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A fractional-order derivative based variational framework for image denoising
The topological gradient method for semi-linear problems and application to edge detection and noise removal
1. | Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France, France |
References:
[1] |
L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Gamma-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036.
doi: 10.1002/cpa.3160430805. |
[2] |
S. Amstutz, Topological sensitivity analysis for some nonlinear PDE system, J. Math. Pures Appl., 85 (2006), 540-557.
doi: 10.1016/j.matpur.2005.10.008. |
[3] |
S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 401-425.
doi: 10.1051/cocv:2005012. |
[4] |
S. Amstutz and J. Fehrenbach, Edge detection using topological gradients: A scale-space approach, J. Math. Imaging Vision, 52 (2015), 249-266.
doi: 10.1007/s10851-015-0558-z. |
[5] |
S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the toplogical gradient method, Control and Cybernetics, 34 (2005), 81-101. |
[6] |
G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal of Applied Mathematics, 68 (2008), 925-946.
doi: 10.1137/060671814. |
[7] |
G. Aubert and A. Drogoul, Topological gradient for a fourth order operator used in image analysis, ESAIM Control Optim. Calc. Var., 21 (2015), 1120-1149.
doi: 10.1051/cocv/2014061. |
[8] |
G. Aubert and A. Drogoul, Topological gradient for fourth order pde and application to the detection of fine structures in 2d images, C. R. Math. Acad. Sci. Paris, 352 (2014), 609-613.
doi: 10.1016/j.crma.2014.06.005. |
[9] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Second edition, Applied Mathematical Sciences, 147, Springer-Verlag, 2006. |
[10] |
D. Auroux, From restoration by topological gradient to medical image segmentation via an asymptotic expansion, Math. Comput. Model., 49 (2009), 2191-2205.
doi: 10.1016/j.mcm.2008.07.002. |
[11] |
D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis, Computational & Applied Mathematics, 25 (2006), 251-267.
doi: 10.1590/S0101-82052006000200008. |
[12] |
D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion, Journal of Mathematical Imaging and Vision, 33 (2009), 122-134.
doi: 10.1007/s10851-008-0121-2. |
[13] |
D. Auroux, M. Masmoudi and L. Jaafar Belaid, Image restoration and classification by topological asymptotic expansion, Variational Formulations in Mechanics: Theory and Applications (eds. E. Taroco, E. A. de Souza Neto and A. A. Novotny), CIMNE, Barcelona, Spain, 2007, 23-42. |
[14] |
H. Ayasso and A. Mohammad-Djafari, Joint image restorationand segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation, IEEE Trans. Image Process, 19 (2010), 2265-2277.
doi: 10.1109/TIP.2010.2047902. |
[15] |
L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, 342 (2006), 313-318.
doi: 10.1016/j.crma.2005.12.009. |
[16] |
L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Application of the topological gradient to image restoration and edge detection, Engineering Analysis with Boundary Elements, 32 (2008), 891-899. |
[17] |
S. Ben Hadj, L. Blanc Féraud and G. Aubert, Space Variant Blind Image Restoration, SIAM Journal on Imaging Sciences, 7 (2014), 2196-2225.
doi: 10.1137/130945776. |
[18] |
S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp.
doi: 10.1088/0266-5611/25/1/015002. |
[19] |
T. Chan and J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005.
doi: 10.1137/1.9780898717877. |
[20] |
N. Dey, L. Blanc-Féraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, 3D Microscopy Deconvolution using Richardson-Lucy Algorithm with Total Variation Regularization, Rapport de recherche RR-5272, INRIA, 2004. |
[21] |
A. Drogoul, Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in 2D imaging, SIAM J. Imaging Sci., 7 (2014), 2700-2731.
doi: 10.1137/140967374. |
[22] |
A. Drogoul, Topological Gradient Method Applied to the Detection of Edges and Fine Structures in Imaging, Ph.D Thesis, University of Nice Sophia Antipolis, 2014. |
[23] |
S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741.
doi: 10.1109/TPAMI.1984.4767596. |
[24] |
T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian recontruction from Poisson data using Gibbs prior, IEEE Trans. Medical imaging, 8 (1989), 194-202. |
[25] |
F. M. Henderson, A. J. Lewis and R. A. Ryerson, eds., Principles and Applications of Imaging Radar. Manual of Remote Sensing, 3rd edition, Wiley, New York, 1998.
doi: 10.1029/99EO00047. |
[26] |
M. Hintermüller, Fast-set based algorithms using shape and topological sensitivity information, Control and Cybernetics, 34 (2005), 305-324. |
[27] |
M. Iguernane, S. A. Nazarov, J.-R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Applied Mathematics and Computer Science, 19 (2009), 191-205.
doi: 10.2478/v10006-009-0016-4. |
[28] |
K. Krissian, R. Kikinis, C.-F. Westin and K. G. Vosburgh, Speckle-constrained filtering of ultrasound images, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, IEEE Computer Society, 2, 2005, 547-552.
doi: 10.1109/CVPR.2005.331. |
[29] |
S. Larnier and J. Fehrenbach, Edge detection and image restoration with anisotropic topological gradient, in Proceedings ICASSP , International Conference on Acoustic Speech and Signal Processing, 2010, 1362-1365.
doi: 10.1109/ICASSP.2010.5495448. |
[30] |
S. Larnier, J. Fehrenbach and M. Masmoudi, The topological gradient method: From optimal design to image processing, Milan Journal of Mathematics, 80 (2012), 411-441.
doi: 10.1007/s00032-012-0196-5. |
[31] |
I. Larrabide, A. A. Novotny, R. A. Feijo and E. Taroco, A medical image enhancement algorithms based on topological derivative and anisotropic diffusion, in Proceedings of the XXVI Iberian Latin-American Congess on Computational Methods in Engineering, 2005. |
[32] |
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998. |
[33] |
M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, GAKUTO Internat. Ser. Math. Appl., 16, Tokyo, Japan, 2001. |
[34] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[35] |
J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences, Springer, New York, 2001.
doi: 10.1007/978-1-4757-4393-7. |
[36] |
J. Pawley, Handbook of Biological Confocal Microscopy, Springer, Berlin, 2006. |
[37] |
A. Sawatzky, D. Tenbrinck, X. Jiang and M. Burger, A variational framework for region-based segmentation incorporating physical noise models, Journal of Mathematical Imaging and Vision, 47 (2013), 179-209.
doi: 10.1007/s10851-013-0419-6. |
[38] |
J. Serra, Image Analysis and Mathematical Morphology, Academic Press, Inc., Orlando, FL, USA, 1984. |
[39] |
J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272.
doi: 10.1137/S0363012997323230. |
[40] |
M. Tur, K. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.
doi: 10.1364/AO.21.001157. |
[41] |
M. N. Wernick and J. N. Aarsvold, eds., Emission Tomography: The Fundamental of PET and SPECT, Elsevier, Amsterdam, 2004. |
show all references
References:
[1] |
L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Gamma-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036.
doi: 10.1002/cpa.3160430805. |
[2] |
S. Amstutz, Topological sensitivity analysis for some nonlinear PDE system, J. Math. Pures Appl., 85 (2006), 540-557.
doi: 10.1016/j.matpur.2005.10.008. |
[3] |
S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 401-425.
doi: 10.1051/cocv:2005012. |
[4] |
S. Amstutz and J. Fehrenbach, Edge detection using topological gradients: A scale-space approach, J. Math. Imaging Vision, 52 (2015), 249-266.
doi: 10.1007/s10851-015-0558-z. |
[5] |
S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the toplogical gradient method, Control and Cybernetics, 34 (2005), 81-101. |
[6] |
G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal of Applied Mathematics, 68 (2008), 925-946.
doi: 10.1137/060671814. |
[7] |
G. Aubert and A. Drogoul, Topological gradient for a fourth order operator used in image analysis, ESAIM Control Optim. Calc. Var., 21 (2015), 1120-1149.
doi: 10.1051/cocv/2014061. |
[8] |
G. Aubert and A. Drogoul, Topological gradient for fourth order pde and application to the detection of fine structures in 2d images, C. R. Math. Acad. Sci. Paris, 352 (2014), 609-613.
doi: 10.1016/j.crma.2014.06.005. |
[9] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Second edition, Applied Mathematical Sciences, 147, Springer-Verlag, 2006. |
[10] |
D. Auroux, From restoration by topological gradient to medical image segmentation via an asymptotic expansion, Math. Comput. Model., 49 (2009), 2191-2205.
doi: 10.1016/j.mcm.2008.07.002. |
[11] |
D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis, Computational & Applied Mathematics, 25 (2006), 251-267.
doi: 10.1590/S0101-82052006000200008. |
[12] |
D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion, Journal of Mathematical Imaging and Vision, 33 (2009), 122-134.
doi: 10.1007/s10851-008-0121-2. |
[13] |
D. Auroux, M. Masmoudi and L. Jaafar Belaid, Image restoration and classification by topological asymptotic expansion, Variational Formulations in Mechanics: Theory and Applications (eds. E. Taroco, E. A. de Souza Neto and A. A. Novotny), CIMNE, Barcelona, Spain, 2007, 23-42. |
[14] |
H. Ayasso and A. Mohammad-Djafari, Joint image restorationand segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation, IEEE Trans. Image Process, 19 (2010), 2265-2277.
doi: 10.1109/TIP.2010.2047902. |
[15] |
L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, 342 (2006), 313-318.
doi: 10.1016/j.crma.2005.12.009. |
[16] |
L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Application of the topological gradient to image restoration and edge detection, Engineering Analysis with Boundary Elements, 32 (2008), 891-899. |
[17] |
S. Ben Hadj, L. Blanc Féraud and G. Aubert, Space Variant Blind Image Restoration, SIAM Journal on Imaging Sciences, 7 (2014), 2196-2225.
doi: 10.1137/130945776. |
[18] |
S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp.
doi: 10.1088/0266-5611/25/1/015002. |
[19] |
T. Chan and J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005.
doi: 10.1137/1.9780898717877. |
[20] |
N. Dey, L. Blanc-Féraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, 3D Microscopy Deconvolution using Richardson-Lucy Algorithm with Total Variation Regularization, Rapport de recherche RR-5272, INRIA, 2004. |
[21] |
A. Drogoul, Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in 2D imaging, SIAM J. Imaging Sci., 7 (2014), 2700-2731.
doi: 10.1137/140967374. |
[22] |
A. Drogoul, Topological Gradient Method Applied to the Detection of Edges and Fine Structures in Imaging, Ph.D Thesis, University of Nice Sophia Antipolis, 2014. |
[23] |
S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741.
doi: 10.1109/TPAMI.1984.4767596. |
[24] |
T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian recontruction from Poisson data using Gibbs prior, IEEE Trans. Medical imaging, 8 (1989), 194-202. |
[25] |
F. M. Henderson, A. J. Lewis and R. A. Ryerson, eds., Principles and Applications of Imaging Radar. Manual of Remote Sensing, 3rd edition, Wiley, New York, 1998.
doi: 10.1029/99EO00047. |
[26] |
M. Hintermüller, Fast-set based algorithms using shape and topological sensitivity information, Control and Cybernetics, 34 (2005), 305-324. |
[27] |
M. Iguernane, S. A. Nazarov, J.-R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Applied Mathematics and Computer Science, 19 (2009), 191-205.
doi: 10.2478/v10006-009-0016-4. |
[28] |
K. Krissian, R. Kikinis, C.-F. Westin and K. G. Vosburgh, Speckle-constrained filtering of ultrasound images, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, IEEE Computer Society, 2, 2005, 547-552.
doi: 10.1109/CVPR.2005.331. |
[29] |
S. Larnier and J. Fehrenbach, Edge detection and image restoration with anisotropic topological gradient, in Proceedings ICASSP , International Conference on Acoustic Speech and Signal Processing, 2010, 1362-1365.
doi: 10.1109/ICASSP.2010.5495448. |
[30] |
S. Larnier, J. Fehrenbach and M. Masmoudi, The topological gradient method: From optimal design to image processing, Milan Journal of Mathematics, 80 (2012), 411-441.
doi: 10.1007/s00032-012-0196-5. |
[31] |
I. Larrabide, A. A. Novotny, R. A. Feijo and E. Taroco, A medical image enhancement algorithms based on topological derivative and anisotropic diffusion, in Proceedings of the XXVI Iberian Latin-American Congess on Computational Methods in Engineering, 2005. |
[32] |
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998. |
[33] |
M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, GAKUTO Internat. Ser. Math. Appl., 16, Tokyo, Japan, 2001. |
[34] |
D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[35] |
J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences, Springer, New York, 2001.
doi: 10.1007/978-1-4757-4393-7. |
[36] |
J. Pawley, Handbook of Biological Confocal Microscopy, Springer, Berlin, 2006. |
[37] |
A. Sawatzky, D. Tenbrinck, X. Jiang and M. Burger, A variational framework for region-based segmentation incorporating physical noise models, Journal of Mathematical Imaging and Vision, 47 (2013), 179-209.
doi: 10.1007/s10851-013-0419-6. |
[38] |
J. Serra, Image Analysis and Mathematical Morphology, Academic Press, Inc., Orlando, FL, USA, 1984. |
[39] |
J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272.
doi: 10.1137/S0363012997323230. |
[40] |
M. Tur, K. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.
doi: 10.1364/AO.21.001157. |
[41] |
M. N. Wernick and J. N. Aarsvold, eds., Emission Tomography: The Fundamental of PET and SPECT, Elsevier, Amsterdam, 2004. |
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