A variational algorithm for the detection of line segments

Elena Beretta; Markus Grasmair; Monika Muszkieta; Otmar Scherzer

doi:10.3934/ipi.2014.8.389

Article Contents

2014, Volume 8, Issue 2: 389-408. Doi: 10.3934/ipi.2014.8.389

This issue Previous Article Minimal partitions and image classification using a gradient-free perimeter approximation Next Article An inner-outer regularizing method for ill-posed problems

A variational algorithm for the detection of line segments

1.
Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano
2.
Norwegian University of Science and Technology, 7491 Trondheim
3.
Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw
4.
Computational Science Center, University of Vienna, Nordbergstrasse 15, 1090 Wien

Received: May 31, 2013

Revised: January 31, 2014

Published: April 2014

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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.

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Mathematics Subject Classification: Primary: 35C20; Secondary: 65K10.

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