Inverse Problems and Imaging (IPI)

A variational algorithm for the detection of line segments

Pages: 389 - 408, Volume 8, Issue 2, May 2014      doi:10.3934/ipi.2014.8.389

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Elena Beretta - Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy (email)
Markus Grasmair - Norwegian University of Science and Technology, 7491 Trondheim, Norway (email)
Monika Muszkieta - Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland (email)
Otmar Scherzer - Computational Science Center, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria (email)

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.

Keywords:  Line segment detection, topological minimization, image segmentation.
Mathematics Subject Classification:  Primary: 35C20; Secondary: 65K10.

Received: June 2013;      Revised: February 2014;      Available Online: May 2014.