### 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) |
---|---|

Pages (from-to) | 389-408 |

Number of pages | 20 |

Journal | Inverse Problems and Imaging |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2014 |

### Fingerprint

### Keywords

- Image segmentation
- Line segment detection
- Topological minimization

### ASJC Scopus subject areas

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

### Cite this

*Inverse Problems and Imaging*,

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

**A variational algorithm for the detection of line segments.** / Beretta, Elena; Grasmair, Markus; Muszkieta, Monika; Scherzer, Otmar.

Research output: Contribution to journal › Article

*Inverse Problems and Imaging*, vol. 8, no. 2, pp. 389-408. https://doi.org/10.3934/ipi.2014.8.389

}

TY - JOUR

T1 - A variational algorithm for the detection of line segments

AU - Beretta, Elena

AU - Grasmair, Markus

AU - Muszkieta, Monika

AU - Scherzer, Otmar

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Image segmentation

KW - Line segment detection

KW - Topological minimization

UR - http://www.scopus.com/inward/record.url?scp=84900389295&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84900389295&partnerID=8YFLogxK

U2 - 10.3934/ipi.2014.8.389

DO - 10.3934/ipi.2014.8.389

M3 - Article

VL - 8

SP - 389

EP - 408

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

SN - 1930-8337

IS - 2

ER -