### Abstract

Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. In [1], an algorithm which performs four of these filling operations has a time complexity of O(N^{2} log N), where N is the size of projections, and leads to a reconstruction algorithm for convex polyominoes running in O(N ^{6} log N)-time. In this paper we first improve the implementation of these four filling operations to a time complexity of O(N^{2}), and additionally we provide an implementation of a fifth filling operation (introduced in [2]) in O(N^{2} log N) that permits to decrease the overall time-complexity of the reconstruction algorithm to O(N^{4} log N). More generally, the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with ℤ^{2}) can be done in O(N^{4} log N)-time.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 98-109 |

Number of pages | 12 |

Volume | 4245 LNCS |

ISBN (Print) | 3540476512, 9783540476511 |

Publication status | Published - 2006 |

Event | 13th International Conference on Discrete Geometry for Computer Imagery, DGCI 2006 - Szeged, Hungary Duration: Oct 25 2006 → Oct 27 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4245 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 13th International Conference on Discrete Geometry for Computer Imagery, DGCI 2006 |
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Country | Hungary |

City | Szeged |

Period | 10/25/06 → 10/27/06 |

### Fingerprint

### Keywords

- Convexity
- Discrete tomography
- Filling operations

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 4245 LNCS, pp. 98-109). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4245 LNCS). Springer Verlag.

**Fast filling operations used in the reconstruction of convex lattice sets.** / Brunetti, Sara; Daurat, Alain; Kuba, A.

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

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 4245 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4245 LNCS, Springer Verlag, pp. 98-109, 13th International Conference on Discrete Geometry for Computer Imagery, DGCI 2006, Szeged, Hungary, 10/25/06.

}

TY - GEN

T1 - Fast filling operations used in the reconstruction of convex lattice sets

AU - Brunetti, Sara

AU - Daurat, Alain

AU - Kuba, A.

PY - 2006

Y1 - 2006

N2 - Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. In [1], an algorithm which performs four of these filling operations has a time complexity of O(N2 log N), where N is the size of projections, and leads to a reconstruction algorithm for convex polyominoes running in O(N 6 log N)-time. In this paper we first improve the implementation of these four filling operations to a time complexity of O(N2), and additionally we provide an implementation of a fifth filling operation (introduced in [2]) in O(N2 log N) that permits to decrease the overall time-complexity of the reconstruction algorithm to O(N4 log N). More generally, the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with ℤ2) can be done in O(N4 log N)-time.

AB - Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. In [1], an algorithm which performs four of these filling operations has a time complexity of O(N2 log N), where N is the size of projections, and leads to a reconstruction algorithm for convex polyominoes running in O(N 6 log N)-time. In this paper we first improve the implementation of these four filling operations to a time complexity of O(N2), and additionally we provide an implementation of a fifth filling operation (introduced in [2]) in O(N2 log N) that permits to decrease the overall time-complexity of the reconstruction algorithm to O(N4 log N). More generally, the reconstruction of Q-convex sets and convex lattice sets (intersection of a convex polygon with ℤ2) can be done in O(N4 log N)-time.

KW - Convexity

KW - Discrete tomography

KW - Filling operations

UR - http://www.scopus.com/inward/record.url?scp=33845216921&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845216921&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33845216921

SN - 3540476512

SN - 9783540476511

VL - 4245 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 98

EP - 109

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -