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A variational approach to edge detection
1. | Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw |
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003. |
[2] |
S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108. |
[3] |
E. Beretta, M. Grasmair, M. Muszkieta and O. Scherzer, A variational algorithm for the detection of line segments, Inverse Problems and Imaging, 8 (2014), 389-408.
doi: 10.3934/ipi.2014.8.389. |
[4] |
J. Canny, A computational approach to edge detection, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 184-203.
doi: 10.1016/B978-0-08-051581-6.50024-6. |
[5] |
Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Mathematical Modeling and Numerical Analysis, 37 (2003), 159-173.
doi: 10.1051/m2an:2003014. |
[6] |
D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595.
doi: 10.1088/0266-5611/14/3/011. |
[7] |
A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle, Journal of Mathematical Imaging and Vision, 14 (2001), 271-284. |
[8] |
A. Desolneux, L. Moisan and J.-M. Morel, From Gestalt Theory to Image Analysis, volume 34 of {Interdisciplinary Applied Mathematics}, Springer New York, 2008.
doi: 10.1007/978-0-387-74378-3. |
[9] |
M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals, Acta Mathematica, 108 (1962), 147-228.
doi: 10.1007/BF02545767. |
[10] |
R. A. Feijóo, A. Novotny, C. Padra and E. Taroco, The topological derivative for the Poisson problem, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1825-1844.
doi: 10.1142/S0218202503003136. |
[11] |
S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control and Optimization, 39 (2000), 1756-1778.
doi: 10.1137/S0363012900369538. |
[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 and Free Boundaries, 15 (2013), 141-166.
doi: 10.4171/IFB/298. |
[13] |
A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989. |
[14] |
D. Marr and E. Hildreth, Theory of edge detection, Proceedings of the Royal Society of London, 207 (1980), 187-217.
doi: 10.1098/rspb.1980.0020. |
[15] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Communications on Pure and Applied Mathematics, 42 (1988), 577-685.
doi: 10.1002/cpa.3160420503. |
[16] |
M. Muszkieta, Optimal edge detection by topological asymptotic analysis, Mathematical Models and Methods in Applied Science, 19 (2009), 2127-2143.
doi: 10.1142/S0218202509004066. |
[17] |
M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives, Journal of the London Mathematical Society, 41 (1990), 526-546.
doi: 10.1112/jlms/s2-41.3.526. |
[18] |
W. K. Pratt, Digital Image Processing, Wiley, New York, 1991. |
[19] |
J. Sokołowski and A. Żochowski, On topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37 (1999), 1251-1272.
doi: 10.1137/S0363012997323230. |
[20] |
M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, Mathematical Modeling and Numerical Analysis, 34 (2000), 723-748.
doi: 10.1051/m2an:2000101. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003. |
[2] |
S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property, Asymptotic Analysis, 49 (2006), 87-108. |
[3] |
E. Beretta, M. Grasmair, M. Muszkieta and O. Scherzer, A variational algorithm for the detection of line segments, Inverse Problems and Imaging, 8 (2014), 389-408.
doi: 10.3934/ipi.2014.8.389. |
[4] |
J. Canny, A computational approach to edge detection, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 184-203.
doi: 10.1016/B978-0-08-051581-6.50024-6. |
[5] |
Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Mathematical Modeling and Numerical Analysis, 37 (2003), 159-173.
doi: 10.1051/m2an:2003014. |
[6] |
D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595.
doi: 10.1088/0266-5611/14/3/011. |
[7] |
A. Desolneux, L. Moisan and J.-M. Morel, Edge detection by Helmholtz principle, Journal of Mathematical Imaging and Vision, 14 (2001), 271-284. |
[8] |
A. Desolneux, L. Moisan and J.-M. Morel, From Gestalt Theory to Image Analysis, volume 34 of {Interdisciplinary Applied Mathematics}, Springer New York, 2008.
doi: 10.1007/978-0-387-74378-3. |
[9] |
M. D. Donsker and J. L. Lions, Fréchet-Volterra variational equations, boundary value problems, and function space integrals, Acta Mathematica, 108 (1962), 147-228.
doi: 10.1007/BF02545767. |
[10] |
R. A. Feijóo, A. Novotny, C. Padra and E. Taroco, The topological derivative for the Poisson problem, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1825-1844.
doi: 10.1142/S0218202503003136. |
[11] |
S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: The elasticity case, SIAM Journal on Control and Optimization, 39 (2000), 1756-1778.
doi: 10.1137/S0363012900369538. |
[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 and Free Boundaries, 15 (2013), 141-166.
doi: 10.4171/IFB/298. |
[13] |
A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989. |
[14] |
D. Marr and E. Hildreth, Theory of edge detection, Proceedings of the Royal Society of London, 207 (1980), 187-217.
doi: 10.1098/rspb.1980.0020. |
[15] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Communications on Pure and Applied Mathematics, 42 (1988), 577-685.
doi: 10.1002/cpa.3160420503. |
[16] |
M. Muszkieta, Optimal edge detection by topological asymptotic analysis, Mathematical Models and Methods in Applied Science, 19 (2009), 2127-2143.
doi: 10.1142/S0218202509004066. |
[17] |
M. Z. Nashed and E. P. Hamilton, Bivariational and singular variational derivatives, Journal of the London Mathematical Society, 41 (1990), 526-546.
doi: 10.1112/jlms/s2-41.3.526. |
[18] |
W. K. Pratt, Digital Image Processing, Wiley, New York, 1991. |
[19] |
J. Sokołowski and A. Żochowski, On topological derivative in shape optimization, SIAM Journal on Control and Optimization, 37 (1999), 1251-1272.
doi: 10.1137/S0363012997323230. |
[20] |
M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, Mathematical Modeling and Numerical Analysis, 34 (2000), 723-748.
doi: 10.1051/m2an:2000101. |
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