipoles). As a result, Glass patterns have a dipole structure.

**Figure 11.**Illustration of Jenkins’ [72] first-order model and the bootstrapping approach of Wagemans et al. [29]. (a) If the symmetry pairs in a perfect symmetry are connected by virtual lines (dashed horizontal lines), two features arise. First of all, orientational uniformity, which means that the virtual lines are parallel with respect to each other. Second, midpoint collinearity (white ellipses), which means that the midpoints of virtual lines lie on a straight line coinciding with the symmetry axis (vertical line). (b) Wagemans et al. expanded Jenkins’ first-order structures to also include higher-order structures, formed by joining pairs of virtual lines into correlation quadrangles. In the bootstrapping process, virtual lines are successively added, as indicated by the arrows, to existing higher-order structures (black dashed lines) until the whole stimulus is parsed.

**Figure 12.**Illustration of Barlow and Reeves’ [46] and Dry’s [119] process models. (a) Sketch of the model by Barlow and Reeves applied to a jittered symmetric dot pattern. The pattern is tiled into a number of rectangular cells (here, 4×4 squares). The model acts by counting the number of elements within each square and comparing dot frequencies for symmetrically positioned cells. Despite the jitter, there is a perfect match in dot frequencies for most symmetric pairs of cells. In the third row, however, there is a mismatch, with the first two cells containing 4 dots and 1 dot, respectively, and their symmetric counterparts containing 3 dots and 2 dots. Consequently, there is a slight deterioration of symmetry compared to perfect symmetry. (b) Dry’s model abandons the somewhat artificial rectangular tiling of the stimulus. Rather, each dot is placed in its Voronoi cell (see text for details), whose boundaries are indicated by dashed lines. (c) If the number of dots is decreased, the size of the Voronoi cells increases. This demonstrates that the model’s jitter tolerance scales with element density.

**Figure 13.**Application of Dakin and Watt’s [107] filtering model to a grayscale image of a Stenopus Hispidus. (a) Original image. (b) Image filtered with a relatively fine-scale filter selective for horizontal orientations. The resulting image is obtained by thresholding the output to a ternary image. The according filter is shown in the top right corner. At this scale, the filter is responsive mainly to the fine-scale details of the figure, not the symmetry. (c) Filtering at a slightly coarser scale. Upon visual inspection, the blobs at the lower tip of the animal and blobs at the head are roughly aligned about the symmetry axis. (d) Filtering with a coarse filter. Again, the lower half of the animal displays some degree of blob alignment.

**Figure 14.**Application of Gurnsey et al.’s model [127]. (a) First stage. Pixel symmetry corrupted with pixel noise is convolved with a Gaussian kernel to enhance low spatial frequencies. (b) Second stage. A global differencing operation is performed in search for a vertical symmetry axis. A black trough appears at its position. (c) Third stage. The symmetry axis is explicitly detected by convolution with a vertical filter (small inset at the right top). (d) Just for visualization purposes, the image in (c) was thresholded to illustrate that filter output is indeed most prominent at the symmetry axis.

© 2010 by the author; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Treder, M.S.
Behind the Looking-Glass: A Review on Human Symmetry Perception. *Symmetry* **2010**, *2*, 1510-1543.
https://doi.org/10.3390/sym2031510

**AMA Style**

Treder MS.
Behind the Looking-Glass: A Review on Human Symmetry Perception. *Symmetry*. 2010; 2(3):1510-1543.
https://doi.org/10.3390/sym2031510

**Chicago/Turabian Style**

Treder, Matthias Sebastian.
2010. "Behind the Looking-Glass: A Review on Human Symmetry Perception" *Symmetry* 2, no. 3: 1510-1543.
https://doi.org/10.3390/sym2031510