## Abstract

Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 ≤ i ≤ m ≤ and ≤ j ≤ n. A problem related to the study of segmentation evaluation of visual images concerns finding a permutation σ of the points of S for which the sum is maximal among all possible permutations of S, where d denotes the Euclidean metric. In this note, we show that this maximum is achieved by exactly those permutations of S for which the line joining the points s and σ(s) passes through the point (m+1/2, n+1/2) for all s ∈ S. In fact, the result applies to any point-symmetric set in any dimension for all L^{p} metrics, 1 ≤ p ≤ ∞. In addition, we provide asymptotic estimates as m and n grow large for the actual maximum value achieved by the above sum.

Original language | English |
---|---|

Pages (from-to) | 193-203 |

Number of pages | 11 |

Journal | Journal of Mathematical Modelling and Algorithms |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2011 |

## Keywords

- Metric
- Segmentation algorithm
- Segmentation evaluation
- Spatial accuracy

## ASJC Scopus subject areas

- Modeling and Simulation
- Applied Mathematics