### Abstract

The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (in) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and Single Source Shortest Path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner Tree problem, which is NP-hard. Nevertheless, an approximate polynomial-time algorithm for this problem exists: we apply a fast implementation of the Goemans-Williamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method.

Original language | English |
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Title of host publication | Proceedings of the IEEE International Conference on Computer Vision |

Pages | 1132-1139 |

Number of pages | 8 |

Volume | II |

DOIs | |

Publication status | Published - 2005 |

Externally published | Yes |

Event | Proceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005 - Beijing Duration: 2005 Oct 17 → 2005 Oct 20 |

### Other

Other | Proceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005 |
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City | Beijing |

Period | 05/10/17 → 05/10/20 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Computer Vision and Pattern Recognition

### Cite this

*Proceedings of the IEEE International Conference on Computer Vision*(Vol. II, pp. 1132-1139). [1544848] https://doi.org/10.1109/ICCV.2005.100

**Finding tree structures by grouping symmetries.** / Ishikawa, Hiroshi; Geiger, Davi; Cole, Richard.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the IEEE International Conference on Computer Vision.*vol. II, 1544848, pp. 1132-1139, Proceedings - 10th IEEE International Conference on Computer Vision, ICCV 2005, Beijing, 05/10/17. https://doi.org/10.1109/ICCV.2005.100

}

TY - GEN

T1 - Finding tree structures by grouping symmetries

AU - Ishikawa, Hiroshi

AU - Geiger, Davi

AU - Cole, Richard

PY - 2005

Y1 - 2005

N2 - The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (in) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and Single Source Shortest Path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner Tree problem, which is NP-hard. Nevertheless, an approximate polynomial-time algorithm for this problem exists: we apply a fast implementation of the Goemans-Williamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method.

AB - The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (in) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and Single Source Shortest Path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner Tree problem, which is NP-hard. Nevertheless, an approximate polynomial-time algorithm for this problem exists: we apply a fast implementation of the Goemans-Williamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method.

UR - http://www.scopus.com/inward/record.url?scp=33745929058&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745929058&partnerID=8YFLogxK

U2 - 10.1109/ICCV.2005.100

DO - 10.1109/ICCV.2005.100

M3 - Conference contribution

SN - 076952334X

SN - 9780769523347

VL - II

SP - 1132

EP - 1139

BT - Proceedings of the IEEE International Conference on Computer Vision

ER -