## Abstract

Separating an object in an image from its background is a central problem (called segmentation) in pattern recognition and computer vision. In this paper, we study the complexity of the segmentation problem, assuming that the object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in an n-pixel grid is NP-hard under the interclass variance, a criterion that is used in discriminant analysis. More importantly, we consider the basic case in which the object is separated by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing exact and approximate optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(n^{2}) time; this algorithm is based on several techniques such as a parametric optimization formulation, a hand-probing algorithm for the convex hull of an unknown point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an ϵ-approximate solution in O(^{ϵ-1}n log L) time, where ϵ is any fixed constant with 1 > ϵ > 0, and L is the total sum of the absolute values of brightness levels of the image.

Original language | English |
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Title of host publication | Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996 |

Publisher | Association for Computing Machinery |

Pages | 104-113 |

Number of pages | 10 |

Volume | Part F129447 |

ISBN (Electronic) | 0898713668 |

Publication status | Published - 1996 Jan 28 |

Externally published | Yes |

Event | 7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996 - Atlanta, United States Duration: 1996 Jan 28 → 1996 Jan 30 |

### Other

Other | 7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996 |
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Country/Territory | United States |

City | Atlanta |

Period | 96/1/28 → 96/1/30 |

## ASJC Scopus subject areas

- Software
- Mathematics(all)