# Perception of 3D Symmetrical and Nearly Symmetrical Shapes

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## Abstract

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## 1. Introduction

## 2. 3D Shapes, 2D Orthographic Projections and 3D Recovery

## 3. Psychophysical Experiment on 3D Shape Recovery

#### 3.1. Stimuli

#### 3.2. Procedure

## 4. Model

## 5. Control Experiment

## 6. Results

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. A 2D Orthographic Image of an Asymmetrical 3D Shape in our Experiment Allows for Recovering 3D Symmetrical Shapes

## References

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**Figure 1.**An example of a symmetrical polyhedron. The “top” of the shape is shown in (

**a**), (

**b**) shows the flat (planar) “bottom” of the shape and (

**c**) shows the shape’s coordinate system.

**Figure 2.**Perceived vs. real asymmetry of shapes for (

**a**) subject EP (binocular); (

**b**) subject EP (monocular); (

**c**) subject VJ (binocular); (

**d**) subject VJ (monocular); (

**e**) subject ZP (binocular) and (

**f**) subject ZP (monocular).

**Figure 3.**Accuracy in identifying symmetrical and asymmetrical shapes in (

**a**) binocular and (

**b**) monocular condition. Recall that there were 30 symmetrical and 60 asymmetrical shapes.

**Figure 4.**Perceived vs. real angles (first column, in radians) and depth (second column) for three different symmetric shapes. Each row represents the corresponding plots for a particular shape. (

**a**) and (

**b**) represent the plots for the shape in (row = 2, column = 5) in set 2; (

**c**) and (

**d**) represent the plots for the shape in (row = 2, column = 6) in set 5 and (

**e**) and (

**f**) represent the plots for the shape in (row = 3, column = 3) in set 3 on the https://lorenz.ecn.purdue.edu/~vthottat/shapeexp/chooseshp.php.

**Figure 5.**Subject shape vs. reference shape depth plots for symmetrical shapes for (

**a**) subject EP (binocular); (

**b**) subject EP (monocular); (

**c**) subject VJ (binocular); (

**d**) subject VJ (monocular); (

**e**) subject ZP (binocular) and (

**f**) subject ZP (monocular). The numbers 45 and 70 indicate viewing directions. The green x marks include both 20${}^{\circ}$ and 70${}^{\circ}$ viewing directions.

**Figure 6.**Perceived vs. real depth plots for asymmetric shapes for subject VJ in (

**a**) binocular and (

**b**) monocular condition.

**Figure 7.**Shape dissimilarity for (

**a**) subject EP (binocular); (

**b**) subject EP (monocular); (

**c**) subject VJ (binocular); (

**d**) subject VJ (monocular); (

**e**) subject ZP (binocular) and (

**f**) subject ZP (monocular).

**Figure 8.**Subject vs. model shape difference, as function of the model weights, for symmetrical shapes, for the subject EP, in the binocular condition. (

**a**) and (

**b**) represent two views of the same shape difference plot.

**Figure 9.**Two views ((

**a**) and (

**b**)) of the subject vs. model shape difference, as function of the model weights, for asymmetrical shapes, for the subject EP, in the binocular condition.

**Figure 10.**Shape difference for asymmetric shapes in the monocular condition as a function of the weight of the symmetry term.

Symmetry | $\mathbf{asym}$ = 0. 00 | 0.01 < $\mathbf{asym}$ < 0.11 | $\mathbf{asym}$ > 0.11 | |
---|---|---|---|---|

Compactness | ||||

$cmp$ > 0.38 | Group 1 | Group 2 | Group 3 | |

0.2 < $cmp$ < 0.38 | Group 4 | Group 5 | Group 6 | |

$cmp$ < 0.20 | Group 7 | Group 8 | Group 9 |

Subject | ${\mathit{W}}_{\mathit{S}}$ |
---|---|

EP | 5.1 |

VJ | 3 |

ZP | 4 |

Subject | ${\mathit{W}}_{\mathit{S}}$ (Symmetrical Case) | ${\mathit{W}}_{\mathit{D}}$ (Symmetrical Case) | ${\mathit{W}}_{\mathit{S}}$ (Asymmetrical Case) | ${\mathit{W}}_{\mathit{D}}$ (Asymmetrical Case) |
---|---|---|---|---|

EP | 9.7 | 8.6 | 10 | 1 |

VJ | 7.6 | 11.5 | 4 | 5.8 |

ZP | 9.0 | 14.0 | 5 | 0.3 |

Subject | Condition | Number of times (out of 90) the model shape was chosen |
---|---|---|

EP | Monocular | 62 (68.8%) |

EP | Binocular | 48 (53.3%) |

VJ | Monocular | 40 (44.4%) |

VJ | Binocular | 46 (51.1%) |

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## Share and Cite

**MDPI and ACS Style**

Jayadevan, V.; Sawada, T.; Delp, E.; Pizlo, Z.
Perception of 3D Symmetrical and Nearly Symmetrical Shapes. *Symmetry* **2018**, *10*, 344.
https://doi.org/10.3390/sym10080344

**AMA Style**

Jayadevan V, Sawada T, Delp E, Pizlo Z.
Perception of 3D Symmetrical and Nearly Symmetrical Shapes. *Symmetry*. 2018; 10(8):344.
https://doi.org/10.3390/sym10080344

**Chicago/Turabian Style**

Jayadevan, Vijai, Tadamasa Sawada, Edward Delp, and Zygmunt Pizlo.
2018. "Perception of 3D Symmetrical and Nearly Symmetrical Shapes" *Symmetry* 10, no. 8: 344.
https://doi.org/10.3390/sym10080344