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Minimal partitions and image classification using a gradient-free perimeter approximation
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 |
References:
[1] |
S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method,, Control and Cybernetics, 34 (2005), 81.
|
[2] |
S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property,, Asymptotic Analysis, 49 (2006), 87.
|
[3] |
L. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion,, C. R. Acad. Sci. Paris, 342 (2006), 313.
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).
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, (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.
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.
|
[8] |
G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions,, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27.
|
[9] |
G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies,, Nonlinear Anal., 38 (1999), 585.
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, (7893), 404. Google Scholar |
[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.
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.
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, (1968).
|
[15] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.
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.
doi: 10.1051/m2an:2000101. |
show all references
References:
[1] |
S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method,, Control and Cybernetics, 34 (2005), 81.
|
[2] |
S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property,, Asymptotic Analysis, 49 (2006), 87.
|
[3] |
L. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion,, C. R. Acad. Sci. Paris, 342 (2006), 313.
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).
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, (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.
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.
|
[8] |
G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions,, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27.
|
[9] |
G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies,, Nonlinear Anal., 38 (1999), 585.
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, (7893), 404. Google Scholar |
[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.
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.
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, (1968).
|
[15] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577.
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.
doi: 10.1051/m2an:2000101. |
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