# American Institute of Mathematical Sciences

May  2014, 8(2): 389-408. doi: 10.3934/ipi.2014.8.389

## 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, Italy 2 Norwegian University of Science and Technology, 7491 Trondheim, Norway 3 Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland 4 Computational Science Center, University of Vienna, Nordbergstrasse 15, 1090 Wien

Received  June 2013 Revised  February 2014 Published  May 2014

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.
Citation: Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems and Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389
##### References:
 [1] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method, Control and Cybernetics, 34 (2005), 81-101. [2] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108. [3] L. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, Ser., 342 (2006), 313-318. doi: 10.1016/j.crma.2005.12.009. [4] E. Beretta, Y. Capdeboscq, F. de Gournay and E. Francini, Thin cylindrical conductivity inclusions in a three-dimensional domain: a polarization tensor and unique determination from boundary data. Inverse Probl., 25 (2009), 065004, 22pp. doi: 10.1088/0266-5611/25/6/065004. [5] A. Braides, Approximation of Free-Discontinuity Problems, volume 1694 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998. [6] Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 37 (2003), 159-173. doi: 10.1051/m2an:2003014. [7] Y. Capdeboscq and M. S. Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptot. Anal., 50 (2006), 175-204. [8] G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49 (1998). [9] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal., 38 (1999),(5, Ser. B: Real World Appl.), 585-604. doi: 10.1016/S0362-546X(98)00132-1. [10] G. Dong, M. Grasmair, S. H. Kang and O. Scherzer, Scale and edge detection with topological derivatives of the Mumford-Shah functional, In A. Kuijper, editor, SSVM'13: Proceedings of the fourth International Conference on Scale Space and Variational Methods in Computer Vision, volume 7893 of Lecture Notes in Computer Science, pages 404-415, Berlin, Heidelberg, 2013. Springer-Verlag. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 2nd ed. Springer, 2001. [12] M. Grasmair, M. Muszkieta and O. Scherzer, An approach to the minimization of the Mumford-Shah functional using $\Gamma$-convergence and topological asymptotic expansion, Interfaces Free Bound., 15 (2013), 141-166. doi: 10.4171/IFB/298. [13] Y. M. Jung, S. H. Kang and J. Shen, Multiphase image segmentation via Modica-Mortola phase transition, SIAM J. Appl. Math., 67 (2007), 1213-1232. doi: 10.1137/060662708. [14] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. [15] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [16] M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), 723-748. doi: 10.1051/m2an:2000101.

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##### References:
 [1] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method, Control and Cybernetics, 34 (2005), 81-101. [2] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108. [3] L. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, Ser., 342 (2006), 313-318. doi: 10.1016/j.crma.2005.12.009. [4] E. Beretta, Y. Capdeboscq, F. de Gournay and E. Francini, Thin cylindrical conductivity inclusions in a three-dimensional domain: a polarization tensor and unique determination from boundary data. Inverse Probl., 25 (2009), 065004, 22pp. doi: 10.1088/0266-5611/25/6/065004. [5] A. Braides, Approximation of Free-Discontinuity Problems, volume 1694 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998. [6] Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 37 (2003), 159-173. doi: 10.1051/m2an:2003014. [7] Y. Capdeboscq and M. S. Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptot. Anal., 50 (2006), 175-204. [8] G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49 (1998). [9] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal., 38 (1999),(5, Ser. B: Real World Appl.), 585-604. doi: 10.1016/S0362-546X(98)00132-1. [10] G. Dong, M. Grasmair, S. H. Kang and O. Scherzer, Scale and edge detection with topological derivatives of the Mumford-Shah functional, In A. Kuijper, editor, SSVM'13: Proceedings of the fourth International Conference on Scale Space and Variational Methods in Computer Vision, volume 7893 of Lecture Notes in Computer Science, pages 404-415, Berlin, Heidelberg, 2013. Springer-Verlag. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 2nd ed. Springer, 2001. [12] M. Grasmair, M. Muszkieta and O. Scherzer, An approach to the minimization of the Mumford-Shah functional using $\Gamma$-convergence and topological asymptotic expansion, Interfaces Free Bound., 15 (2013), 141-166. doi: 10.4171/IFB/298. [13] Y. M. Jung, S. H. Kang and J. Shen, Multiphase image segmentation via Modica-Mortola phase transition, SIAM J. Appl. Math., 67 (2007), 1213-1232. doi: 10.1137/060662708. [14] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. [15] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [16] M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), 723-748. doi: 10.1051/m2an:2000101.
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