## Abstract

We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high probability. Our method uses a carefully designed sampling of the transformation space, where each transformation is efficiently evaluated using a sublinear algorithm. We prove that the density of the sampling depends on the total variation of the shape, allowing us to derive formal bounds on the algorithm's complexity and approximation quality. We further investigate different volumetric shape representations (in the form of truncated distance transforms), and in such a way control the total variation of the shape and hence the sampling density and the runtime of the algorithm. A comprehensive set of experiments assesses the proposed method, including an evaluation on the eight categories of the COSEG data set. This is the first large-scale evaluation of any symmetry detection technique that we are aware of. We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high probability. Our method uses a carefully designed sampling of the transformation space, where each transformation is efficiently evaluated using a sublinear algorithm. We prove that the density of the sampling depends on the total variation of the shape, allowing us to derive formal bounds on the algorithm's complexity and approximation quality.

Original language | English |
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Pages (from-to) | 2-13 |

Number of pages | 12 |

Journal | Computer Graphics Forum |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2015 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2014 The Eurographics Association and John Wiley & Sons Ltd.

## Keywords

- Computational geometry
- I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling
- modeling

## ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design