## Computational Symmetry in Computer Vision and Computer GraphicsIn the arts and sciences, as well as in our daily lives, symmetry has made a profound and lasting impact. Likewise, a computational treatment of symmetry and group theory (the ultimate mathematical formalization of symmetry) has the potential to play an important role in computational sciences. Though the term Computational Symmetry was formally defined a decade ago by the first author, referring to algorithmic treatment of symmetries, seeking symmetry from digital data has been attempted for over four decades. Computational symmetry on real world data turns out to be challenging enough that, after decades of effort, a fully automated symmetry-savvy system remains elusive for real world applications. The recent resurging interests in computational symmetry for computer vision and computer graphics applications have shown promising results. Recognizing the fundamental relevance and potential power that computational symmetry affords, we offer this survey to the computer vision and computer graphics communities. This survey provides a succinct summary of the relevant mathematical theory, a historic perspective of some important symmetry-related ideas, a partial yet timely report on the state of the arts symmetry detection algorithms along with its first quantitative benchmark, a diverse set of real world applications, suggestions for future directions and a comprehensive reference list. |

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### Contents

Introduction | 1 |

Symmetry and Symmetry Groups | 13 |

Symmetry Detection | 59 |

Near Regular Texture NRT | 105 |

Continuous Symmetry | 141 |

Symmetry in Graphics | 159 |

Summary | 175 |

181 | |

### Common terms and phrases

17 wallpaper groups affine transformations analysis applied automatically building blocks computational symmetry computer graphics computer vision constraints contour corresponding curve cyclic defined dihedral group dynamic NRT Escher Euclidean group Euclidean space evaluation example extraction facial asymmetry Figure finite fixed lines fixed points fixed structures frieze groups frieze pattern gait geometry glide-reflection symmetries group theory hyperbolic plane inpainting input lattice detection local symmetry measure method metry mirror symmetry multiple near-regular texture object occlusion orbits pair parameter parameterisation periodic pattern planar primitive symmetries real images real world reflection axes reflection symmetry region regular texture robust rotation centers rotation symmetry rotation symmetry detection rotation symmetry group segment skewed symmetry spatial symme symmetry axis symmetry detection algorithms symmetry group synthesis t1 and t2 texels tiling tion topological tracking translation symmetry vectors vision and computer voting scheme wallpaper groups wallpaper pattern