### Abstract

In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford-Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times.

Original language | English (US) |
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Pages (from-to) | 389-408 |

Number of pages | 20 |

Journal | Inverse Problems and Imaging |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - May 2014 |

### Keywords

- Image segmentation
- Line segment detection
- Topological minimization

### ASJC Scopus subject areas

- Analysis
- Modeling and Simulation
- Discrete Mathematics and Combinatorics
- Control and Optimization

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## Cite this

*Inverse Problems and Imaging*,

*8*(2), 389-408. https://doi.org/10.3934/ipi.2014.8.389